{"id":96,"date":"2021-12-15T09:53:28","date_gmt":"2021-12-15T09:53:28","guid":{"rendered":"https:\/\/wp-prd.let.ethz.ch\/analysis19\/chapter\/taylor-approximation\/"},"modified":"2021-12-15T09:53:28","modified_gmt":"2021-12-15T09:53:28","slug":"taylor-approximation","status":"publish","type":"chapter","link":"https:\/\/wp-prd.let.ethz.ch\/analysis19\/chapter\/taylor-approximation\/","title":{"raw":"Taylor Approximation","rendered":"Taylor Approximation"},"content":{"raw":"\n<style>.cmr-5{font-size:50%;}\n.cmr-7{font-size:70%;}\n.cmmi-5{font-size:50%;font-style: italic;}\n.cmmi-7{font-size:70%;font-style: italic;}\n.cmmi-10{font-style: italic;}\n.cmsy-5{font-size:50%;}\n.cmsy-7{font-size:70%;}\n.cmbx-10{ font-weight: bold;}\n.cmbsy-10{font-weight: bold;}\n.cmbsy-10{font-weight: bold;}\n.cmbsy-10{font-weight: bold;}\n.cmbsy-7{font-size:70%;font-weight: bold;}\n.cmbsy-7{font-weight: bold;}\n.cmbsy-7{font-weight: bold;}\n.cmbsy-5{font-size:50%;font-weight: bold;}\n.cmbsy-5{font-weight: bold;}\n.cmbsy-5{font-weight: bold;}\n.cmex-7{font-size:70%;}\n.cmex-7x-x-71{font-size:49%;}\n.msam-7{font-size:70%;}\n.msam-5{font-size:50%;}\n.msbm-7{font-size:70%;}\n.msbm-5{font-size:50%;}\n.cmr-17{font-size:170%;}\n.cmr-12{font-size:120%;}\n.cmti-10{ font-style: italic;}\np{margin-top:0;margin-bottom:0}\np.indent{text-indent:0;}\np + p{margin-top:1em;}\np + div, p + pre {margin-top:1em;}\ndiv + p, pre + p {margin-top:1em;}\n@media print {div.crosslinks {visibility:hidden;}}\na img { border-top: 0; 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\n}\ndiv.proof p:first-of-type {\n\tmargin: 0px;\n}\ndiv.qed {\n\tmargin-top: -25px;\n\tmargin-bottom: -7px;\n\ttext-align: right;\n}\ntable.equation+div.qed {\n\tmargin-top: -65px;\n}\n\n\/* The following is making also math-formulas inside the headers of Lemmas, etc., white. *\/\ndiv.melemma h4 span {\n    color: white;\n}\ndiv.metheorem h4 span {\n    color: white;\n}\n\n\/* The following are used to avoid fullstop, period, colon, semicolon, and endquote (broader) to move by itself to the next line after a formula.\n   The math-environment before needs to be wrapped in span.maperiod and the fullstop etc. in a span.period --- together they achieve what we want.  *\/\nspan.maperiod {\n       margin-right: 5px;\n}\nspan.period {\n       display: inline-block;\n       width: 0px;\n       margin-left: -5px;\n       margin-right: 4.9px;\n\t   text-indent: 0px;\n}\nspan.maendquote {\n       margin-right: 8px;\n}\nspan.endquote {\n       display: inline-block;\n       width: 0px;\n       margin-left: -8px;\n       margin-right: 7.9px;\n}\n\n\n\/* The following is removing an extra space left of the equation side in aligned equations *\/\nspan.mjx-mtd {\n    padding-left: 0em !important;\n}\n\n\/* The following fixes the weird problem that math appears smaller if it was rendered while the details tag was closed. *\/\ndetails span.mjx-chtml, details span.MathJax_CHTML {\n font-size: 100% !important;\n}\n\n\/* trying to fix line breaks in verbatim, new lines are missing *\/\npre.verbatim {\n\twhite-space: pre-wrap;\n\tfont-size: small;\n}\n<\/style><h3 id=\"z26169359a53f\" class=\"sectionHead\"><span class=\"titlemark\">9.4 <\/span> <a id=\"x1-2760004\"><\/a>Taylor Approximation<\/h3> <p class=\"noindent\">Wir erinnern daran, dass die Ableitung&nbsp;<math display=\"inline\"><msup><mrow><mi>f<\/mi><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><\/math> einer reellwertigen differenzierbaren Funktion&nbsp;<math display=\"inline\"><mi>f<\/mi><\/math> auf einem Intervall die Steigung der Tangente <\/p><table id=\"z9f621b391178\" class=\"equation-star\"><tr><td> <math class=\"equation\" display=\"block\"> <mi>y<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">+<\/mo> <msup><mrow><mi>f<\/mi><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow> <mn>0<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo> <\/math><\/td><\/tr><\/table> <p class=\"indent\">des Graphen von&nbsp;<math display=\"inline\"><mi>f<\/mi><\/math> bei&nbsp;<math display=\"inline\"><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn> <\/mrow> <\/msub> <\/math> angibt, wobei die Tangente den Graphen gut approximiert (im Sinne von&nbsp;<math display=\"inline\"><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <mi>y<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">+<\/mo> <mi>o<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><\/math> f\u00fcr&nbsp;<math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2192<\/mo> <msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><\/math>). Wir wollen hier die G\u00fcte der Approximation erh\u00f6hen indem wir statt linearen Approximationen auch noch h\u00f6here polynomiale Approximationen erlauben. <\/p><p class=\"indent\">Sei also <math display=\"inline\"><mi>f<\/mi> <mo class=\"MathClass-punc\">:<\/mo> <mo class=\"MathClass-open\">(<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>b<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">\u2192<\/mo> <mi>\u2102<\/mi><\/math> eine <math display=\"inline\"><mi>n<\/mi><\/math>-mal differenzierbare Funktion auf einem offenen, nicht-leeren (m\u00f6glicherweise unbeschr\u00e4nkten) Intervall <span class=\"maperiod\"><math display=\"inline\"><mo class=\"MathClass-open\">(<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo> <mi>b<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">\u2286<\/mo> <mi>\u211d<\/mi><\/math><\/span><span class=\"period\">.<\/span> Die <math display=\"inline\"><mstyle><mi>n<\/mi><\/mstyle><\/math><span class=\"ecbx-1095\">-te Taylor-Approximation<\/span> von <math display=\"inline\"><mi>f<\/mi><\/math> um einen Punkt <math display=\"inline\"><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub> <mo class=\"MathClass-rel\">\u2208<\/mo> <mo class=\"MathClass-open\">(<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>b<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math> ist das Polynom                                                                                                                                                                           <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><msubsup><mrow><mi>P<\/mi><\/mrow><mrow><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><mo class=\"MathClass-punc\">,<\/mo><mi>n<\/mi><\/mrow><mrow><mi>f<\/mi><\/mrow><\/msubsup> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-rel\">=<\/mo><munderover accent=\"false\" accentunder=\"false\"><mrow><mo> \u2211<\/mo> <\/mrow><mrow><mi>k<\/mi><mo class=\"MathClass-rel\">=<\/mo><mn>0<\/mn><\/mrow><mrow><mi>n<\/mi><\/mrow><\/munderover><mfrac><mrow><msup><mrow><mi>f<\/mi><\/mrow><mrow><mo class=\"MathClass-open\">(<\/mo><mi>k<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow><\/msup><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><\/mrow> <mrow><mi>k<\/mi><mo class=\"MathClass-punc\">!<\/mo><\/mrow><\/mfrac> <msup><mrow><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mi>k<\/mi><\/mrow><\/msup><mo class=\"MathClass-punc\">.<\/mo><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">Die Koeffizienten wurden dabei so gew\u00e4hlt, dass <math display=\"inline\"><msubsup><mrow><mi>P<\/mi><\/mrow><mrow><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-punc\">,<\/mo><mi>n<\/mi> <\/mrow> <mrow> <mi>f<\/mi> <\/mrow> <\/msubsup> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn> <\/mrow> <\/msub> <\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-rel\">=<\/mo> <mi>f<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><\/math> und allgemeiner <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><msup><mrow><mo class=\"MathClass-open\">(<\/mo><msubsup><mrow><mi>P<\/mi><\/mrow><mrow><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><mo class=\"MathClass-punc\">,<\/mo><mi>n<\/mi><\/mrow><mrow><mi>f<\/mi><\/mrow><\/msubsup><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mo class=\"MathClass-open\">(<\/mo><mi>k<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow><\/msup> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><msub><mrow><mi>x<\/mi><\/mrow><mrow> <mn>0<\/mn><\/mrow><\/msub><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-rel\">=<\/mo> <msup><mrow><mi>f<\/mi><\/mrow><mrow><mo class=\"MathClass-open\">(<\/mo><mi>k<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow><\/msup> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><msub><mrow><mi>x<\/mi><\/mrow><mrow> <mn>0<\/mn><\/mrow><\/msub><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">f\u00fcr <math display=\"inline\"><mi>k<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> {<\/mo><mrow><mn>0<\/mn><mo class=\"MathClass-punc\">,<\/mo> <mi class=\"MathClass-op\">\u2026<\/mi><mo> <\/mo><mo class=\"MathClass-punc\">,<\/mo><mi>n<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">}<\/mo><\/mrow><\/math> gilt (wieso?). Die vielleicht naive Hoffnung ist hierbei, dass die Taylor-Approximation, wie der Name sagt, die Funktion <math display=\"inline\"><mi>f<\/mi><\/math> approximiert. Falls <math display=\"inline\"><mi>f<\/mi><\/math> glatt ist, dann ist die <span class=\"ecbx-1095\">Taylorreihe <\/span>von <math display=\"inline\"><mi>f<\/mi><\/math> um <math display=\"inline\"><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-rel\">\u2208<\/mo> <mo class=\"MathClass-open\">(<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>b<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math> definiert als die Potenzreihe <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><msubsup><mrow><mi>T<\/mi><\/mrow><mrow><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><\/mrow><mrow><mi>f<\/mi><\/mrow><\/msubsup> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-rel\">=<\/mo><munderover accent=\"false\" accentunder=\"false\"><mrow><mo> \u2211<\/mo> <\/mrow><mrow><mi>k<\/mi><mo class=\"MathClass-rel\">=<\/mo><mn>0<\/mn><\/mrow><mrow><mi>\u221e<\/mi><\/mrow><\/munderover><mfrac><mrow><msup><mrow><mi>f<\/mi><\/mrow><mrow><mo class=\"MathClass-open\">(<\/mo><mi>k<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow><\/msup><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><\/mrow> <mrow><mi>k<\/mi><mo class=\"MathClass-punc\">!<\/mo><\/mrow><\/mfrac> <msup><mrow><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mi>k<\/mi><\/mrow><\/msup><mo class=\"MathClass-punc\">.<\/mo><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">Damit wir hier von einer Potenzreihe sprechen d\u00fcrfen, erweitern wir die Definition <a href=\"..\/..\/chapter\/potenzreihen#x1-199001r54\">7.54<\/a> wie folgt. Eine <span class=\"ecbx-1095\">Potenzreihe um einen Punkt<\/span> <math display=\"inline\"><msub><mrow><mi>z<\/mi><\/mrow><mrow><mn>0<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2102<\/mi><\/math> in der Variable <math display=\"inline\"><mi>z<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2102<\/mi><\/math> ist ein formaler Ausdruck der Form <span class=\"maperiod\"><math display=\"inline\"><msubsup><mrow><mi class=\"MathClass-op\"> \u2211<\/mi><mo> <\/mo> <\/mrow><mrow><mi>n<\/mi><mo class=\"MathClass-rel\">=<\/mo><mn>0<\/mn><\/mrow><mrow><mi>\u221e<\/mi><\/mrow><\/msubsup><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><msup><mrow><mo class=\"MathClass-open\">(<\/mo><mi>z<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <msub><mrow><mi>z<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msup><\/math><\/span><span class=\"period\">,<\/span> wobei <math display=\"inline\"><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi> <\/mrow> <\/msub> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2102<\/mi><\/math> f\u00fcr alle <math display=\"inline\"><mi>n<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2115<\/mi><\/math> die Koeffizienten der Potenzreihe sind. Der Konvergenzradius <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mi>R<\/mi> <mo class=\"MathClass-rel\">=<\/mo><msup><mrow> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><munder class=\"msub\"><mrow><mi class=\"qopname\">limsup<\/mi><mo>  <\/mo><\/mrow><mrow><mi>n<\/mi><mo class=\"MathClass-rel\">\u2192<\/mo><mi>\u221e<\/mi><\/mrow><\/munder><mroot><mrow><mo class=\"MathClass-rel\">|<\/mo><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><mo class=\"MathClass-rel\">|<\/mo><\/mrow><mrow><mi>n<\/mi><\/mrow><\/mroot><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><\/mrow><mrow><mo class=\"MathClass-bin\">\u2212<\/mo><mn>1<\/mn><\/mrow><\/msup><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">ist wie in Abschnitt <a href=\"..\/..\/chapter\/potenzreihen#x1-2000001\">7.4.1<\/a> definiert und die Potenzreihe <math display=\"inline\"><msubsup><mrow><mi class=\"MathClass-op\">\u2211<\/mi><mo> <\/mo> <\/mrow><mrow><mi>n<\/mi><mo class=\"MathClass-rel\">=<\/mo><mn>0<\/mn><\/mrow><mrow><mi>\u221e<\/mi><\/mrow><\/msubsup><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><msup><mrow><mo class=\"MathClass-open\">(<\/mo><mi>z<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <msub><mrow><mi>z<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msup><\/math> konvergiert f\u00fcr alle <math display=\"inline\"><mi>z<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2102<\/mi><\/math> mit Abstand kleiner <math display=\"inline\"><mi>R<\/mi><\/math> von <math display=\"inline\"><msub><mrow><mi>z<\/mi><\/mrow><mrow><mn>0<\/mn> <\/mrow> <\/msub> <\/math> und divergiert f\u00fcr alle <math display=\"inline\"><mi>z<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2102<\/mi><\/math> mit <math display=\"inline\"><mo class=\"MathClass-rel\">|<\/mo><mi>z<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <msub><mrow><mi>z<\/mi><\/mrow><mrow><mn>0<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-rel\">|<\/mo> <mo class=\"MathClass-rel\">&gt;<\/mo> <mi>R<\/mi><\/math> (was aus Satz <a href=\"..\/..\/chapter\/potenzreihen#x1-200002r56\">7.56<\/a> folgt, wie?). <\/p><p class=\"indent\">Soweit ist nicht klar, was der Konvergenzradius der Taylor-Reihe einer glatten Funktion <math display=\"inline\"><mi>f<\/mi><\/math> um einen Punkt <math display=\"inline\"><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn> <\/mrow> <\/msub> <\/math> im Definitionsbereich ist und ob er positiv ist. Die naive Hoffnung ist, dass die Taylor-Reihe, wo definiert, gleich der Funktion <math display=\"inline\"><mi>f<\/mi><\/math> ist, da die Taylorreihe bei <math display=\"inline\"><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn> <\/mrow> <\/msub> <\/math> die gleichen Ableitungen wie <math display=\"inline\"><mi>f<\/mi><\/math> hat. Dies ist in der Tat f\u00fcr viele Funktionen der Fall, doch, wie folgendes Beispiel zeigt, nicht immer. <\/p> <div class=\"me meexample\"> <p class=\"indent\"><\/p><h4 id=\"zb9b320cd2ad8\"> <a id=\"x1-276001r45\"><\/a> <span class=\"ecbx-1095\">Beispiel 9.45 <\/span>(Verschwindende Taylorreihe)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Wir betrachten die Funktion<\/span> <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mi>\u03c8<\/mi> <mo class=\"MathClass-punc\">:<\/mo> <mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u211d<\/mi><mo class=\"MathClass-rel\">\u21a6<\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> {<\/mo><mrow> <mtable align=\"axis\" class=\"array\" columnlines=\"none\" equalcolumns=\"false\" equalrows=\"false\"> <mtr><mtd class=\"array\" columnalign=\"center\"><mi class=\"qopname\">exp<\/mi><mo>  <\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mo class=\"MathClass-bin\">\u2212<\/mo><mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mi>x<\/mi><\/mrow><\/mfrac> <\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><\/mtd><mtd class=\"array\" columnalign=\"center\"><mstyle class=\"text\"><mtext>falls&nbsp;<\/mtext><\/mstyle><mi>x<\/mi> <mo class=\"MathClass-rel\">&gt;<\/mo> <mn>0<\/mn><\/mtd><\/mtr> <mtr><mtd class=\"array\" columnalign=\"center\"> <mn>0<\/mn> <\/mtd> <mtd class=\"array\" columnalign=\"center\"><mstyle class=\"text\"><mtext>falls&nbsp;<\/mtext><\/mstyle> <mi>x<\/mi> <mo class=\"MathClass-rel\">\u2264<\/mo> <mn>0<\/mn><\/mtd> <\/mtr> <\/mtable> <\/mrow><mo fence=\"true\" form=\"postfix\" \/><\/mrow><mo class=\"MathClass-punc\">,<\/mo><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\"><span class=\"ecti-1095\">die nach Beispiel <\/span><a href=\"..\/..\/chapter\/die-ableitung#x1-231004r23\"><span class=\"ecti-1095\">8.23<\/span><\/a> <span class=\"ecti-1095\">glatt ist und <\/span><math display=\"inline\"><msup><mrow><mi>\u03c8<\/mi><\/mrow><mrow><mo class=\"MathClass-open\">(<\/mo><mi>n<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow><\/msup><mo class=\"MathClass-open\">(<\/mo><mn>0<\/mn><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <mn>0<\/mn><\/math> <span class=\"ecti-1095\">f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r alle <\/span><math display=\"inline\"><mi>n<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <msub><mrow><mi>\u2115<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><\/math> <span class=\"ecti-1095\">erf<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">llt. Wir<\/span> <span class=\"ecti-1095\">zeigen, dass sich <\/span><math display=\"inline\"><mi>\u03c8<\/mi><\/math> <span class=\"ecti-1095\">nicht<\/span> <span class=\"ecti-1095\">durch eine Potenzreihe um <\/span><math display=\"inline\"><mn>0<\/mn><\/math> <span class=\"ecti-1095\">darstellen l<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">sst.<\/span> <\/p><p class=\"indent\"><span class=\"ecti-1095\">Wir nehmen also indirekt an, dass es eine Potenzreihe<\/span> <math display=\"inline\"><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">=<\/mo><msubsup><mrow><mi class=\"MathClass-op\"> \u2211<\/mi><mo> <\/mo> <\/mrow><mrow><mi>n<\/mi><mo class=\"MathClass-rel\">=<\/mo><mn>0<\/mn><\/mrow><mrow><mi>\u221e<\/mi><\/mrow><\/msubsup><msub><mrow><mi>c<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><msup><mrow><mi>x<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msup><\/math> <span class=\"ecti-1095\">mit reellen Koeffizienten<\/span> <span class=\"ecti-1095\">und Konvergenzradius <\/span><math display=\"inline\"><mi>R<\/mi> <mo class=\"MathClass-rel\">&gt;<\/mo> <mn>0<\/mn><\/math> <span class=\"ecti-1095\">gibt, so dass <\/span><math display=\"inline\"><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <mi>\u03c8<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math> <span class=\"ecti-1095\">f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r alle<\/span> <math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mo class=\"MathClass-open\">(<\/mo><mo class=\"MathClass-bin\">\u2212<\/mo><mi>R<\/mi><mo class=\"MathClass-punc\">,<\/mo> <mi>R<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math><span class=\"ecti-1095\">. Nach Korollar <\/span><a href=\"..\/..\/chapter\/der-fundamentalsatz-der-integral--und-differentialrechnung#x1-260001r11\"><span class=\"ecti-1095\">9.11<\/span><\/a> <span class=\"ecti-1095\">(oder<\/span> <span class=\"ecti-1095\">genauer <\/span><span class=\"ecti-1095\">\u00dc<\/span><span class=\"ecti-1095\">bung <\/span><a href=\"..\/..\/chapter\/der-fundamentalsatz-der-integral--und-differentialrechnung#x1-260002r12\"><span class=\"ecti-1095\">9.12<\/span><\/a><span class=\"ecti-1095\">) ist <\/span><span class=\"maperiod\"><math display=\"inline\"><msup><mrow><mi>f<\/mi><\/mrow><mrow><mo class=\"MathClass-open\">(<\/mo><mi>n<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow><\/msup><mo class=\"MathClass-open\">(<\/mo><mn>0<\/mn><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <msub><mrow><mi>c<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><mi>n<\/mi><mo class=\"MathClass-punc\">!<\/mo><\/math><\/span><span class=\"period\">.<\/span> <span class=\"ecti-1095\">Aber da <\/span><math display=\"inline\"><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <mi>\u03c8<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math> <span class=\"ecti-1095\">f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r alle <\/span><math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mo class=\"MathClass-open\">(<\/mo><mo class=\"MathClass-bin\">\u2212<\/mo><mi>R<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>R<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math> <span class=\"ecti-1095\">angenommen wurde und da Ableiten eine lokale Operation ist, schliessen wir, dass<\/span> <math display=\"inline\"><msup><mrow><mi>f<\/mi><\/mrow><mrow><mo class=\"MathClass-open\">(<\/mo><mi>n<\/mi><mo class=\"MathClass-close\">)<\/mo> <\/mrow> <\/msup> <mo class=\"MathClass-open\">(<\/mo><mn>0<\/mn><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <mn>0<\/mn><\/math> <span class=\"ecti-1095\">und somit<\/span> <math display=\"inline\"><msub><mrow><mi>c<\/mi><\/mrow><mrow><mi>n<\/mi> <\/mrow> <\/msub> <mo class=\"MathClass-rel\">=<\/mo> <mn>0<\/mn><\/math> <span class=\"ecti-1095\">f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r alle<\/span> <math display=\"inline\"><mi>n<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <msub><mrow><mi>\u2115<\/mi><\/mrow><mrow><mn>0<\/mn> <\/mrow> <\/msub> <\/math><span class=\"ecti-1095\">. Dies widerspricht<\/span> <span class=\"ecti-1095\">jedoch <\/span><math display=\"inline\"><mi>\u03c8<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">&gt;<\/mo> <mn>0<\/mn><\/math> <span class=\"ecti-1095\">f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r<\/span> <span class=\"ecti-1095\">alle <\/span><math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">&gt;<\/mo> <mn>0<\/mn><\/math><span class=\"ecti-1095\">. Also<\/span> <span class=\"ecti-1095\">kann <\/span><math display=\"inline\"><mi>\u03c8<\/mi><\/math> <span class=\"ecti-1095\">in keiner<\/span> <span class=\"ecti-1095\">Umgebung von <\/span><math display=\"inline\"><mn>0<\/mn><\/math> <span class=\"ecti-1095\">durch eine Potenzreihe dargestellt werden.<\/span> <\/p> <\/div> <p class=\"indent\">Folgender Satz liefert nun einen direkten Vergleich zwischen einer Funktion und ihrer Taylor-Approximationen. Er impliziert f\u00fcr viele Funktionen, dass sie mit ihrer Taylorreihe \u00fcbereinstimmen. <\/p> <div class=\"me metheorem\"> <p class=\"indent\"><\/p><h4 id=\"z2d3e739c4657\"> <a id=\"x1-276002r46\"><\/a> <span class=\"ecbx-1095\">Theorem 9.46 <\/span>(Taylor-Approximation)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Sei <\/span><math display=\"inline\"><mo class=\"MathClass-open\">(<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo> <mi>b<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">\u2286<\/mo> <mi>\u211d<\/mi><\/math> <span class=\"ecti-1095\">ein offenes, nicht-leeres<\/span> <span class=\"ecti-1095\">Intervall und sei <\/span><math display=\"inline\"><mi>f<\/mi> <mo class=\"MathClass-punc\">:<\/mo> <mo class=\"MathClass-open\">(<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>b<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">\u2192<\/mo> <mi>\u2102<\/mi><\/math> <span class=\"ecti-1095\">eine<\/span> <math display=\"inline\"><mo class=\"MathClass-open\">(<\/mo><mi>n<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mn>1<\/mn><mo class=\"MathClass-close\">)<\/mo><\/math><span class=\"ecti-1095\">-mal stetig differenzierbare<\/span> <span class=\"ecti-1095\">Funktion. Sei <\/span><span class=\"maperiod\"><math display=\"inline\"><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub> <mo class=\"MathClass-rel\">\u2208<\/mo> <mo class=\"MathClass-open\">(<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>b<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math><\/span><span class=\"period\">.<\/span> <span class=\"ecti-1095\">Dann gilt f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r alle <\/span><math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mo class=\"MathClass-open\">(<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>b<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math> <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mi>f<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-rel\">=<\/mo> <msubsup><mrow><mi>P<\/mi><\/mrow><mrow><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><mo class=\"MathClass-punc\">,<\/mo><mi>n<\/mi><\/mrow><mrow><mi>f<\/mi><\/mrow><\/msubsup> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-bin\">+<\/mo> <msubsup><mrow><mi>R<\/mi><\/mrow><mrow><msub><mrow> <mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><mo class=\"MathClass-punc\">,<\/mo><mi>n<\/mi><\/mrow><mrow><mi>f<\/mi><\/mrow><\/msubsup> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><mo class=\"MathClass-punc\">,<\/mo><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\"><span class=\"ecti-1095\">wobei <\/span><math display=\"inline\"><msubsup><mrow><mi>P<\/mi><\/mrow><mrow><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-punc\">,<\/mo><mi>n<\/mi> <\/mrow> <mrow> <mi>f<\/mi><\/mrow><\/msubsup><\/math> <span class=\"ecti-1095\">die<\/span> <math display=\"inline\"><mi>n<\/mi><\/math><span class=\"ecti-1095\">-te Taylor-Approximation<\/span> <span class=\"ecti-1095\">ist und wir den Fehlerterm <\/span><math display=\"inline\"><msubsup><mrow><mi>R<\/mi><\/mrow><mrow><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><mo class=\"MathClass-punc\">,<\/mo><mi>n<\/mi><\/mrow><mrow><mi>f<\/mi><\/mrow><\/msubsup><\/math> <span class=\"ecti-1095\">durch das sogenannte <\/span><span class=\"ecbi-1095\">Integral-Restglied<\/span> <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>b<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><mo class=\"MathClass-rel\">\u21a6<\/mo><msubsup><mrow><mi>R<\/mi><\/mrow><mrow><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><mo class=\"MathClass-punc\">,<\/mo><mi>n<\/mi><\/mrow><mrow><mi>f<\/mi><\/mrow><\/msubsup> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-rel\">=<\/mo><msubsup><mrow><mo> \u222b  <\/mo><\/mrow><mrow><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><\/mrow><mrow><mi>x<\/mi><\/mrow><\/msubsup><msup><mrow><mi>f<\/mi><\/mrow><mrow><mo class=\"MathClass-open\">(<\/mo><mi>n<\/mi><mo class=\"MathClass-bin\">+<\/mo><mn>1<\/mn><mo class=\"MathClass-close\">)<\/mo><\/mrow><\/msup> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>t<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mfrac><mrow><msup><mrow><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>t<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msup><\/mrow> <mrow><mi>n<\/mi><mo class=\"MathClass-punc\">!<\/mo><\/mrow><\/mfrac> <mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>t<\/mi><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\"><span class=\"ecti-1095\">darstellen k<\/span><span class=\"ecti-1095\">\u00f6<\/span><span class=\"ecti-1095\">nnen. Dies gilt auch f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r Funktionen auf<\/span> <math display=\"inline\"><mo class=\"MathClass-open\">[<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-punc\">,<\/mo> <mi>b<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math> <span class=\"ecti-1095\">und<\/span> <span class=\"ecti-1095\">Punkte <\/span><math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mo class=\"MathClass-open\">[<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><mo class=\"MathClass-punc\">,<\/mo><mi>b<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math> <span class=\"ecti-1095\">(beziehungsweise <\/span><math display=\"inline\"><mo class=\"MathClass-open\">(<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">]<\/mo><\/math> <span class=\"ecti-1095\">und <\/span><math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mo class=\"MathClass-open\">(<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo> <msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">]<\/mo><\/math><span class=\"ecti-1095\">).<\/span> <\/p> <\/div> <p class=\"indent\">Die Annahme im Theorem, dass <math display=\"inline\"><mi>f<\/mi><\/math> eine <math display=\"inline\"><mo class=\"MathClass-open\">(<\/mo><mi>n<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mn>1<\/mn><mo class=\"MathClass-close\">)<\/mo><\/math>-mal stetig differenzierbare Funktion ist, ist essentiell f\u00fcr obige Formulierung. In der Tat ist damit <math display=\"inline\"><msup><mrow><mi>f<\/mi><\/mrow><mrow><mo class=\"MathClass-open\">(<\/mo><mi>n<\/mi><mo class=\"MathClass-bin\">+<\/mo><mn>1<\/mn><mo class=\"MathClass-close\">)<\/mo> <\/mrow> <\/msup> <\/math> eine stetige Funktion und das Integral der stetigen Funktion <math display=\"inline\"><mi>t<\/mi><mo class=\"MathClass-rel\">\u21a6<\/mo> <msup><mrow><mi>f<\/mi><\/mrow><mrow><mo class=\"MathClass-open\">(<\/mo><mi>n<\/mi><mo class=\"MathClass-bin\">+<\/mo><mn>1<\/mn><mo class=\"MathClass-close\">)<\/mo> <\/mrow> <\/msup> <mo class=\"MathClass-open\">(<\/mo><mi>t<\/mi><mo class=\"MathClass-close\">)<\/mo><mfrac><mrow><msup><mrow><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-bin\">\u2212<\/mo><mi>t<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msup><\/mrow> <mrow><mi>n<\/mi><mo class=\"MathClass-punc\">!<\/mo><\/mrow><\/mfrac> <\/math> im Integral-Restglied existiert. <\/p><p class=\"indent\"> <\/p> <div class=\"proof\"> <p class=\"indent\"><span class=\"head\"><\/span><\/p><details open><summary><b>Beweis von Satz <a href=\"..\/..\/chapter\/taylor-approximation#x1-276002r46\">9.46<\/a>.<\/b><\/summary><p class=\"indent\" style=\"margin-top: 10\"> Das Theorem ergibt sich mit Induktion \u00fcber <math display=\"inline\"><mi>n<\/mi><\/math> und partieller Integration (die ebenso f\u00fcr komplexwertige Funktionen auf <math display=\"inline\"><mo class=\"MathClass-open\">(<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo> <mi>b<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math> gilt, wieso?). Ist <math display=\"inline\"><mi>n<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mn>0<\/mn><\/math> und <math display=\"inline\"><mi>f<\/mi> <mo class=\"MathClass-punc\">:<\/mo> <mo class=\"MathClass-open\">(<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo> <mi>b<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">\u2192<\/mo> <mi>\u211d<\/mi><\/math> stetig differenzierbar, so gilt nach dem Fundamentalsatz der Differential- und Integralrechnung (genauer Korollar <a href=\"..\/..\/chapter\/der-fundamentalsatz-der-integral--und-differentialrechnung#x1-259008r5\">9.5<\/a>) <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mi>f<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-rel\">=<\/mo> <mi>f<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-bin\">+<\/mo><msubsup><mrow><mo> \u222b  <\/mo><\/mrow><mrow><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><\/mrow><mrow><mi>x<\/mi><\/mrow><\/msubsup><msup><mrow><mi>f<\/mi><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup><mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>t<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>t<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <msubsup><mrow><mi>P<\/mi><\/mrow><mrow><msub><mrow> <mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><mo class=\"MathClass-punc\">,<\/mo><mn>0<\/mn><\/mrow><mrow><mi>f<\/mi><\/mrow><\/msubsup> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-bin\">+<\/mo> <msubsup><mrow><mi>R<\/mi><\/mrow><mrow><msub><mrow> <mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><mo class=\"MathClass-punc\">,<\/mo><mn>0<\/mn><\/mrow><mrow><mi>f<\/mi><\/mrow><\/msubsup> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><mo class=\"MathClass-punc\">.<\/mo><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">Ist <math display=\"inline\"><mi>f<\/mi><\/math> nun zweimal stetig differenzierbar (also <math display=\"inline\"><mi>n<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mn>1<\/mn><\/math>), so k\u00f6nnen wir auf obiges Integral partielle Integration mit <math display=\"inline\"><mi>u<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>t<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <msup><mrow><mi>f<\/mi><\/mrow><mrow><mo>\u2032<\/mo> <\/mrow> <\/msup> <mo class=\"MathClass-open\">(<\/mo><mi>t<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math> und <math display=\"inline\"><mi>v<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>t<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <mi>t<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>x<\/mi><\/math> anwenden und erhalten <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mtd> <mtd class=\"align-even\"> <mo class=\"MathClass-rel\">=<\/mo> <mi>f<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-bin\">+<\/mo><msubsup><mrow> <mrow><mo fence=\"true\" form=\"prefix\"> [<\/mo><mrow><msup><mrow><mi>f<\/mi><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup><mo class=\"MathClass-open\">(<\/mo><mi>t<\/mi><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-open\">(<\/mo><mi>t<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow><mo fence=\"true\" form=\"postfix\">]<\/mo><\/mrow><\/mrow><mrow> <mi>t<\/mi><mo class=\"MathClass-rel\">=<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><\/mrow><mrow><mi>t<\/mi><mo class=\"MathClass-rel\">=<\/mo><mi>x<\/mi><\/mrow><\/msubsup> <mo class=\"MathClass-bin\">\u2212<\/mo><msubsup><mrow><mo>\u222b  <\/mo><\/mrow><mrow><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><\/mrow><mrow><mi>x<\/mi><\/mrow><\/msubsup><msup><mrow><mi>f<\/mi><\/mrow><mrow><mo>\u2033<\/mo><\/mrow><\/msup> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>t<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>t<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>t<\/mi><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\" \/> <mtd class=\"align-label\"> <mspace width=\"2em\" \/><\/mtd><\/mtr><mtr><mtd class=\"align-odd\" columnalign=\"right\" \/> <mtd class=\"align-even\"> <mo class=\"MathClass-rel\">=<\/mo> <mi>f<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-bin\">+<\/mo> <msup><mrow><mi>f<\/mi><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup><mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><msub><mrow><mi>x<\/mi><\/mrow><mrow> <mn>0<\/mn><\/mrow><\/msub><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-bin\">+<\/mo><msubsup><mrow><mo> \u222b  <\/mo><\/mrow><mrow><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><\/mrow><mrow><mi>x<\/mi><\/mrow><\/msubsup><msup><mrow><mi>f<\/mi><\/mrow><mrow><mo>\u2033<\/mo><\/mrow><\/msup> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>t<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mfrac><mrow><msup><mrow><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>t<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msup><\/mrow> <mrow><mn>1<\/mn><mo class=\"MathClass-punc\">!<\/mo><\/mrow><\/mfrac> <mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>t<\/mi><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\" \/> <mtd class=\"align-label\"> <mspace width=\"2em\" \/><\/mtd><\/mtr><mtr><mtd class=\"align-odd\" columnalign=\"right\" \/> <mtd class=\"align-even\"> <mo class=\"MathClass-rel\">=<\/mo> <msubsup><mrow><mi>P<\/mi><\/mrow><mrow><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><mo class=\"MathClass-punc\">,<\/mo><mn>1<\/mn><\/mrow><mrow><mi>f<\/mi><\/mrow><\/msubsup> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-bin\">+<\/mo> <msubsup><mrow><mi>R<\/mi><\/mrow><mrow><msub><mrow> <mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><mo class=\"MathClass-punc\">,<\/mo><mn>1<\/mn><\/mrow><mrow><mi>f<\/mi><\/mrow><\/msubsup> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><mo class=\"MathClass-punc\">.<\/mo><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\" \/> <mtd class=\"align-label\"> <mspace width=\"2em\" \/><\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">Wir sehen also wie sich ausgehend vom Fundamentalsatz mittels partieller Integration der n\u00e4chste Fall des Satzes ergibt. <\/p><p class=\"indent\">Angenommen die Aussage des Satzes stimmt f\u00fcr <math display=\"inline\"><mi>n<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <mn>1<\/mn> <mo class=\"MathClass-rel\">\u2265<\/mo> <mn>0<\/mn><\/math> und sei <math display=\"inline\"><mi>f<\/mi> <mo class=\"MathClass-punc\">:<\/mo> <mo class=\"MathClass-open\">(<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo> <mi>b<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">\u2192<\/mo> <mi>\u211d<\/mi><\/math> eine <math display=\"inline\"><mo class=\"MathClass-open\">(<\/mo><mi>n<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mn>1<\/mn><mo class=\"MathClass-close\">)<\/mo><\/math>-mal stetig differenzierbare Funktion. Dann gilt auf Grund der Induktionsvorraussetzung                                                                                                                                                                           <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mi>f<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-rel\">=<\/mo><munderover accent=\"false\" accentunder=\"false\"><mrow><mo> \u2211<\/mo> <\/mrow><mrow><mi>k<\/mi><mo class=\"MathClass-rel\">=<\/mo><mn>0<\/mn><\/mrow><mrow><mi>n<\/mi><mo class=\"MathClass-bin\">\u2212<\/mo><mn>1<\/mn><\/mrow><\/munderover><mfrac><mrow><msup><mrow><mi>f<\/mi><\/mrow><mrow><mo class=\"MathClass-open\">(<\/mo><mi>k<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow><\/msup><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><\/mrow> <mrow><mi>k<\/mi><mo class=\"MathClass-punc\">!<\/mo><\/mrow><\/mfrac> <msup><mrow><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mi>k<\/mi><\/mrow><\/msup> <mo class=\"MathClass-bin\">+<\/mo><msubsup><mrow><mo> \u222b  <\/mo><\/mrow><mrow><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><\/mrow><mrow><mi>x<\/mi><\/mrow><\/msubsup><msup><mrow><mi>f<\/mi><\/mrow><mrow><mo class=\"MathClass-open\">(<\/mo><mi>n<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow><\/msup> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>t<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mfrac><mrow><msup><mrow><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>t<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mi>n<\/mi><mo class=\"MathClass-bin\">\u2212<\/mo><mn>1<\/mn><\/mrow><\/msup><\/mrow> <mrow><mo class=\"MathClass-open\">(<\/mo><mi>n<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <mn>1<\/mn><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-punc\">!<\/mo><\/mrow><\/mfrac> <mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>t<\/mi><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">f\u00fcr alle <span class=\"maperiod\"><math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mo class=\"MathClass-open\">(<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>b<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math><\/span><span class=\"period\">.<\/span> <\/p><p class=\"indent\">Wir setzen <math display=\"inline\"><mi>u<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>t<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <msup><mrow><mi>f<\/mi><\/mrow><mrow><mo class=\"MathClass-open\">(<\/mo><mi>n<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow><\/msup><mo class=\"MathClass-open\">(<\/mo><mi>t<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math> und <span class=\"maperiod\"><math display=\"inline\"><mi>v<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>t<\/mi> <\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-rel\">=<\/mo> <mo class=\"MathClass-bin\">\u2212<\/mo><mfrac><mrow><msup><mrow><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-bin\">\u2212<\/mo><mi>t<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msup><\/mrow> <mrow><mi>n<\/mi><mo class=\"MathClass-punc\">!<\/mo><\/mrow><\/mfrac> <\/math><\/span><span class=\"period\">,<\/span> bemerken <math display=\"inline\"><msup><mrow><mi>v<\/mi><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup><mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>t<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-rel\">=<\/mo> <mfrac><mrow><msup><mrow><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-bin\">\u2212<\/mo><mi>t<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mi>n<\/mi><mo class=\"MathClass-bin\">\u2212<\/mo><mn>1<\/mn><\/mrow><\/msup><\/mrow> <mrow><mo class=\"MathClass-open\">(<\/mo><mi>n<\/mi><mo class=\"MathClass-bin\">\u2212<\/mo><mn>1<\/mn><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-punc\">!<\/mo><\/mrow><\/mfrac> <\/math> und wenden partielle Integration an, um <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mtd> <mtd class=\"align-even\"> <mo class=\"MathClass-rel\">=<\/mo><munderover accent=\"false\" accentunder=\"false\"><mrow><mo> \u2211<\/mo> <\/mrow><mrow><mi>k<\/mi><mo class=\"MathClass-rel\">=<\/mo><mn>0<\/mn><\/mrow><mrow><mi>n<\/mi><mo class=\"MathClass-bin\">\u2212<\/mo><mn>1<\/mn><\/mrow><\/munderover><mfrac><mrow><msup><mrow><mi>f<\/mi><\/mrow><mrow><mo class=\"MathClass-open\">(<\/mo><mi>k<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow><\/msup><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><\/mrow> <mrow><mi>k<\/mi><mo class=\"MathClass-punc\">!<\/mo><\/mrow><\/mfrac> <msup><mrow><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mi>k<\/mi><\/mrow><\/msup> <mo class=\"MathClass-bin\">\u2212<\/mo><msubsup><mrow><mrow><mo fence=\"true\" form=\"prefix\"> [<\/mo><mrow><msup><mrow><mi>f<\/mi><\/mrow><mrow><mo class=\"MathClass-open\">(<\/mo><mi>n<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow><\/msup><mo class=\"MathClass-open\">(<\/mo><mi>t<\/mi><mo class=\"MathClass-close\">)<\/mo><mfrac><mrow><msup><mrow><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>t<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msup><\/mrow> <mrow><mi>n<\/mi><mo class=\"MathClass-punc\">!<\/mo><\/mrow><\/mfrac> <\/mrow><mo fence=\"true\" form=\"postfix\">]<\/mo><\/mrow><\/mrow><mrow><mi>t<\/mi><mo class=\"MathClass-rel\">=<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><\/mrow><mrow><mi>t<\/mi><mo class=\"MathClass-rel\">=<\/mo><mi>x<\/mi><\/mrow><\/msubsup> <mo class=\"MathClass-bin\">+<\/mo><msubsup><mrow><mo> \u222b  <\/mo><\/mrow><mrow><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><\/mrow><mrow><mi>x<\/mi><\/mrow><\/msubsup><msup><mrow><mi>f<\/mi><\/mrow><mrow><mo class=\"MathClass-open\">(<\/mo><mi>n<\/mi><mo class=\"MathClass-bin\">+<\/mo><mn>1<\/mn><mo class=\"MathClass-close\">)<\/mo><\/mrow><\/msup> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>t<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mfrac><mrow><msup><mrow><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>t<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msup><\/mrow> <mrow><mi>n<\/mi><mo class=\"MathClass-punc\">!<\/mo><\/mrow><\/mfrac> <mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>t<\/mi><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\" \/> <mtd class=\"align-label\"> <mspace width=\"2em\" \/><\/mtd><\/mtr><mtr><mtd class=\"align-odd\" columnalign=\"right\" \/> <mtd class=\"align-even\"> <mo class=\"MathClass-rel\">=<\/mo><munderover accent=\"false\" accentunder=\"false\"><mrow><mo> \u2211<\/mo> <\/mrow><mrow><mi>k<\/mi><mo class=\"MathClass-rel\">=<\/mo><mn>0<\/mn><\/mrow><mrow><mi>n<\/mi><\/mrow><\/munderover><mfrac><mrow><msup><mrow><mi>f<\/mi><\/mrow><mrow><mo class=\"MathClass-open\">(<\/mo><mi>k<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow><\/msup><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><\/mrow> <mrow><mi>k<\/mi><mo class=\"MathClass-punc\">!<\/mo><\/mrow><\/mfrac> <msup><mrow><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mi>k<\/mi><\/mrow><\/msup> <mo class=\"MathClass-bin\">+<\/mo><msubsup><mrow><mo> \u222b  <\/mo><\/mrow><mrow><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><\/mrow><mrow><mi>x<\/mi><\/mrow><\/msubsup><msup><mrow><mi>f<\/mi><\/mrow><mrow><mo class=\"MathClass-open\">(<\/mo><mi>n<\/mi><mo class=\"MathClass-bin\">+<\/mo><mn>1<\/mn><mo class=\"MathClass-close\">)<\/mo><\/mrow><\/msup> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>t<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mfrac><mrow><msup><mrow><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>t<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msup><\/mrow> <mrow><mi>n<\/mi><mo class=\"MathClass-punc\">!<\/mo><\/mrow><\/mfrac> <mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>t<\/mi><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\" \/> <mtd class=\"align-label\"> <mspace width=\"2em\" \/><\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">zu erhalten. Dies beweist den Induktionsschritt und damit den Satz. <span>&nbsp;&nbsp;<\/span><\/p><div class=\"qed\">\u25a0<\/div><\/details><\/div> <p class=\"indent\">Oft werden wir das Theorem von Taylor (Theorem <a href=\"..\/..\/chapter\/taylor-approximation#x1-276002r46\">9.46<\/a>) in folgender Form verwenden. <\/p> <div class=\"me metheorem\"> <p class=\"indent\"><\/p><h4 id=\"zad53dae02d37\"> <a id=\"x1-276003r47\"><\/a> <span class=\"ecbx-1095\">Korollar 9.47 <\/span>(Taylor-Absch\u00e4tzung)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Sei <\/span><math display=\"inline\"><mo class=\"MathClass-open\">(<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo> <mi>b<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">\u2286<\/mo> <mi>\u211d<\/mi><\/math> <span class=\"ecti-1095\">ein offenes,<\/span> <span class=\"ecti-1095\">nicht-leeres Intervall, sei <\/span><math display=\"inline\"><mi>f<\/mi> <mo class=\"MathClass-punc\">:<\/mo> <mo class=\"MathClass-open\">(<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>b<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">\u2192<\/mo> <mi>\u2102<\/mi><\/math> <span class=\"ecti-1095\">eine<\/span> <math display=\"inline\"><mo class=\"MathClass-open\">(<\/mo><mi>n<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mn>1<\/mn><mo class=\"MathClass-close\">)<\/mo><\/math><span class=\"ecti-1095\">-mal stetig differenzierbare<\/span> <span class=\"ecti-1095\">Funktion und seien <\/span><math display=\"inline\"><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><mo class=\"MathClass-punc\">,<\/mo><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mo class=\"MathClass-open\">(<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>b<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math> <span class=\"ecti-1095\">zwei<\/span> <span class=\"ecti-1095\">Punkte. Wir setzen <\/span><span class=\"maperiod\"><math display=\"inline\"><msub><mrow><mi>M<\/mi><\/mrow><mrow><mi>n<\/mi><mo class=\"MathClass-bin\">+<\/mo><mn>1<\/mn><\/mrow><\/msub> <mo class=\"MathClass-rel\">=<\/mo><mi class=\"qopname\"> max<\/mi><mo>  <\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> {<\/mo><mrow><mo class=\"MathClass-rel\">|<\/mo><msup><mrow><mi>f<\/mi><\/mrow><mrow><mo class=\"MathClass-open\">(<\/mo><mi>n<\/mi><mo class=\"MathClass-bin\">+<\/mo><mn>1<\/mn><mo class=\"MathClass-close\">)<\/mo><\/mrow><\/msup><mo class=\"MathClass-open\">(<\/mo><mi>t<\/mi><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-rel\">|<\/mo><mo class=\"MathClass-rel\">\u2223<\/mo><mi>t<\/mi><mstyle class=\"text\"><mtext>&nbsp;zwischen&nbsp;<\/mtext><\/mstyle><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><mstyle class=\"text\"><mtext>&nbsp;und&nbsp;<\/mtext><\/mstyle><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">}<\/mo><\/mrow><\/math><\/span><span class=\"period\">.<\/span> <span class=\"ecti-1095\">Dann gilt<\/span> <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"> <mrow><mo fence=\"true\" form=\"prefix\"> |<\/mo><mrow><mi>f<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-bin\">\u2212<\/mo> <msubsup><mrow><mi>P<\/mi><\/mrow><mrow><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><mo class=\"MathClass-punc\">,<\/mo><mi>n<\/mi><\/mrow><mrow><mi>f<\/mi><\/mrow><\/msubsup> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><\/mrow><mo fence=\"true\" form=\"postfix\">|<\/mo><\/mrow> <mo class=\"MathClass-rel\">=<\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> |<\/mo><mrow><msubsup><mrow><mi>R<\/mi><\/mrow><mrow><msub><mrow> <mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><mo class=\"MathClass-punc\">,<\/mo><mi>n<\/mi><\/mrow><mrow><mi>f<\/mi><\/mrow><\/msubsup> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><\/mrow><mo fence=\"true\" form=\"postfix\">|<\/mo><\/mrow> <mo class=\"MathClass-rel\">\u2264<\/mo> <mfrac><mrow><msub><mrow><mi>M<\/mi><\/mrow><mrow><mi>n<\/mi><mo class=\"MathClass-bin\">+<\/mo><mn>1<\/mn><\/mrow><\/msub><mo class=\"MathClass-rel\">|<\/mo><mi>x<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><msup><mrow><mo class=\"MathClass-rel\">|<\/mo><\/mrow><mrow><mi>n<\/mi><mo class=\"MathClass-bin\">+<\/mo><mn>1<\/mn><\/mrow><\/msup><\/mrow> <mrow><mo class=\"MathClass-open\">(<\/mo><mi>n<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mn>1<\/mn><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-punc\">!<\/mo><\/mrow><\/mfrac> <mo class=\"MathClass-punc\">.<\/mo><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\"><span class=\"ecti-1095\">Insbesondere ist <\/span><math display=\"inline\"><mi>f<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-rel\">=<\/mo> <msubsup><mrow><mi>P<\/mi><\/mrow><mrow><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><mo class=\"MathClass-punc\">,<\/mo><mi>n<\/mi><\/mrow><mrow><mi>f<\/mi><\/mrow><\/msubsup> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-bin\">+<\/mo> <mi>O<\/mi><mo class=\"MathClass-open\">(<\/mo><msup><mrow><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mi>n<\/mi><mo class=\"MathClass-bin\">+<\/mo><mn>1<\/mn><\/mrow><\/msup><mo class=\"MathClass-close\">)<\/mo><\/math> <span class=\"ecti-1095\">f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r <\/span><span class=\"maperiod\"><math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2192<\/mo> <msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn> <\/mrow> <\/msub> <\/math><\/span><span class=\"period\">.<\/span> <\/p> <\/div> <p class=\"indent\"> <\/p> <div class=\"proof\"> <p class=\"indent\"><span class=\"head\"><\/span><\/p><details open><summary><b>Beweis.<\/b><\/summary><p class=\"indent\" style=\"margin-top: 10\">Angenommen <span class=\"maperiod\"><math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2265<\/mo> <msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><\/math><\/span><span class=\"period\">.<\/span> Dann gilt <\/p><math display=\"block\"> <mtable class=\"multline-star\"> <mtr><mtd class=\"multline-star\"> <mrow><mo fence=\"true\" form=\"prefix\"> |<\/mo><mrow><msubsup><mrow><mi>R<\/mi><\/mrow><mrow><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><mo class=\"MathClass-punc\">,<\/mo><mi>n<\/mi><\/mrow><mrow><mi>f<\/mi><\/mrow><\/msubsup> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><\/mrow><mo fence=\"true\" form=\"postfix\">|<\/mo><\/mrow> <mo class=\"MathClass-rel\">\u2264<\/mo><msubsup><mrow><mo>\u222b  <\/mo><\/mrow><mrow><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><\/mrow><mrow><mi>x<\/mi><\/mrow><\/msubsup> <mrow><mo fence=\"true\" form=\"prefix\"> |<\/mo><mrow><msup><mrow><mi>f<\/mi><\/mrow><mrow><mo class=\"MathClass-open\">(<\/mo><mi>n<\/mi><mo class=\"MathClass-bin\">+<\/mo><mn>1<\/mn><mo class=\"MathClass-close\">)<\/mo><\/mrow><\/msup> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>t<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><\/mrow><mo fence=\"true\" form=\"postfix\">|<\/mo><\/mrow> <mfrac><mrow><msup><mrow><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>t<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msup><\/mrow> <mrow><mi>n<\/mi><mo class=\"MathClass-punc\">!<\/mo><\/mrow><\/mfrac> <mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>t<\/mi> <mo class=\"MathClass-rel\">\u2264<\/mo> <mfrac><mrow><msub><mrow><mi>M<\/mi><\/mrow><mrow><mi>n<\/mi><mo class=\"MathClass-bin\">+<\/mo><mn>1<\/mn><\/mrow><\/msub><\/mrow> <mrow><mi>n<\/mi><mo class=\"MathClass-punc\">!<\/mo><\/mrow><\/mfrac> <msubsup><mrow><mo>\u222b  <\/mo><\/mrow><mrow><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><\/mrow><mrow><mi>x<\/mi><\/mrow><\/msubsup><msup><mrow><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>t<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msup><mspace class=\"thinspace\" width=\"0.17em\" \/> <mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>t<\/mi> <\/mtd><\/mtr><mtr><mtd class=\"multline-star\"> <mo class=\"MathClass-rel\">=<\/mo> <mfrac><mrow><msub><mrow><mi>M<\/mi><\/mrow><mrow><mi>n<\/mi><mo class=\"MathClass-bin\">+<\/mo><mn>1<\/mn><\/mrow><\/msub><\/mrow> <mrow><mi>n<\/mi><mo class=\"MathClass-punc\">!<\/mo><\/mrow><\/mfrac><msubsup><mrow> <mrow><mo fence=\"true\" form=\"prefix\"> [<\/mo><mrow><mo class=\"MathClass-bin\">\u2212<\/mo> <mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mi>n<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mn>1<\/mn><\/mrow><\/mfrac><msup><mrow><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>t<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mi>n<\/mi><mo class=\"MathClass-bin\">+<\/mo><mn>1<\/mn><\/mrow><\/msup><\/mrow><mo fence=\"true\" form=\"postfix\">]<\/mo><\/mrow> <\/mrow><mrow><msub><mrow> <mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><\/mrow><mrow><mi>x<\/mi><\/mrow><\/msubsup> <mo class=\"MathClass-rel\">=<\/mo> <mfrac><mrow><msub><mrow><mi>M<\/mi><\/mrow><mrow><mi>n<\/mi><mo class=\"MathClass-bin\">+<\/mo><mn>1<\/mn><\/mrow><\/msub><msup><mrow><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mi>n<\/mi><mo class=\"MathClass-bin\">+<\/mo><mn>1<\/mn><\/mrow><\/msup><\/mrow> <mrow><mo class=\"MathClass-open\">(<\/mo><mi>n<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mn>1<\/mn><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-punc\">!<\/mo><\/mrow><\/mfrac> <mo class=\"MathClass-punc\">.<\/mo> <\/mtd><\/mtr><\/mtable> <\/math> <p class=\"nopar\"> Der Beweis f\u00fcr <math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">&lt;<\/mo> <msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><\/math> ist analog ausgehend von                                                                                                                                                                           <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"> <mrow><mo fence=\"true\" form=\"prefix\"> |<\/mo><mrow><msubsup><mrow><mi>R<\/mi><\/mrow><mrow><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><mo class=\"MathClass-punc\">,<\/mo><mi>n<\/mi><\/mrow><mrow><mi>f<\/mi><\/mrow><\/msubsup> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><\/mrow><mo fence=\"true\" form=\"postfix\">|<\/mo><\/mrow> <mo class=\"MathClass-rel\">=<\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> |<\/mo><mrow><msubsup><mrow><mo>\u222b  <\/mo><\/mrow><mrow><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><\/mrow><mrow><mi>x<\/mi><\/mrow><\/msubsup><msup><mrow><mi>f<\/mi><\/mrow><mrow><mo class=\"MathClass-open\">(<\/mo><mi>n<\/mi><mo class=\"MathClass-bin\">+<\/mo><mn>1<\/mn><mo class=\"MathClass-close\">)<\/mo><\/mrow><\/msup> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>t<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mfrac><mrow><msup><mrow><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>t<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msup><\/mrow> <mrow><mi>n<\/mi><mo class=\"MathClass-punc\">!<\/mo><\/mrow><\/mfrac> <mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>t<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">|<\/mo><\/mrow> <mo class=\"MathClass-rel\">\u2264<\/mo><msubsup><mrow><mo>\u222b  <\/mo><\/mrow><mrow><mi>x<\/mi><\/mrow><mrow><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub> <\/mrow><\/msubsup><mo class=\"MathClass-rel\">|<\/mo><msup><mrow><mi>f<\/mi><\/mrow><mrow><mo class=\"MathClass-open\">(<\/mo><mi>n<\/mi><mo class=\"MathClass-bin\">+<\/mo><mn>1<\/mn><mo class=\"MathClass-close\">)<\/mo><\/mrow><\/msup> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>t<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><mo class=\"MathClass-rel\">|<\/mo><mfrac><mrow><msup><mrow><mo class=\"MathClass-open\">(<\/mo><mi>t<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msup><\/mrow> <mrow><mi>n<\/mi><mo class=\"MathClass-punc\">!<\/mo><\/mrow><\/mfrac> <mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>t<\/mi><mo class=\"MathClass-punc\">.<\/mo><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"><mstyle class=\"maketag\"><mtext \/><\/mstyle><mspace class=\"nbsp\" width=\"0.33em\" \/><mstyle class=\"label\" id=\"x1-276005r8\" \/> <\/mtd><\/mtr><\/mtable><\/math> <span>&nbsp;&nbsp;<\/span><div class=\"qed\">\u25a0<\/div><\/details><\/div> <div class=\"me meexample\"> <p class=\"indent\"><\/p><h4 id=\"zeefd0afc37dd\"> <span class=\"ecti-1095\">Bemerkung.<\/span><\/h4> <p class=\"indent\">Wir h\u00e4tten andere Versionen des obigen Satz zu Taylor-Approximationen bereits in Abschnitt <a href=\"..\/..\/chapter\/zentrale-saetze-der-differentialrechnung#x1-2320002\">8.2<\/a> f\u00fcr reellwertige Funktionen beweisen k\u00f6nnen. Dies sogar unter der etwas schw\u00e4cheren Annahme an die <math display=\"inline\"><mo class=\"MathClass-open\">(<\/mo><mi>n<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mn>1<\/mn><mo class=\"MathClass-close\">)<\/mo><\/math>-te Ableitung, dass diese bloss zwischen <math display=\"inline\"><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><\/math> und <math display=\"inline\"><mi>x<\/mi><\/math> existieren soll und nicht unbedingt stetig sein muss. Unter dieser Voraussetzung gibt es ein <math display=\"inline\"><msub><mrow><mi>\u03be<\/mi><\/mrow><mrow><mi>C<\/mi> <\/mrow> <\/msub> <\/math> zwischen <math display=\"inline\"><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn> <\/mrow> <\/msub> <\/math> und <span class=\"maperiod\"><math display=\"inline\"><mi>x<\/mi><\/math><\/span><span class=\"period\">,<\/span> so dass das sogenannte <span class=\"ecbx-1095\">Restglied nach Cauchy <\/span>durch <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><msubsup><mrow><mi>R<\/mi><\/mrow><mrow><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><mo class=\"MathClass-punc\">,<\/mo><mi>n<\/mi><\/mrow><mrow><mi>f<\/mi><\/mrow><\/msubsup> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-rel\">=<\/mo> <mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mi>n<\/mi><mo class=\"MathClass-punc\">!<\/mo><\/mrow><\/mfrac><msup><mrow><mi>f<\/mi><\/mrow><mrow><mo class=\"MathClass-open\">(<\/mo><mi>n<\/mi><mo class=\"MathClass-bin\">+<\/mo><mn>1<\/mn><mo class=\"MathClass-close\">)<\/mo><\/mrow><\/msup> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><msub><mrow><mi>\u03be<\/mi><\/mrow><mrow> <mi>C<\/mi><\/mrow><\/msub><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <msup><mrow><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <msub><mrow><mi>\u03be<\/mi><\/mrow><mrow><mi>C<\/mi><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msup> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <msub><mrow><mi>x<\/mi><\/mrow><mrow> <mn>0<\/mn><\/mrow><\/msub><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">gegeben ist. Es gibt unter denselben Voraussetzungen auch ein <math display=\"inline\"><msub><mrow><mi>\u03be<\/mi><\/mrow><mrow><mi>L<\/mi> <\/mrow> <\/msub> <\/math> zwischen <math display=\"inline\"><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn> <\/mrow> <\/msub> <\/math> und <span class=\"maperiod\"><math display=\"inline\"><mi>x<\/mi><\/math><\/span><span class=\"period\">,<\/span> so dass das <span class=\"ecbx-1095\">Restglied nach Lagrange <\/span>durch <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><msubsup><mrow><mi>R<\/mi><\/mrow><mrow><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><mo class=\"MathClass-punc\">,<\/mo><mi>n<\/mi><\/mrow><mrow><mi>f<\/mi><\/mrow><\/msubsup> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-rel\">=<\/mo> <mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mo class=\"MathClass-open\">(<\/mo><mi>n<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mn>1<\/mn><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-punc\">!<\/mo><\/mrow><\/mfrac><msup><mrow><mi>f<\/mi><\/mrow><mrow><mo class=\"MathClass-open\">(<\/mo><mi>n<\/mi><mo class=\"MathClass-bin\">+<\/mo><mn>1<\/mn><mo class=\"MathClass-close\">)<\/mo><\/mrow><\/msup> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><msub><mrow><mi>\u03be<\/mi><\/mrow><mrow> <mi>L<\/mi><\/mrow><\/msub><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <msup><mrow><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mi>n<\/mi><mo class=\"MathClass-bin\">+<\/mo><mn>1<\/mn><\/mrow><\/msup><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">gegeben ist. Wir verweisen daf\u00fcr auf folgende \u00dcbung. <\/p> <\/div> <div class=\"me meexample\"> <p class=\"indent\"><\/p><h4 id=\"z6682e7ccd91f\"> <a id=\"x1-276006r48\"><\/a> <span class=\"ecbx-1095\">\u00dc<\/span><span class=\"ecbx-1095\">bung 9.48 <\/span>(Andere Formeln f\u00fcr das Restglied)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Gegeben sei ein nicht-leeres Intervall <\/span><span class=\"maperiod\"><math display=\"inline\"><mo class=\"MathClass-open\">(<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>b<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">\u2286<\/mo> <mi>\u211d<\/mi><\/math><\/span><span class=\"period\">,<\/span> <span class=\"ecti-1095\">Zahlen <\/span><math display=\"inline\"><mi>x<\/mi><mo class=\"MathClass-punc\">,<\/mo> <msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn> <\/mrow><\/msub> <mo class=\"MathClass-rel\">\u2208<\/mo> <mo class=\"MathClass-open\">(<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>b<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math> <span class=\"ecti-1095\">und eine<\/span> <math display=\"inline\"><mo class=\"MathClass-open\">(<\/mo><mi>n<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mn>1<\/mn><mo class=\"MathClass-close\">)<\/mo><\/math><span class=\"ecti-1095\">-mal differenzierbare<\/span> <span class=\"ecti-1095\">Funktion <\/span><span class=\"maperiod\"><math display=\"inline\"><mi>f<\/mi> <mo class=\"MathClass-punc\">:<\/mo> <mo class=\"MathClass-open\">(<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>b<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">\u2192<\/mo> <mi>\u211d<\/mi><\/math><\/span><span class=\"period\">.<\/span> <\/p><dl class=\"enumerate\"><dt class=\"enumerate\"> <span class=\"ecti-1095\">(a)<\/span><\/dt><dd class=\"enumerate\"><span class=\"ecti-1095\">Zeigen Sie, dass die Ableitung der Funktion<\/span> <math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mi>F<\/mi> <mo class=\"MathClass-punc\">:<\/mo> <mi>t<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>b<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><mo class=\"MathClass-rel\">\u21a6<\/mo><munderover accent=\"false\" accentunder=\"false\"><mrow><mo>\u2211<\/mo> <\/mrow><mrow><mi>k<\/mi><mo class=\"MathClass-rel\">=<\/mo><mn>0<\/mn><\/mrow><mrow><mi>n<\/mi><\/mrow><\/munderover><mfrac><mrow><msup><mrow><mi>f<\/mi><\/mrow><mrow><mo class=\"MathClass-open\">(<\/mo><mi>k<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow><\/msup><mo class=\"MathClass-open\">(<\/mo><mi>t<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow> <mrow><mi>k<\/mi><mo class=\"MathClass-punc\">!<\/mo><\/mrow><\/mfrac> <msup><mrow><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>t<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mi>k<\/mi><\/mrow><\/msup><mo class=\"MathClass-punc\">,<\/mo><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\"><span class=\"ecti-1095\">durch <\/span><math display=\"inline\"><mi>t<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>b<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><mo class=\"MathClass-rel\">\u21a6<\/mo><mfrac><mrow><msup><mrow><mi>f<\/mi><\/mrow><mrow><mo class=\"MathClass-open\">(<\/mo><mi>n<\/mi><mo class=\"MathClass-bin\">+<\/mo><mn>1<\/mn><mo class=\"MathClass-close\">)<\/mo><\/mrow><\/msup><mo class=\"MathClass-open\">(<\/mo><mi>t<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow> <mrow><mi>n<\/mi><mo class=\"MathClass-punc\">!<\/mo><\/mrow><\/mfrac> <msup><mrow><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>t<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msup><\/math> <span class=\"ecti-1095\">gegeben ist.<\/span> <\/p><\/dd><dt class=\"enumerate\"> <span class=\"ecti-1095\">(b)<\/span><\/dt><dd class=\"enumerate\"><span class=\"ecti-1095\">Wenden Sie den Mittelwertsatz (Theorem <\/span><a href=\"..\/..\/chapter\/zentrale-saetze-der-differentialrechnung#x1-233002r29\"><span class=\"ecti-1095\">8.29<\/span><\/a><span class=\"ecti-1095\">) auf die obige Funktion<\/span> <math display=\"inline\"><mi>F<\/mi><\/math> <span class=\"ecti-1095\">an, um die<\/span> <span class=\"ecti-1095\">Formel <\/span><math display=\"inline\"><msubsup><mrow><mi>R<\/mi><\/mrow><mrow><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><mo class=\"MathClass-punc\">,<\/mo><mi>n<\/mi><\/mrow><mrow><mi>f<\/mi><\/mrow><\/msubsup> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-rel\">=<\/mo> <mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mi>n<\/mi><mo class=\"MathClass-punc\">!<\/mo><\/mrow><\/mfrac><msup><mrow><mi>f<\/mi><\/mrow><mrow><mo class=\"MathClass-open\">(<\/mo><mi>n<\/mi><mo class=\"MathClass-bin\">+<\/mo><mn>1<\/mn><mo class=\"MathClass-close\">)<\/mo><\/mrow><\/msup> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><msub><mrow><mi>\u03be<\/mi><\/mrow><mrow><mi>C<\/mi><\/mrow><\/msub><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><msup><mrow> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <msub><mrow><mi>\u03be<\/mi><\/mrow><mrow><mi>C<\/mi><\/mrow><\/msub><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <\/mrow><mrow><mi>n<\/mi><\/mrow><\/msup> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><\/math> <span class=\"ecti-1095\">f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r das Restglied nach Cauchy zu beweisen.<\/span> <\/dd><dt class=\"enumerate\"> <span class=\"ecti-1095\">(c)<\/span><\/dt><dd class=\"enumerate\"><span class=\"ecti-1095\">Verwenden Sie den verallgemeinerten Mittelwertsatz (Satz <\/span><a href=\"..\/..\/chapter\/zentrale-saetze-der-differentialrechnung#x1-236001r48\"><span class=\"ecti-1095\">8.48<\/span><\/a><span class=\"ecti-1095\">) f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r obiges<\/span> <math display=\"inline\"><mi>F<\/mi><\/math> <span class=\"ecti-1095\">und die<\/span> <span class=\"ecti-1095\">Funktion <\/span><math display=\"inline\"><mi>g<\/mi> <mo class=\"MathClass-punc\">:<\/mo> <mi>t<\/mi><mo class=\"MathClass-rel\">\u21a6<\/mo><msup><mrow><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>t<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mi>n<\/mi><mo class=\"MathClass-bin\">+<\/mo><mn>1<\/mn><\/mrow><\/msup><\/math><span class=\"ecti-1095\">, um<\/span> <span class=\"ecti-1095\">die Formel <\/span><math display=\"inline\"><msubsup><mrow><mi>R<\/mi><\/mrow><mrow><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><mo class=\"MathClass-punc\">,<\/mo><mi>n<\/mi><\/mrow><mrow><mi>f<\/mi><\/mrow><\/msubsup> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-rel\">=<\/mo> <mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mo class=\"MathClass-open\">(<\/mo><mi>n<\/mi><mo class=\"MathClass-bin\">+<\/mo><mn>1<\/mn><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-punc\">!<\/mo><\/mrow><\/mfrac><msup><mrow><mi>f<\/mi><\/mrow><mrow><mo class=\"MathClass-open\">(<\/mo><mi>n<\/mi><mo class=\"MathClass-bin\">+<\/mo><mn>1<\/mn><mo class=\"MathClass-close\">)<\/mo><\/mrow><\/msup> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><msub><mrow><mi>\u03be<\/mi><\/mrow><mrow><mi>L<\/mi><\/mrow><\/msub><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <msup><mrow><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mi>n<\/mi><mo class=\"MathClass-bin\">+<\/mo><mn>1<\/mn><\/mrow><\/msup><\/math> <span class=\"ecti-1095\">f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r das Restglied nach Lagrange zu beweisen.<\/span><\/dd><\/dl> <\/div> <div class=\"me meexample\"> <p class=\"indent\"><\/p><h4 id=\"zb051224d12d3\"> <a id=\"x1-276010r49\"><\/a> <span class=\"ecbx-1095\">Beispiel 9.49 <\/span>(Extremwerte)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Wir wollen die Taylor-Approximation verwenden um die Diskussion in Abschnitt <\/span><a href=\"..\/..\/chapter\/die-ableitung#x1-2290003\"><span class=\"ecti-1095\">8.1.3<\/span><\/a> <span class=\"ecti-1095\">und Korollar <\/span><a href=\"..\/..\/chapter\/zentrale-saetze-der-differentialrechnung#x1-234005r37\"><span class=\"ecti-1095\">8.37<\/span><\/a> <span class=\"ecti-1095\">zu verfeinern.<\/span> <span class=\"ecti-1095\">Sei <\/span><math display=\"inline\"><mo class=\"MathClass-open\">(<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo> <mi>b<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">\u2286<\/mo> <mi>\u211d<\/mi><\/math> <span class=\"ecti-1095\">ein offenes, nicht-leeres<\/span> <span class=\"ecti-1095\">Intervall und sei <\/span><math display=\"inline\"><mi>f<\/mi> <mo class=\"MathClass-punc\">:<\/mo> <mo class=\"MathClass-open\">(<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>b<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">\u2192<\/mo> <mi>\u2102<\/mi><\/math> <span class=\"ecti-1095\">eine<\/span> <math display=\"inline\"><mi>n<\/mi><\/math><span class=\"ecti-1095\">-mal stetig differenzierbare<\/span> <span class=\"ecti-1095\">Funktion. Angenommen <\/span><math display=\"inline\"><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub> <mo class=\"MathClass-rel\">\u2208<\/mo> <mo class=\"MathClass-open\">(<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>b<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math> <span class=\"ecti-1095\">erf<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">llt<\/span> <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><msup><mrow><mi>f<\/mi><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow> <mn>0<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <mi class=\"MathClass-op\">\u2026<\/mi><mo> <\/mo> <mo class=\"MathClass-rel\">=<\/mo> <msup><mrow><mi>f<\/mi><\/mrow><mrow><mo class=\"MathClass-open\">(<\/mo><mi>n<\/mi><mo class=\"MathClass-bin\">\u2212<\/mo><mn>1<\/mn><mo class=\"MathClass-close\">)<\/mo><\/mrow><\/msup><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow> <mn>0<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <mn>0<\/mn><mo class=\"MathClass-punc\">.<\/mo><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\"><span class=\"ecti-1095\">Dann gelten folgende Implikationen.<\/span> <\/p> <div class=\"custom-itemize\"><div class=\"item-head\"> <span class=\"tcrm-1095\">\u2022<\/span><\/div><div class=\"item-content\"><span class=\"ecti-1095\">Falls <\/span><math display=\"inline\"><msup><mrow><mi>f<\/mi><\/mrow><mrow><mo class=\"MathClass-open\">(<\/mo><mi>n<\/mi><mo class=\"MathClass-close\">)<\/mo> <\/mrow> <\/msup> <mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">&lt;<\/mo> <mn>0<\/mn><\/math> <span class=\"ecti-1095\">ist und <\/span><math display=\"inline\"><mi>n<\/mi><\/math> <span class=\"ecti-1095\">gerade ist, so nimmt <\/span><math display=\"inline\"><mi>f<\/mi><\/math> <span class=\"ecti-1095\">in <\/span><math display=\"inline\"><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn> <\/mrow> <\/msub> <\/math> <span class=\"ecti-1095\">ein lokales Maximum an.<\/span> <\/div><div class=\"item-head\"> <span class=\"tcrm-1095\">\u2022<\/span><\/div><div class=\"item-content\"><span class=\"ecti-1095\">Falls <\/span><math display=\"inline\"><msup><mrow><mi>f<\/mi><\/mrow><mrow><mo class=\"MathClass-open\">(<\/mo><mi>n<\/mi><mo class=\"MathClass-close\">)<\/mo> <\/mrow> <\/msup> <mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">&gt;<\/mo> <mn>0<\/mn><\/math> <span class=\"ecti-1095\">ist und <\/span><math display=\"inline\"><mi>n<\/mi><\/math> <span class=\"ecti-1095\">gerade ist, so nimmt <\/span><math display=\"inline\"><mi>f<\/mi><\/math> <span class=\"ecti-1095\">in <\/span><math display=\"inline\"><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn> <\/mrow> <\/msub> <\/math> <span class=\"ecti-1095\">ein lokales Minimum an.<\/span> <\/div><div class=\"item-head\"> <span class=\"tcrm-1095\">\u2022<\/span><\/div><div class=\"item-content\"><span class=\"ecti-1095\">Falls <\/span><math display=\"inline\"><msup><mrow><mi>f<\/mi><\/mrow><mrow><mo class=\"MathClass-open\">(<\/mo><mi>n<\/mi><mo class=\"MathClass-close\">)<\/mo> <\/mrow> <\/msup> <mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-rel\">\u2260<\/mo><mn>0<\/mn><\/math> <span class=\"ecti-1095\">und <\/span><math display=\"inline\"><mi>n<\/mi><\/math> <span class=\"ecti-1095\">ungerade ist, so nimmt <\/span><math display=\"inline\"><mi>f<\/mi><\/math> <span class=\"ecti-1095\">in <\/span><math display=\"inline\"><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn> <\/mrow> <\/msub> <\/math> <span class=\"ecti-1095\">kein lokales Extremum an.<\/span><\/div><\/div> <p class=\"indent\"><span class=\"ecti-1095\">Alle drei Aussagen folgen aus der Taylor-Approximation in Theorem <\/span><a href=\"..\/..\/chapter\/taylor-approximation#x1-276002r46\"><span class=\"ecti-1095\">9.46<\/span><\/a><span class=\"ecti-1095\">, die in diesem Fall f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r<\/span> <math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mo class=\"MathClass-open\">(<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo> <mi>b<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math> <span class=\"ecti-1095\">die<\/span> <span class=\"ecti-1095\">Form<\/span> <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mi>f<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-rel\">=<\/mo> <mi>f<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-bin\">+<\/mo><msubsup><mrow><mo> \u222b  <\/mo><\/mrow><mrow><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><\/mrow><mrow><mi>x<\/mi><\/mrow><\/msubsup><msup><mrow><mi>f<\/mi><\/mrow><mrow><mo class=\"MathClass-open\">(<\/mo><mi>n<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow><\/msup> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>t<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mfrac><mrow><msup><mrow><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>t<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mi>n<\/mi><mo class=\"MathClass-bin\">\u2212<\/mo><mn>1<\/mn><\/mrow><\/msup><\/mrow> <mrow><mo class=\"MathClass-open\">(<\/mo><mi>n<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <mn>1<\/mn><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-punc\">!<\/mo><\/mrow><\/mfrac> <mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>t<\/mi><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\"><span class=\"ecti-1095\">annimmt. Falls <\/span><math display=\"inline\"><msup><mrow><mi>f<\/mi><\/mrow><mrow><mo class=\"MathClass-open\">(<\/mo><mi>n<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow><\/msup><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">&gt;<\/mo> <mn>0<\/mn><\/math> <span class=\"ecti-1095\">ist, existiert ein <\/span><math display=\"inline\"><mi>\u03b4<\/mi> <mo class=\"MathClass-rel\">&gt;<\/mo> <mn>0<\/mn><\/math> <span class=\"ecti-1095\">mit <\/span><math display=\"inline\"><msup><mrow><mi>f<\/mi><\/mrow><mrow><mo class=\"MathClass-open\">(<\/mo><mi>n<\/mi><mo class=\"MathClass-close\">)<\/mo> <\/mrow> <\/msup> <mo class=\"MathClass-open\">(<\/mo><mi>t<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">&gt;<\/mo> <mn>0<\/mn><\/math> <span class=\"ecti-1095\">f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r alle<\/span> <math display=\"inline\"><mi>t<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>\u03b4<\/mi><mo class=\"MathClass-punc\">,<\/mo> <msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub> <mo class=\"MathClass-bin\">+<\/mo> <mi>\u03b4<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math><span class=\"ecti-1095\">. Wir betrachten nun mehrere<\/span> <span class=\"ecti-1095\">F<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">lle. Ist <\/span><math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><mo class=\"MathClass-punc\">,<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub> <mo class=\"MathClass-bin\">+<\/mo> <mi>\u03b4<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math><span class=\"ecti-1095\">, so ist obiges<\/span> <span class=\"ecti-1095\">Integral positiv und damit <\/span><span class=\"maperiod\"><math display=\"inline\"><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">&gt;<\/mo> <mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><\/math><\/span><span class=\"period\">.<\/span> <span class=\"ecti-1095\">Ist <\/span><math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>\u03b4<\/mi><mo class=\"MathClass-punc\">,<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><\/math> <span class=\"ecti-1095\">und<\/span> <math display=\"inline\"><mi>n<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <mn>1<\/mn><\/math> <span class=\"ecti-1095\">gerade, so<\/span> <span class=\"ecti-1095\">ist obiges Integral (auf Grund der umgekehrten Reihenfolge der Integrationsgrenzen) negativ, womit<\/span> <math display=\"inline\"><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">&lt;<\/mo> <mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><\/math> <span class=\"ecti-1095\">gilt<\/span> <span class=\"ecti-1095\">und <\/span><math display=\"inline\"><mi>f<\/mi><\/math> <span class=\"ecti-1095\">in<\/span> <math display=\"inline\"><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn> <\/mrow> <\/msub> <\/math> <span class=\"ecti-1095\">kein lokales Extremum<\/span> <span class=\"ecti-1095\">annimmt. Ist hingegen <\/span><math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>\u03b4<\/mi><mo class=\"MathClass-punc\">,<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><\/math> <span class=\"ecti-1095\">und <\/span><math display=\"inline\"><mi>n<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <mn>1<\/mn><\/math> <span class=\"ecti-1095\">ungerade, so ergibt<\/span> <span class=\"ecti-1095\">sich auf dieselbe Weise <\/span><span class=\"maperiod\"><math display=\"inline\"><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">&gt;<\/mo> <mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><\/math><\/span><span class=\"period\">,<\/span> <span class=\"ecti-1095\">womit <\/span><math display=\"inline\"><mi>f<\/mi><\/math> <span class=\"ecti-1095\">ein lokales<\/span> <span class=\"ecti-1095\">Minimum in <\/span><math display=\"inline\"><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><\/math> <span class=\"ecti-1095\">annimmt.<\/span> <span class=\"ecti-1095\">F<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r <\/span><math display=\"inline\"><msup><mrow><mi>f<\/mi><\/mrow><mrow><mo class=\"MathClass-open\">(<\/mo><mi>n<\/mi><mo class=\"MathClass-close\">)<\/mo> <\/mrow> <\/msup> <mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">&lt;<\/mo> <mn>0<\/mn><\/math> <span class=\"ecti-1095\">k<\/span><span class=\"ecti-1095\">\u00f6<\/span><span class=\"ecti-1095\">nnen wir<\/span> <span class=\"ecti-1095\">obige Diskussion auf <\/span><math display=\"inline\"> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>f<\/mi><\/math> <span class=\"ecti-1095\">anwenden.<\/span> <\/p> <\/div> <a id=\"x1-276011r275\"><\/a> <h4 id=\"z2f16eff2c7c4\" class=\"subsectionHead\"><span class=\"titlemark\">9.4.1 <\/span> <a id=\"x1-2770001\"><\/a>Analytische Funktionen<\/h4> <div class=\"me meexample\"> <p class=\"indent\"><\/p><h4 id=\"zf64cf6a02dcc\"> <a id=\"x1-277001r50\"><\/a> <span class=\"ecbx-1095\">Applet 9.50 <\/span>(Taylor-Approximationen)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\"><\/p><div class=\"geoapplet\" style=\"width: 688px\"><iframe height=\"416px\" scrolling=\"no\" src=\"https:\/\/www.geogebra.org\/material\/iframe\/id\/Vh93DD4U\/width\/688\/height\/416\/border\/888888\/rc\/false\/ai\/false\/sdz\/true\/smb\/false\/stb\/false\/stbh\/false\/ld\/false\/sri\/false\" style=\"border:0px\"><\/iframe><\/div><p class=\"indent\"><span class=\"ecti-1095\">Wir stellen einige Taylor-Approximationen bei verschiebbaren Fusspunkten zu bekannten<\/span> <span class=\"ecti-1095\">Funktionen dar.<\/span> <\/p> <\/div> <p class=\"indent\">Betrachtet man eine glatte Funktion <math display=\"inline\"><mi>f<\/mi><\/math> in Theorem <a href=\"..\/..\/chapter\/taylor-approximation#x1-276002r46\">9.46<\/a>, f\u00fcr die der Wert der Ableitung von <math display=\"inline\"><mi>f<\/mi><\/math> nicht zu wild wird f\u00fcr hohe Ableitungen, dann geht das Restglied zu einem fest gew\u00e4hlten <math display=\"inline\"><mi>x<\/mi><\/math> f\u00fcr                                                                                                                                                                           <math display=\"inline\"><mi>n<\/mi> <mo class=\"MathClass-rel\">\u2192<\/mo> <mi>\u221e<\/mi><\/math> gegen Null und die Funktion <math display=\"inline\"><mi>f<\/mi><\/math> wird durch ihre Taylorreihe beschrieben. Wie bereits gesehen, muss dies jedoch nicht unbedingt der Fall sein. Eine m\u00f6gliche Absch\u00e4tzung, die ausreichen w\u00fcrde, ist <\/p><table id=\"zeb0349350474\" class=\"equation-star\"><tr><td> <math class=\"equation\" display=\"block\"> <munder class=\"msub\"><mrow><mi class=\"qopname\">max<\/mi><mo>  <\/mo><\/mrow><mrow><mi>t<\/mi><mo class=\"MathClass-rel\">\u2208<\/mo><mo class=\"MathClass-open\">[<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><mo class=\"MathClass-punc\">,<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">]<\/mo><\/mrow><\/munder> <mrow><mo fence=\"true\" form=\"prefix\"> |<\/mo><mrow><msup><mrow><mi>f<\/mi><\/mrow><mrow><mo class=\"MathClass-open\">(<\/mo><mi>n<\/mi><mo class=\"MathClass-bin\">+<\/mo><mn>1<\/mn><mo class=\"MathClass-close\">)<\/mo><\/mrow><\/msup> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>t<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><\/mrow><mo fence=\"true\" form=\"postfix\">|<\/mo><\/mrow> <mo class=\"MathClass-rel\">\u2264<\/mo> <mi>c<\/mi><msup><mrow><mi>A<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msup> <\/math><\/td><\/tr><\/table> <p class=\"indent\">f\u00fcr alle&nbsp;<math display=\"inline\"><mi>n<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2115<\/mi><\/math> und zwei Konstanten&nbsp;<span class=\"maperiod\"><math display=\"inline\"><mi>c<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>A<\/mi> <mo class=\"MathClass-rel\">\u2265<\/mo> <mn>1<\/mn><\/math><\/span><span class=\"period\">.<\/span> Eine Absch\u00e4tzung dieser Art findet man beispielsweise f\u00fcr <math display=\"inline\"><mi class=\"qopname\">exp<\/mi><mo>  <\/mo><mo class=\"MathClass-punc\">,<\/mo><mi class=\"qopname\"> sin<\/mi><mo>  <\/mo> <mo class=\"MathClass-punc\">,<\/mo><mi class=\"qopname\"> cos<\/mi><mo>  <\/mo> <\/math> und Kombinationen dieser Funktionen. <\/p> <div class=\"me meexample\"> <p class=\"indent\"><\/p><h4 id=\"ze538c7206e48\"> <a id=\"x1-277002r51\"><\/a> <span class=\"ecbx-1095\">\u00dc<\/span><span class=\"ecbx-1095\">bung 9.51.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Sei <\/span><math display=\"inline\"><mo class=\"MathClass-open\">[<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo> <mi>b<\/mi><mo class=\"MathClass-close\">]<\/mo> <mo class=\"MathClass-rel\">\u2286<\/mo> <mi>\u211d<\/mi><\/math> <span class=\"ecti-1095\">ein kompaktes<\/span> <span class=\"ecti-1095\">Intervall mit Endpunkten <\/span><span class=\"maperiod\"><math display=\"inline\"><mi>a<\/mi> <mo class=\"MathClass-rel\">&lt;<\/mo> <mi>b<\/mi><\/math><\/span><span class=\"period\">.<\/span> <\/p><dl class=\"enumerate\"><dt class=\"enumerate\"> <span class=\"ecti-1095\">(a)<\/span><\/dt><dd class=\"enumerate\"><span class=\"ecti-1095\">Sei <\/span><math display=\"inline\"><mi>f<\/mi> <mo class=\"MathClass-punc\">:<\/mo> <mo class=\"MathClass-open\">[<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo> <mi>b<\/mi><mo class=\"MathClass-close\">]<\/mo> <mo class=\"MathClass-rel\">\u2192<\/mo> <mi>\u211d<\/mi><\/math> <span class=\"ecti-1095\">ein Polynom oder eine<\/span> <span class=\"ecti-1095\">der Funktionen <\/span><math display=\"inline\"><mi class=\"qopname\">exp<\/mi><mo>  <\/mo><mo class=\"MathClass-punc\">,<\/mo><mi class=\"qopname\">sin<\/mi><mo>  <\/mo><mo class=\"MathClass-punc\">,<\/mo><mi class=\"qopname\">cos<\/mi><mo>  <\/mo><\/math><span class=\"ecti-1095\">. Zeigen<\/span> <span class=\"ecti-1095\">Sie, dass Konstanten <\/span><math display=\"inline\"><mi>c<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>A<\/mi> <mo class=\"MathClass-rel\">\u2265<\/mo> <mn>1<\/mn><\/math> <span class=\"ecti-1095\">mit<\/span> <math display=\"block\"><mtable class=\"align\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><munder class=\"msub\"><mrow><mi class=\"qopname\">max<\/mi><mo>  <\/mo><\/mrow><mrow><mi>t<\/mi><mo class=\"MathClass-rel\">\u2208<\/mo><mo class=\"MathClass-open\">[<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>b<\/mi><mo class=\"MathClass-close\">]<\/mo><\/mrow><\/munder> <mrow><mo fence=\"true\" form=\"prefix\"> |<\/mo><mrow><msup><mrow><mi>f<\/mi><\/mrow><mrow><mo class=\"MathClass-open\">(<\/mo><mi>n<\/mi><mo class=\"MathClass-bin\">+<\/mo><mn>1<\/mn><mo class=\"MathClass-close\">)<\/mo><\/mrow><\/msup> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>t<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><\/mrow><mo fence=\"true\" form=\"postfix\">|<\/mo><\/mrow> <mo class=\"MathClass-rel\">\u2264<\/mo> <mi>c<\/mi><msup><mrow><mi>A<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msup><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"><mstyle class=\"label\" id=\"x1-277004r9\" \/><mstyle class=\"maketag\"><mtext>(9.9)<\/mtext><\/mstyle><mspace class=\"nbsp\" width=\"0.33em\" \/> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\"><span class=\"ecti-1095\">existieren.<\/span> <\/p><\/dd><dt class=\"enumerate\"> <span class=\"ecti-1095\">(b)<\/span><\/dt><dd class=\"enumerate\"><span class=\"ecti-1095\">Zeigen Sie, dass falls <\/span><math display=\"inline\"><mi>f<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>g<\/mi> <mo class=\"MathClass-punc\">:<\/mo> <mo class=\"MathClass-open\">[<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>b<\/mi><mo class=\"MathClass-close\">]<\/mo> <mo class=\"MathClass-rel\">\u2192<\/mo> <mi>\u211d<\/mi><\/math> <span class=\"ecti-1095\">Funktionen<\/span> <span class=\"ecti-1095\">mit der Eigenschaft<\/span> (<a href=\"..\/..\/chapter\/taylor-approximation#x1-277004r9\">9.9<\/a>) <span class=\"ecti-1095\">auch <\/span><math display=\"inline\"><mi>f<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mi>g<\/mi><\/math> <span class=\"ecti-1095\">und <\/span><math display=\"inline\"><mi>f<\/mi> <mo class=\"MathClass-bin\">\u22c5<\/mo> <mi>g<\/mi><\/math> <span class=\"ecti-1095\">diese Eigenschaften besitzen.<\/span><\/dd><\/dl> <\/div> <p class=\"indent\">Eine Funktion, die sich um jeden Punkt im Definitionsbereich als Potenzreihe darstellen l\u00e4sst, nennen wir analytisch. Genauer sei <math display=\"inline\"><mo class=\"MathClass-open\">(<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo> <mi>b<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">\u2286<\/mo> <mi>\u211d<\/mi><\/math> ein offenes, nicht-leeres Intervall und <math display=\"inline\"><mi>f<\/mi> <mo class=\"MathClass-punc\">:<\/mo> <mo class=\"MathClass-open\">(<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo> <mi>b<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">\u2192<\/mo> <mi>\u2102<\/mi><\/math> eine glatte Funktion. Die Funktion <math display=\"inline\"><mi>f<\/mi><\/math> heisst <span class=\"ecbx-1095\">analytisch<\/span>, falls die Taylorreihe von <math display=\"inline\"><mi>f<\/mi><\/math> um jeden Punkt in <math display=\"inline\"><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub> <mo class=\"MathClass-rel\">\u2208<\/mo> <mo class=\"MathClass-open\">(<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>b<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math> positiven Konvergenzradius&nbsp;<math display=\"inline\"><mi>R<\/mi><\/math> hat und auf <math display=\"inline\"><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>R<\/mi><mo class=\"MathClass-punc\">,<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub> <mo class=\"MathClass-bin\">+<\/mo> <mi>R<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">\u2229<\/mo> <mo class=\"MathClass-open\">(<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>b<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math> mit <math display=\"inline\"><mi>f<\/mi><\/math> \u00fcbereinstimmt. <\/p><p class=\"indent\">Wir wollen ein weiteres Beispiel, wo sich eine glatte Funktion mit ihrer Taylorreihe vergleichen l\u00e4sst, Interessierten als \u00dcbung \u00fcberlassen (vergleichen Sie dies mit \u00dcbung <a href=\"..\/..\/chapter\/der-fundamentalsatz-der-integral--und-differentialrechnung#x1-260004r14\">9.14<\/a>). <\/p> <div class=\"me meexample\"> <p class=\"indent\"><\/p><h4 id=\"zb2696af92b1c\"> <a id=\"x1-277006r52\"><\/a> <span class=\"ecbx-1095\">\u00dc<\/span><span class=\"ecbx-1095\">bung 9.52.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Sei <\/span><span class=\"maperiod\"><math display=\"inline\"><mi>\u03b1<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2102<\/mi><\/math><\/span><span class=\"period\">.<\/span> <span class=\"ecti-1095\">Zeigen Sie, dass<\/span> <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><msup><mrow><mo class=\"MathClass-open\">(<\/mo><mn>1<\/mn> <mo class=\"MathClass-bin\">+<\/mo> <mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mi>\u03b1<\/mi><\/mrow><\/msup> <mo class=\"MathClass-rel\">=<\/mo><munderover accent=\"false\" accentunder=\"false\"><mrow><mo> \u2211<\/mo> <\/mrow><mrow><mi>n<\/mi><mo class=\"MathClass-rel\">=<\/mo><mn>0<\/mn><\/mrow><mrow><mi>\u221e<\/mi><\/mrow><\/munderover><mfenced close=\")\" open=\"(\" separators><mfrac linethickness=\"0.0pt\"><mrow><mi>\u03b1<\/mi><\/mrow> <mrow><mi>n<\/mi><\/mrow><\/mfrac><\/mfenced><msup><mrow><mi>x<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msup><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\"><span class=\"ecti-1095\">f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r alle <\/span><span class=\"maperiod\"><math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mo class=\"MathClass-open\">(<\/mo><mo class=\"MathClass-bin\">\u2212<\/mo><mn>1<\/mn><mo class=\"MathClass-punc\">,<\/mo><mn>1<\/mn><mo class=\"MathClass-close\">)<\/mo><\/math><\/span><span class=\"period\">,<\/span> <span class=\"ecti-1095\">wobei f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r <\/span><math display=\"inline\"><mi>n<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <msub><mrow><mi>\u2115<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><\/math> <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mfenced close=\")\" open=\"(\" separators><mfrac linethickness=\"0.0pt\"><mrow><mi>\u03b1<\/mi><\/mrow> <mrow><mi>n<\/mi><\/mrow><\/mfrac><\/mfenced> <mo class=\"MathClass-rel\">=<\/mo> <mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mi>n<\/mi><mo class=\"MathClass-punc\">!<\/mo><\/mrow><\/mfrac><munderover accent=\"false\" accentunder=\"false\"><mrow><mo>\u220f<\/mo> <\/mrow><mrow><mi>k<\/mi><mo class=\"MathClass-rel\">=<\/mo><mn>0<\/mn><\/mrow><mrow><mi>n<\/mi><mo class=\"MathClass-bin\">\u2212<\/mo><mn>1<\/mn><\/mrow><\/munderover> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>\u03b1<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>k<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-rel\">=<\/mo> <mfrac><mrow><mi>\u03b1<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>\u03b1<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <mn>1<\/mn><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-rel\">\u22ef<\/mo><mo class=\"MathClass-open\">(<\/mo><mi>\u03b1<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>n<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mn>1<\/mn><mo class=\"MathClass-close\">)<\/mo><\/mrow> <mrow><mi>n<\/mi><mo class=\"MathClass-punc\">!<\/mo><\/mrow><\/mfrac> <mo class=\"MathClass-punc\">.<\/mo><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\"><\/p><details><summary style=\"color:#FF7F00\"><span class=\"ecti-1095\">Hinweis.<\/span><\/summary><p class=\"indent\" style=\"margin-top: 0\"><span class=\"ecti-1095\">Sei <\/span><span class=\"maperiod\"><math display=\"inline\"><mi>f<\/mi> <mo class=\"MathClass-punc\">:<\/mo> <mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mo class=\"MathClass-open\">(<\/mo><mo class=\"MathClass-bin\">\u2212<\/mo><mn>1<\/mn><mo class=\"MathClass-punc\">,<\/mo><mn>1<\/mn><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-rel\">\u21a6<\/mo><msup><mrow><mo class=\"MathClass-open\">(<\/mo><mn>1<\/mn> <mo class=\"MathClass-bin\">+<\/mo> <mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mi>\u03b1<\/mi><\/mrow><\/msup><\/math><\/span><span class=\"period\">.<\/span> <span class=\"ecti-1095\">Berechnen Sie zuerst <\/span><math display=\"inline\"><msup><mrow><mi>f<\/mi><\/mrow><mrow><mo class=\"MathClass-open\">(<\/mo><mi>n<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow><\/msup><\/math> <span class=\"ecti-1095\">f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r alle <\/span><math display=\"inline\"><mi>n<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2115<\/mi><\/math> <span class=\"ecti-1095\">und geben Sie<\/span> <span class=\"ecti-1095\">die Taylorreihe von <\/span><math display=\"inline\"><mi>f<\/mi><\/math> <span class=\"ecti-1095\">um Null an. Zeigen Sie dann, dass das Restglied<\/span> <span class=\"ecti-1095\">\u201e<\/span><span class=\"ecti-1095\">klein<\/span><span class=\"ecti-1095\">\u201c<\/span> <span class=\"ecti-1095\">wird.<\/span><\/p><\/details>  <\/div> <div class=\"me meexample\"> <p class=\"indent\"><\/p><h4 id=\"zdbec45531e83\"> <span class=\"ecti-1095\">Bemerkung.<\/span><\/h4> <p class=\"indent\">Analytische  Funktionen  haben  im  Zusammenhang  mit  \u201eholomorphen  Funktionen\u201c  auf offenen Teilmengen von <math display=\"inline\"><mi>\u2102<\/mi><\/math> nach <math display=\"inline\"><mi>\u2102<\/mi><\/math> eine sch\u00f6ne, geschlossene Behandlung, die in der Vorlesung \u201eFunktionentheorie\u201c im zweiten Studienjahr thematisiert wird. Der komplexe Blickwinkel erkl\u00e4rt zum Beispiel, warum die analytische Funktion&nbsp;<math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u211d<\/mi> <mo class=\"MathClass-rel\">\u2192<\/mo> <mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mn>1<\/mn><mo class=\"MathClass-bin\">+<\/mo><msup><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><\/mrow><\/mfrac><\/math> bei&nbsp;<math display=\"inline\"><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-rel\">=<\/mo> <mn>0<\/mn><\/math> eine Taylorreihe mit Konvergenzradius&nbsp;<math display=\"inline\"><mn>1<\/mn><\/math> besitzt, obwohl die Funktion auf ganz&nbsp;<math display=\"inline\"><mi>\u211d<\/mi><\/math>                                                                                                                                                                           definiert ist. <\/p> <\/div> <a id=\"x1-277007r277\"><\/a> <h4 id=\"zc3d5550b5f20\" class=\"subsectionHead\"><span class=\"titlemark\">9.4.2 <\/span> <a id=\"x1-2780002\"><\/a>Konvergenzgeschwindigkeit des Newton-Verfahrens*<\/h4> <p class=\"noindent\">Als Anwendung des Satzes von Taylor (Theorem <a href=\"..\/..\/chapter\/taylor-approximation#x1-276002r46\">9.46<\/a>) m\u00f6chten wir in diesem Teilabschnitt das Newton-Verfahren zur approximativen Berechnung einer Nullstelle einer gegebenen Funktion diskutieren. Weitere Anwendung werden in den n\u00e4chsten Abschnitten dieses Kapitels folgen. <\/p> <div class=\"me meexample\"> <p class=\"indent\"><\/p><h4 id=\"z4801a04426fa\"> <a id=\"x1-278001r53\"><\/a> <span class=\"ecbx-1095\">Beispiel 9.53.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Sei <\/span><math display=\"inline\"><mo class=\"MathClass-open\">[<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo> <mi>b<\/mi><mo class=\"MathClass-close\">]<\/mo> <mo class=\"MathClass-rel\">\u2286<\/mo> <mi>\u211d<\/mi><\/math> <span class=\"ecti-1095\">ein kompaktes<\/span> <span class=\"ecti-1095\">Intervall mit Endpunkten <\/span><math display=\"inline\"><mi>a<\/mi> <mo class=\"MathClass-rel\">&lt;<\/mo> <mi>b<\/mi><\/math> <span class=\"ecti-1095\">und <\/span><math display=\"inline\"><mi>f<\/mi> <mo class=\"MathClass-punc\">:<\/mo> <mo class=\"MathClass-open\">[<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo> <mi>b<\/mi><mo class=\"MathClass-close\">]<\/mo> <mo class=\"MathClass-rel\">\u2192<\/mo> <mi>\u211d<\/mi><\/math> <span class=\"ecti-1095\">eine stetig<\/span> <span class=\"ecti-1095\">differenzierbare Funktion mit <\/span><math display=\"inline\"><msup><mrow><mi>f<\/mi><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">&gt;<\/mo> <mn>0<\/mn><\/math> <span class=\"ecti-1095\">f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r alle <\/span><span class=\"maperiod\"><math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mo class=\"MathClass-open\">[<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>b<\/mi><mo class=\"MathClass-close\">]<\/mo><\/math><\/span><span class=\"period\">.<\/span> <span class=\"ecti-1095\">(Wir erinnern daran, dass dies im Allgemeinen nicht dasselbe wie strenge Monotonie von<\/span> <math display=\"inline\"><mi>f<\/mi><\/math> <span class=\"ecti-1095\">ist, aber strenge Monotonie impliziert.) Angenommen es gilt<\/span> <math display=\"inline\"><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>a<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">&lt;<\/mo> <mn>0<\/mn><\/math> <span class=\"ecti-1095\">und<\/span> <span class=\"maperiod\"><math display=\"inline\"><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>b<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">&gt;<\/mo> <mn>0<\/mn><\/math><\/span><span class=\"period\">.<\/span> <span class=\"ecti-1095\">Dann gibt es auf Grund des Zwischenwertsatzes (Satz <\/span><a href=\"..\/..\/chapter\/der-zwischenwertsatz#x1-96001r58\"><span class=\"ecti-1095\">3.58<\/span><\/a><span class=\"ecti-1095\">) ein<\/span> <math display=\"inline\"><mi>z<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mo class=\"MathClass-open\">[<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo> <mi>b<\/mi><mo class=\"MathClass-close\">]<\/mo><\/math> <span class=\"ecti-1095\">mit<\/span> <math display=\"inline\"><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>z<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <mn>0<\/mn><\/math><span class=\"ecti-1095\">, das wegen der<\/span> <span class=\"ecti-1095\">strengen Monotonie von <\/span><math display=\"inline\"><mi>f<\/mi><\/math> <span class=\"ecti-1095\">eindeutig bestimmt ist. Beginnend mit einem gew<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">hlten Punkt<\/span> <math display=\"inline\"><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-rel\">\u2208<\/mo> <mo class=\"MathClass-open\">[<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo> <mi>b<\/mi><mo class=\"MathClass-close\">]<\/mo><\/math> <span class=\"ecti-1095\">definieren wir rekursiv<\/span> <\/p><math display=\"block\"><mtable class=\"align\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><msub><mrow><mi>x<\/mi><\/mrow><mrow><mi>n<\/mi><mo class=\"MathClass-bin\">+<\/mo><mn>1<\/mn><\/mrow><\/msub> <mo class=\"MathClass-rel\">=<\/mo> <msub><mrow><mi>x<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub> <mo class=\"MathClass-bin\">\u2212<\/mo> <mfrac><mrow><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><\/mrow> <mrow><msup><mrow><mi>f<\/mi><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><\/mrow><\/mfrac><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"><mstyle class=\"label\" id=\"x1-278002r10\" \/><mstyle class=\"maketag\"><mtext>(9.10)<\/mtext><\/mstyle><mspace class=\"nbsp\" width=\"0.33em\" \/> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\"><span class=\"ecti-1095\">f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r alle <\/span><math display=\"inline\"><mi>n<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <msub><mrow><mi>\u2115<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><\/math> <span class=\"ecti-1095\">(falls <\/span><math display=\"inline\"><msub><mrow><mi>x<\/mi><\/mrow><mrow><mi>n<\/mi> <\/mrow> <\/msub> <mo class=\"MathClass-rel\">\u2208<\/mo> <mo class=\"MathClass-open\">[<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>b<\/mi><mo class=\"MathClass-close\">]<\/mo><\/math><span class=\"ecti-1095\">).<\/span> <span class=\"ecti-1095\">Die Idee hinter diesem Verfahren m<\/span><span class=\"ecti-1095\">\u00f6<\/span><span class=\"ecti-1095\">chten wir in folgendem Bild erkl<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">ren.<\/span> <\/p> <div class=\"center\"> <p class=\"noindent\"> <\/p><p class=\"noindent\"><\/p><div class=\"mefigcentered\" id=\"wpsize=634&amp;url=Pictures\/fundsatz\/newtonverfahren.pdf\"><img id=\"zf983508b420b\" alt=\"PIC\" src=\"https:\/\/people.math.ethz.ch\/~einsiedl\/Pictures\/fundsatz\/newtonverfahren.svg\" width=\"634\"><\/div> <a id=\"x1-278003r2\"><\/a> <a id=\"x1-278004\"><\/a> <br><div class=\"caption\"><span class=\"id\">&nbsp;&nbsp;&nbsp;&nbsp;              Figur&nbsp;9.2:     <\/span><span class=\"content\">In     einem     ersten     Schritt     approximiert     man               <math display=\"inline\"><mi>f<\/mi><\/math>             mit der Tangenten <math display=\"inline\"><msub><mrow><mi>g<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub> <mo class=\"MathClass-punc\">:<\/mo> <mi>t<\/mi><mo class=\"MathClass-rel\">\u21a6<\/mo><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">+<\/mo> <msup><mrow><mi>f<\/mi><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-open\">(<\/mo><mi>t<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><\/math>               bei <span class=\"maperiod\"><math display=\"inline\"><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><\/math><\/span><span class=\"period\">.<\/span>               Von    dieser    berechnet    man    nun    die    eindeutige    Nullstelle               <span class=\"maperiod\"><math display=\"inline\"><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub> <mo class=\"MathClass-rel\">=<\/mo> <msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub> <mo class=\"MathClass-bin\">\u2212<\/mo> <mfrac><mrow><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><\/mrow> <mrow><msup><mrow><mi>f<\/mi><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><\/mrow><\/mfrac><\/math><\/span><span class=\"period\">.<\/span>               In  einem  zweiten  Schritt  verwendet  man  nun  diesen  neuen  Punkt               <math display=\"inline\"><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><\/math>                   und            berechnet            die            eindeutige            Nullstelle               <math display=\"inline\"><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><\/math>                   der Tangenten <math display=\"inline\"><msub><mrow><mi>g<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><\/math>                   bei <span class=\"maperiod\"><math display=\"inline\"><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><\/math><\/span><span class=\"period\">.<\/span>               Dies f\u00fchrt man iterativ so fort.                                                   &nbsp;&nbsp;&nbsp;&nbsp; <\/span><\/div> <\/div> <p class=\"indent\"><span class=\"ecti-1095\">Falls die Folge <\/span><math display=\"inline\"><msub><mrow><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><\/math> <span class=\"ecti-1095\">definiert ist (das heisst, <\/span><math display=\"inline\"><msub><mrow><mi>x<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub> <mo class=\"MathClass-rel\">\u2208<\/mo> <mo class=\"MathClass-open\">[<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>b<\/mi><mo class=\"MathClass-close\">]<\/mo><\/math> <span class=\"ecti-1095\">f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r alle <\/span><math display=\"inline\"><mi>n<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2115<\/mi><\/math><span class=\"ecti-1095\">)<\/span> <span class=\"ecti-1095\">und konvergiert, dann ist ihr Grenzwert eine Nullstelle von<\/span> <math display=\"inline\"><mi>f<\/mi><\/math> <span class=\"ecti-1095\">und somit unter unseren<\/span> <span class=\"ecti-1095\">Annahmen gleich der Nullstelle <\/span><span class=\"maperiod\"><math display=\"inline\"><mi>z<\/mi><\/math><\/span><span class=\"period\">.<\/span> <span class=\"ecti-1095\">In der Tat gilt f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r <\/span><math display=\"inline\"><msup><mrow><mi>z<\/mi><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup> <mo class=\"MathClass-rel\">=<\/mo><munder class=\"msub\"><mrow><mi class=\"qopname\"> lim<\/mi><mo>  <\/mo><\/mrow><mrow><mi>n<\/mi><mo class=\"MathClass-rel\">\u2192<\/mo><mi>\u221e<\/mi><\/mrow><\/munder><msub><mrow><mi>x<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><\/math> <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><msup><mrow><mi>z<\/mi><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup> <mo class=\"MathClass-rel\">=<\/mo><munder class=\"msub\"><mrow><mi class=\"qopname\"> lim<\/mi><mo>  <\/mo><\/mrow><mrow> <mi>n<\/mi><mo class=\"MathClass-rel\">\u2192<\/mo><mi>\u221e<\/mi><\/mrow><\/munder><msub><mrow><mi>x<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub> <mo class=\"MathClass-rel\">=<\/mo><munder class=\"msub\"><mrow><mi class=\"qopname\"> lim<\/mi><mo>  <\/mo><\/mrow><mrow><mi>n<\/mi><mo class=\"MathClass-rel\">\u2192<\/mo><mi>\u221e<\/mi><\/mrow><\/munder><msub><mrow><mi>x<\/mi><\/mrow><mrow><mi>n<\/mi><mo class=\"MathClass-bin\">+<\/mo><mn>1<\/mn><\/mrow><\/msub> <mo class=\"MathClass-rel\">=<\/mo><munder class=\"msub\"><mrow><mi class=\"qopname\"> lim<\/mi><mo>  <\/mo><\/mrow><mrow><mi>n<\/mi><mo class=\"MathClass-rel\">\u2192<\/mo><mi>\u221e<\/mi><\/mrow><\/munder><mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><msub><mrow><mi>x<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub> <mo class=\"MathClass-bin\">\u2212<\/mo> <mfrac><mrow><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><\/mrow> <mrow><msup><mrow><mi>f<\/mi><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><\/mrow><\/mfrac> <\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-rel\">=<\/mo> <msup><mrow><mi>z<\/mi><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup><mo class=\"MathClass-bin\">\u2212<\/mo> <mfrac><mrow><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><msup><mrow><mi>z<\/mi><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup><mo class=\"MathClass-close\">)<\/mo><\/mrow> <mrow><msup><mrow><mi>f<\/mi><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup><mo class=\"MathClass-open\">(<\/mo><msup><mrow><mi>z<\/mi><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup><mo class=\"MathClass-close\">)<\/mo><\/mrow><\/mfrac><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\"><span class=\"ecti-1095\">und somit <\/span><math display=\"inline\"><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><msup><mrow><mi>z<\/mi><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <mn>0<\/mn><\/math> <span class=\"ecti-1095\">und <\/span><span class=\"maperiod\"><math display=\"inline\"><msup><mrow><mi>z<\/mi><\/mrow><mrow><mo>\u2032<\/mo> <\/mrow> <\/msup> <mo class=\"MathClass-rel\">=<\/mo> <mi>z<\/mi><\/math><\/span><span class=\"period\">.<\/span> <\/p><p class=\"indent\"><span class=\"ecti-1095\">Das Konvergenzverhalten der Folge <\/span><math display=\"inline\"><msub><mrow><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><\/math> <span class=\"ecti-1095\">ist im Allgemeinen<\/span> <span class=\"ecti-1095\">\u201e<\/span><span class=\"ecti-1095\">sehr chaotisch<\/span><span class=\"ecti-1095\">\u201c<\/span><span class=\"ecti-1095\">. Unter etwas st<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">rkeren Annahmen m<\/span><span class=\"ecti-1095\">\u00f6<\/span><span class=\"ecti-1095\">chten wir nun aber zeigen, dass<\/span> <span class=\"ecti-1095\">die Folge <\/span><math display=\"inline\"><msub><mrow><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><\/math> <span class=\"ecti-1095\">konvergiert und die Konvergenzgeschwindigkeit untersuchen.<\/span> <\/p><p class=\"indent\"><span class=\"ecti-1095\">Wir nehmen zus<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">tzlich an, dass <\/span><math display=\"inline\"><mi>f<\/mi><\/math> <span class=\"ecti-1095\">zweimal stetig differenzierbar ist und definieren<\/span> <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mi>M<\/mi><\/mtd> <mtd class=\"align-even\"> <mo class=\"MathClass-rel\">=<\/mo><mi class=\"qopname\"> max<\/mi><mo>  <\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> {<\/mo><mrow><mo class=\"MathClass-rel\">|<\/mo><msup><mrow><mi>f<\/mi><\/mrow><mrow><mi class=\"qopname\">\u2033<\/mi><mo>  <\/mo><\/mrow><\/msup><mo class=\"MathClass-open\">(<\/mo><mi>t<\/mi><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-rel\">|<\/mo><mo class=\"MathClass-rel\">\u2223<\/mo><mi>t<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mo class=\"MathClass-open\">[<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>b<\/mi><mo class=\"MathClass-close\">]<\/mo><\/mrow><mo fence=\"true\" form=\"postfix\">}<\/mo><\/mrow><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\" \/> <mtd class=\"align-label\"> <mspace width=\"2em\" \/><\/mtd><\/mtr><mtr><mtd class=\"align-odd\" columnalign=\"right\"><mi>m<\/mi><\/mtd> <mtd class=\"align-even\"> <mo class=\"MathClass-rel\">=<\/mo><mi class=\"qopname\"> min<\/mi><mo>  <\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> {<\/mo><mrow><mo class=\"MathClass-rel\">|<\/mo><msup><mrow><mi>f<\/mi><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup><mo class=\"MathClass-open\">(<\/mo><mi>t<\/mi><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-rel\">|<\/mo><mo class=\"MathClass-rel\">\u2223<\/mo><mi>t<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mo class=\"MathClass-open\">[<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>b<\/mi><mo class=\"MathClass-close\">]<\/mo><\/mrow><mo fence=\"true\" form=\"postfix\">}<\/mo><\/mrow><mo class=\"MathClass-punc\">.<\/mo><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\" \/> <mtd class=\"align-label\"> <mspace width=\"2em\" \/><\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\"><span class=\"ecti-1095\">Nun wenden wir Korollar <\/span><a href=\"..\/..\/chapter\/taylor-approximation#x1-276003r47\"><span class=\"ecti-1095\">9.47<\/span><\/a> <span class=\"ecti-1095\">an, um den Fehlerterm<\/span> <math display=\"inline\"><msubsup><mrow><mi>R<\/mi><\/mrow><mrow><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-punc\">,<\/mo><mn>1<\/mn> <\/mrow> <mrow> <mi>f<\/mi> <\/mrow> <\/msubsup><\/math> <span class=\"ecti-1095\">bei der Approximation<\/span> <span class=\"ecti-1095\">von <\/span><math display=\"inline\"><mi>f<\/mi><\/math> <span class=\"ecti-1095\">durch die Tangente<\/span> <span class=\"ecti-1095\">um <\/span><math display=\"inline\"><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn> <\/mrow> <\/msub> <\/math> <span class=\"ecti-1095\">abzusch<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">tzen.<\/span> <span class=\"ecti-1095\">Bei der Nullstelle <\/span><math display=\"inline\"><mi>z<\/mi><\/math> <span class=\"ecti-1095\">ergibt dies<\/span> <\/p><math display=\"block\"><mtable class=\"align\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"> <mrow><mo fence=\"true\" form=\"prefix\"> |<\/mo><mrow><msubsup><mrow><mi>R<\/mi><\/mrow><mrow><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><mo class=\"MathClass-punc\">,<\/mo><mn>1<\/mn><\/mrow><mrow><mi>f<\/mi><\/mrow><\/msubsup> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>z<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><\/mrow><mo fence=\"true\" form=\"postfix\">|<\/mo><\/mrow> <mo class=\"MathClass-rel\">\u2264<\/mo> <mfrac><mrow><mi>M<\/mi><mo class=\"MathClass-rel\">|<\/mo><mi>z<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><msup><mrow><mo class=\"MathClass-rel\">|<\/mo><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><\/mrow> <mrow><mn>2<\/mn><\/mrow><\/mfrac> <\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"><mstyle class=\"label\" id=\"x1-278005r11\" \/><mstyle class=\"maketag\"><mtext>(9.11)<\/mtext><\/mstyle><mspace class=\"nbsp\" width=\"0.33em\" \/> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\"><span class=\"ecti-1095\">Des Weiteren gilt per Definition<\/span> <\/p><math display=\"block\"><mtable class=\"align\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mn>0<\/mn> <mo class=\"MathClass-rel\">=<\/mo> <mi>f<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>z<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-rel\">=<\/mo> <mi>f<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-bin\">+<\/mo> <msup><mrow><mi>f<\/mi><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup><mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><msub><mrow><mi>x<\/mi><\/mrow><mrow> <mn>0<\/mn><\/mrow><\/msub><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>z<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-bin\">+<\/mo> <msubsup><mrow><mi>R<\/mi><\/mrow><mrow><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><mo class=\"MathClass-punc\">,<\/mo><mn>1<\/mn><\/mrow><mrow><mi>f<\/mi><\/mrow><\/msubsup> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>z<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><mo class=\"MathClass-punc\">.<\/mo><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"><mstyle class=\"label\" id=\"x1-278006r12\" \/><mstyle class=\"maketag\"><mtext>(9.12)<\/mtext><\/mstyle><mspace class=\"nbsp\" width=\"0.33em\" \/> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\"><span class=\"ecti-1095\">Wir dividieren Gleichung<\/span> (<a href=\"..\/..\/chapter\/taylor-approximation#x1-278006r12\">9.12<\/a>) <span class=\"ecti-1095\">durch <\/span><math display=\"inline\"><msup><mrow><mi>f<\/mi><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><\/math> <span class=\"ecti-1095\">und erhalten<\/span> <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mn>0<\/mn> <mo class=\"MathClass-rel\">=<\/mo> <mi>z<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo><munder><mrow><munder accentunder=\"false\"><mrow><mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub> <mo class=\"MathClass-bin\">\u2212<\/mo> <mfrac><mrow><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><\/mrow> <mrow><msup><mrow><mi>f<\/mi><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><\/mrow><\/mfrac> <\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><\/mrow><mo>\ufe38<\/mo><\/munder><\/mrow><mrow> <mo class=\"MathClass-rel\">=<\/mo> <msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><\/mrow><\/munder> <mo class=\"MathClass-bin\">+<\/mo> <mfrac><mrow><msubsup><mrow><mi>R<\/mi><\/mrow><mrow><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><mo class=\"MathClass-punc\">,<\/mo><mn>1<\/mn><\/mrow><mrow><mi>f<\/mi><\/mrow><\/msubsup><mo class=\"MathClass-open\">(<\/mo><mi>z<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow> <mrow><msup><mrow><mi>f<\/mi><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><\/mrow><\/mfrac> <mo class=\"MathClass-punc\">.<\/mo><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\"><span class=\"ecti-1095\">Mit der Absch<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">tzung<\/span> (<a href=\"..\/..\/chapter\/taylor-approximation#x1-278005r11\">9.11<\/a>) <span class=\"ecti-1095\">ergibt dies<\/span> <\/p><math display=\"block\"><mtable class=\"align\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"> <mrow><mo fence=\"true\" form=\"prefix\"> |<\/mo><mrow><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>z<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">|<\/mo><\/mrow> <mo class=\"MathClass-rel\">=<\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> |<\/mo><mrow><mfrac><mrow><msubsup><mrow><mi>R<\/mi><\/mrow><mrow><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><mo class=\"MathClass-punc\">,<\/mo><mn>1<\/mn><\/mrow><mrow><mi>f<\/mi><\/mrow><\/msubsup><mo class=\"MathClass-open\">(<\/mo><mi>z<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow> <mrow><msup><mrow><mi>f<\/mi><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><\/mrow><\/mfrac> <\/mrow><mo fence=\"true\" form=\"postfix\">|<\/mo><\/mrow> <mo class=\"MathClass-rel\">\u2264<\/mo><mfrac><mrow> <mi>M<\/mi><\/mrow> <mrow><mn>2<\/mn><mi>m<\/mi><\/mrow><\/mfrac><mspace class=\"thinspace\" width=\"0.17em\" \/><msup><mrow> <mrow><mo fence=\"true\" form=\"prefix\"> |<\/mo><mrow><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><mo class=\"MathClass-bin\">\u2212<\/mo><\/mrow><mo fence=\"true\" form=\"postfix\">|<\/mo><\/mrow><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"><mstyle class=\"label\" id=\"x1-278007r13\" \/><mstyle class=\"maketag\"><mtext>(9.13)<\/mtext><\/mstyle><mspace class=\"nbsp\" width=\"0.33em\" \/> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\"><span class=\"ecti-1095\">Falls nun <\/span><math display=\"inline\"><mi>\ud835\udf00<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mo class=\"MathClass-open\">(<\/mo><mn>0<\/mn><mo class=\"MathClass-punc\">,<\/mo> <mfrac><mrow><mi>m<\/mi><\/mrow> <mrow><mi>M<\/mi><\/mrow><\/mfrac><mo class=\"MathClass-close\">)<\/mo><\/math> <span class=\"ecti-1095\">klein<\/span> <span class=\"ecti-1095\">genug ist, so dass <\/span><math display=\"inline\"><msub><mrow><mi>B<\/mi><\/mrow><mrow><mi>\ud835\udf00<\/mi><\/mrow><\/msub><mo class=\"MathClass-open\">(<\/mo><mi>z<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">\u2286<\/mo> <mo class=\"MathClass-open\">[<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>b<\/mi><mo class=\"MathClass-close\">]<\/mo><\/math> <span class=\"ecti-1095\">gilt, und falls <\/span><math display=\"inline\"><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub> <mo class=\"MathClass-rel\">\u2208<\/mo> <msub><mrow><mi>B<\/mi><\/mrow><mrow><mi>\ud835\udf00<\/mi><\/mrow><\/msub><mo class=\"MathClass-open\">(<\/mo><mi>z<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math> <span class=\"ecti-1095\">ist, dann folgt aus<\/span> (<a href=\"..\/..\/chapter\/taylor-approximation#x1-278007r13\">9.13<\/a>)<span class=\"ecti-1095\">, dass<\/span> <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mo class=\"MathClass-rel\">|<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>z<\/mi><mo class=\"MathClass-rel\">|<\/mo><mo class=\"MathClass-rel\">\u2264<\/mo><mfrac><mrow> <mi>M<\/mi><\/mrow> <mrow><mn>2<\/mn><mi>m<\/mi><\/mrow><\/mfrac><mspace class=\"thinspace\" width=\"0.17em\" \/><mo class=\"MathClass-rel\">|<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>z<\/mi><msup><mrow><mo class=\"MathClass-rel\">|<\/mo><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup> <mo class=\"MathClass-rel\">\u2264<\/mo><mfrac><mrow> <mi>M<\/mi><\/mrow> <mrow><mn>2<\/mn><mi>m<\/mi><\/mrow><\/mfrac><mspace class=\"thinspace\" width=\"0.17em\" \/><mo class=\"MathClass-rel\">|<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>z<\/mi><mo class=\"MathClass-rel\">|<\/mo><mi>\ud835\udf00<\/mi> <mo class=\"MathClass-rel\">\u2264<\/mo><mfrac><mrow> <mn>1<\/mn><\/mrow> <mrow><mn>2<\/mn><\/mrow><\/mfrac><mo class=\"MathClass-rel\">|<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>z<\/mi><mo class=\"MathClass-rel\">|<\/mo> <mo class=\"MathClass-rel\">&lt;<\/mo> <mfrac><mrow><mi>\ud835\udf00<\/mi><\/mrow> <mrow><mn>2<\/mn><\/mrow><\/mfrac><mo class=\"MathClass-punc\">.<\/mo><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\"><span class=\"ecti-1095\">Daher liegt <\/span><math display=\"inline\"><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><\/math> <span class=\"ecti-1095\">auch<\/span> <span class=\"ecti-1095\">in <\/span><math display=\"inline\"><mo class=\"MathClass-open\">[<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo> <mi>b<\/mi><mo class=\"MathClass-close\">]<\/mo><\/math> <span class=\"ecti-1095\">und ist bereits<\/span> <span class=\"ecti-1095\">n<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">her an <\/span><math display=\"inline\"><mi>z<\/mi><\/math> <span class=\"ecti-1095\">als<\/span> <math display=\"inline\"><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn> <\/mrow> <\/msub> <\/math><span class=\"ecti-1095\">. Komplett analog beweist<\/span> <span class=\"ecti-1095\">man, dass f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r alle <\/span><math display=\"inline\"><mi>n<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <msub><mrow><mi>\u2115<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><\/math> <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mo class=\"MathClass-rel\">|<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mi>n<\/mi><mo class=\"MathClass-bin\">+<\/mo><mn>1<\/mn><\/mrow><\/msub> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>z<\/mi><mo class=\"MathClass-rel\">|<\/mo><mo class=\"MathClass-rel\">\u2264<\/mo><mfrac><mrow> <mn>1<\/mn><\/mrow> <mrow><mn>2<\/mn><\/mrow><\/mfrac><mspace class=\"thinspace\" width=\"0.17em\" \/><mo class=\"MathClass-rel\">|<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>z<\/mi><mo class=\"MathClass-rel\">|<\/mo><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\"><span class=\"ecti-1095\">gilt, was mit vollst<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">ndiger Induktion auch<\/span> <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mo class=\"MathClass-rel\">|<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>z<\/mi><mo class=\"MathClass-rel\">|<\/mo><mo class=\"MathClass-rel\">\u2264<\/mo> <msup><mrow><mn>2<\/mn><\/mrow><mrow><mo class=\"MathClass-bin\">\u2212<\/mo><mi>n<\/mi><\/mrow><\/msup><mo class=\"MathClass-rel\">|<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow> <mn>0<\/mn><\/mrow><\/msub> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>z<\/mi><mo class=\"MathClass-rel\">|<\/mo><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\"><span class=\"ecti-1095\">f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r alle <\/span><math display=\"inline\"><mi>n<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <msub><mrow><mi>\u2115<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><\/math> <span class=\"ecti-1095\">zeigt. Insbesondere<\/span> <span class=\"ecti-1095\">konvergiert die Folge <\/span><math display=\"inline\"><msub><mrow><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><\/math> <span class=\"ecti-1095\">und wir k<\/span><span class=\"ecti-1095\">\u00f6<\/span><span class=\"ecti-1095\">nnen das Newton-Verfahren (unter geeigneter Wahl eines Startpunktes<\/span> <math display=\"inline\"><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn> <\/mrow> <\/msub> <\/math><span class=\"ecti-1095\">)<\/span> <span class=\"ecti-1095\">verwenden, um die Nullstelle mit hoher Genauigkeit zu approximieren. Die Konvergenz ist noch<\/span> <span class=\"ecti-1095\">schneller als diese Absch<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">tzung beweist; man spricht von quadratischer Konvergenz.<\/span> <\/p> <\/div> <div class=\"me meexample\"> <p class=\"indent\"><\/p><h4 id=\"z56f6c93a40bd\"> <a id=\"x1-278008r54\"><\/a> <span class=\"ecbx-1095\">\u00dc<\/span><span class=\"ecbx-1095\">bung 9.54 <\/span>(Quadratische Konvergenz)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Wir m<\/span><span class=\"ecti-1095\">\u00f6<\/span><span class=\"ecti-1095\">chten die Konvergenzgeschwindigkeit des Newton-Verfahrens hier genauer analysieren. In<\/span> <span class=\"ecti-1095\">obigem Beispiel wurde gezeigt, dass sich der Fehler in jedem Schritt mindestens halbiert. Wir<\/span> <span class=\"ecti-1095\">betrachten nun das Argument etwas genauer (und behalten dementsprechend die Notation). Sei<\/span> <math display=\"inline\"><mi>\u03b2<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mfrac> <mrow> <mi>M<\/mi><\/mrow> <mrow><mn>2<\/mn><mi>m<\/mi><\/mrow><\/mfrac><\/math><span class=\"ecti-1095\">. Zeigen Sie f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r den<\/span> <span class=\"ecti-1095\">mit <\/span><math display=\"inline\"><mi>\u03b2<\/mi><\/math> <span class=\"ecti-1095\">gewichteten<\/span> <span class=\"ecti-1095\">Abstand<\/span><span class=\"ecti-1095\">&nbsp;<\/span><math display=\"inline\"><mi>\u03b2<\/mi><mo class=\"MathClass-rel\">|<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>z<\/mi><mo class=\"MathClass-rel\">|<\/mo><\/math> <span class=\"ecti-1095\">die Absch<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">tzung<\/span> <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mi>\u03b2<\/mi><mo class=\"MathClass-rel\">|<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>z<\/mi><mo class=\"MathClass-rel\">|<\/mo><mo class=\"MathClass-rel\">\u2264<\/mo> <msup><mrow><mo class=\"MathClass-open\">(<\/mo><mi>\u03b2<\/mi><mo class=\"MathClass-rel\">|<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>z<\/mi><mo class=\"MathClass-rel\">|<\/mo><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><msup><mrow><mn>2<\/mn><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msup> <\/mrow><\/msup><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\"><span class=\"ecti-1095\">f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r jedes <\/span><math display=\"inline\"><mi>n<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2115<\/mi><\/math> <span class=\"ecti-1095\">gilt.<\/span> <\/p> <\/div> <div class=\"me meexample\"> <p class=\"indent\"><\/p><h4 id=\"z7bc48d837598\"> <a id=\"x1-278009r55\"><\/a> <span class=\"ecbx-1095\">\u00dc<\/span><span class=\"ecbx-1095\">bung 9.55.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Verwenden                Sie                das                Newton-Verfahren,                um<\/span> <math display=\"inline\"><msqrt><mrow> <mn>2<\/mn><\/mrow><\/msqrt><\/math> <span class=\"ecti-1095\">(oder alternativ eine andere irrationale, reelle algebraische Zahl) auf drei Dezimalstellen genau<\/span> <span class=\"ecti-1095\">zu berechnen. F<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r die genaue, numerische Berechnung des Resultats d<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">rfen Sie einen Rechner<\/span> <span class=\"ecti-1095\">oder ein Computer-Programm benutzen.<\/span> <\/p> <\/div> <a id=\"x1-278010r276\"><\/a> \n","rendered":"\n<style scoped=\"scoped\">.cmr-5{font-size:50%;}\n.cmr-7{font-size:70%;}\n.cmmi-5{font-size:50%;font-style: italic;}\n.cmmi-7{font-size:70%;font-style: italic;}\n.cmmi-10{font-style: italic;}\n.cmsy-5{font-size:50%;}\n.cmsy-7{font-size:70%;}\n.cmbx-10{ font-weight: bold;}\n.cmbsy-10{font-weight: bold;}\n.cmbsy-10{font-weight: bold;}\n.cmbsy-10{font-weight: bold;}\n.cmbsy-7{font-size:70%;font-weight: bold;}\n.cmbsy-7{font-weight: bold;}\n.cmbsy-7{font-weight: bold;}\n.cmbsy-5{font-size:50%;font-weight: bold;}\n.cmbsy-5{font-weight: bold;}\n.cmbsy-5{font-weight: bold;}\n.cmex-7{font-size:70%;}\n.cmex-7x-x-71{font-size:49%;}\n.msam-7{font-size:70%;}\n.msam-5{font-size:50%;}\n.msbm-7{font-size:70%;}\n.msbm-5{font-size:50%;}\n.cmr-17{font-size:170%;}\n.cmr-12{font-size:120%;}\n.cmti-10{ font-style: italic;}\np{margin-top:0;margin-bottom:0}\np.indent{text-indent:0;}\np + p{margin-top:1em;}\np + div, p + pre {margin-top:1em;}\ndiv + p, pre + p {margin-top:1em;}\n@media print {div.crosslinks {visibility:hidden;}}\na img { border-top: 0; 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width:125%;}\ndt {text-align:right; font-weight:bold; clear:left; float:left;}\ndd {width:100%; padding-left:1em; padding-top: 0px; clear:right;}\ndd + dd {float:right; clear:both;}\ndd + dt {clear:both;}\ndt + dt {width: 100%; float: none; padding: 0 70% 0 0;}\ndt + dt + dd {margin-top: -2em;}\ndt + dt + dd + dt {margin-top: 2em;}\n<\/style>\n<style scoped=\"scoped\">\n\/* CSS Analysis-Skript D-Math ETHZ *\/\n\n\/* Uniform Font, also for headers *\/\nh3 {\n\tfont-family: \"Times New Roman\", serif;\n\tmargin-bottom: 35px;\n}\nh4 {\n\tfont-family: \"Times New Roman\", serif;\n}\nh5 {\n\tfont-family: \"Times New Roman\", serif;\n}\n\n\/* Bold font, e.g. for definitions *\/\n.ecbx-1095 {font-weight: 550 ;}\n\n\n\/* Uniform spacing, indent: larger, noindent, enumerate, itemize *\/\np.indent {\n\tmargin: 25px 0px 0px 0px;\n\ttext-indent: 0px; \n}\np.noindent {\n\tmargin: 15px 0px 0px 0px;\n\ttext-indent: 0px; \n}\ndl.enumerate {\n\tmargin: 0px 0px 0px 0px;\n}\ndl.enumerate dt, dl.enumerate dd {\n\tmargin-top: 15px;\n\tmargin-bottom: 0px;\n}\ndiv.custom-itemize {\n\tmargin: 0px 0px 0px 0px;\n}\ndiv.custom-itemize div.item-head {\n\tmargin-top: 15px;\n\tmargin-bottom: 0px;\n\ttext-align: center;\n}\ndiv.custom-itemize div.item-head:first-of-type {\n\tmargin-top: 0px;\n} \ndiv.custom-itemize div.item-content {\n\tmargin-top: 15px;\n\tmargin-bottom: 0px;\n}\n.MJXc-display {\n\tmargin: 15px 0px 0px 0px;\n}\n\n\n\n\/* green metheorem\/melemma CSS class for more\/medium important latex-theorem-environments *\/\n\/* metheorem box+header *\/\ndiv.metheorem {\n    margin-bottom: 40px;\n    margin-top: 40px;\n\tpadding: 0px 15px 15px 15px;\n    border: 1px solid #333;\n    border-color: #4eb79e;\n    background: #c7e4da;\n}\ndiv.metheorem h4 {\n    background: #4eb79e;\n    color: white;\n\tmargin-top: 12px;\n\tmargin-left: -15px;\n\tmargin-right: -15px;\n\tpadding: 0px 15px 0px 15px;\n}\n\/* melemma box+header *\/\ndiv.melemma {\n    margin-bottom: 40px;\n    margin-top: 40px;\n\tpadding: 0px 15px 15px 15px;\n    border: 1px solid #333;\n    border-color: #4eb79e;\n    background: #F2F2F2;\n}\ndiv.melemma h4 {\n    background: #4eb79e;\n    color: white;\n\tmargin-top: 12px;\n\tmargin-left: -15px;\n\tmargin-right: -15px;\n\tpadding: 0px 15px 0px 15px;\n}\n\/* meexample box+header *\/\ndiv.meexample {\n    margin-bottom: 30px;\n    margin-top: 30px;\n\tpadding: 0px 15px 15px 15px;\n\tborder-color: gainsboro;\n\tborder-style: solid;\n\tborder-width: thin;\n}\ndiv.meexample h4 {\n\tfont-size: inherit;\n\tfont-weight: bold;\n    padding: 15px 0px 0px 0px;\n\tmargin-top: 0px;\n\tmargin-bottom: 5px;\n}\ndiv.meexample h4+p.noindent, div.meexample h4+p.indent {\n\tmargin-top: 5px;\n\ttext-indent: 0px;\n}\n\/* padding and margins for stuff inside these boxes, CSS-selector &gt; doesn't work in WP *\/\ndiv.me details {\n\tmargin: 10px 0px 0px 0px;\n}\ndiv.me dd {\n    width: calc(100% - 30px);\n}\t\n\n\n\/* fixing background of pictures *\/\nimg {\n\tbackground: white;\n}\n\n\/* div-container for centered geoapplet *\/\ndiv.geoapplet {\n\tmargin-left: auto;\n\tmargin-right: auto;\n\tmargin-top: 15px;\n\tmax-width: 100%;\n}\ndiv.geoapplet iframe {\n\tborder-style: none;\n\tmax-height: 110vw;\n}\n\n\/* div-container for centered squeezed tables *\/\ndiv.websqueeze {\n\tmargin-left: auto;\n\tmargin-right: auto;\n}\n\n\/* two containers for squeezing text sizes *\/\ndiv.mesmalltext, div.mesmalltext * {\n\tfont-size: 15px;\n}\nspan.metinytext, span.metinytext * {\n\tfont-size: 12px;\n}\n\n\n\/* removing grid lines in equations *\/\n#content table.equation tr td, #content table.equation tr th {\n    border: none;\n}\n#content table.equation {\n    border: none;\n}\n\n\/* hover\/click-solution for short inline explanations and footnotes *\/\n.hover-text {    \/* hidden part *\/\n    display: none;\n}\n.marginpar {     \/* style for footnote as marginpar *\/\n\ttext-decoration: none;\n\tborder: solid;\n\tborder-width: 1pt;\n\tpadding: 3pt;\t\n\twidth: 30%;\n\tbackground: white;\n}\n.hover-trigger { \/* style for hover\/click-trigger text\/symbol *\/\n\tbackground: none;\n\tborder: none;\n\tpadding: 0;\n\toutline: inherit;\t\n\ttext-transform: none;\n\tfont: inherit;\n\tposition: inherit;\n\tvertical-align: baseline;\n    color: #FF7F00;\n\tcursor: help;\n}\n.hover-trigger:hover +.hover-text{\n    display: inline;\n}\n.hover-trigger:active +.hover-text{\n    display: inline;\n}\n\n\/* simplifying style of details\/summary, removing triangle *\/\ndetails summary {\n  background: none;\n  list-style: none;\n  outline: none;\n  cursor: pointer;\n}\ndetails summary::-webkit-details-marker { \n  display: inline;\n  display: none;\n}\n\n\/* MC-True\/False as inline details\/summary *\/\ndetails.mcquest, div.me details.mcquest {\n\tdisplay: inline;\n\tmargin-top: 0px;\n}\nsummary.mcquest {\n\tdisplay: inline;\n\tcolor: #FF7F00;\n\tcursor: help;\n}\n\n\/* proof style: simple black box with gray background \n                little black square at the end on the right *\/\ndiv.proof {\n\tborder-color: black;\n\tborder-style: solid;\n\tborder-width: thin;\n\tbackground-color: #F2F2F2;\n\tpadding: 15px;\n\tmargin-top: 1em; \n}\ndiv.proof p:first-of-type {\n\tmargin: 0px;\n}\ndiv.qed {\n\tmargin-top: -25px;\n\tmargin-bottom: -7px;\n\ttext-align: right;\n}\ntable.equation+div.qed {\n\tmargin-top: -65px;\n}\n\n\/* The following is making also math-formulas inside the headers of Lemmas, etc., white. *\/\ndiv.melemma h4 span {\n    color: white;\n}\ndiv.metheorem h4 span {\n    color: white;\n}\n\n\/* The following are used to avoid fullstop, period, colon, semicolon, and endquote (broader) to move by itself to the next line after a formula.\n   The math-environment before needs to be wrapped in span.maperiod and the fullstop etc. in a span.period --- together they achieve what we want.  *\/\nspan.maperiod {\n       margin-right: 5px;\n}\nspan.period {\n       display: inline-block;\n       width: 0px;\n       margin-left: -5px;\n       margin-right: 4.9px;\n\t   text-indent: 0px;\n}\nspan.maendquote {\n       margin-right: 8px;\n}\nspan.endquote {\n       display: inline-block;\n       width: 0px;\n       margin-left: -8px;\n       margin-right: 7.9px;\n}\n\n\n\/* The following is removing an extra space left of the equation side in aligned equations *\/\nspan.mjx-mtd {\n    padding-left: 0em !important;\n}\n\n\/* The following fixes the weird problem that math appears smaller if it was rendered while the details tag was closed. *\/\ndetails span.mjx-chtml, details span.MathJax_CHTML {\n font-size: 100% !important;\n}\n\n\/* trying to fix line breaks in verbatim, new lines are missing *\/\npre.verbatim {\n\twhite-space: pre-wrap;\n\tfont-size: small;\n}\n<\/style><h3 id=\"z26169359a53f\" class=\"sectionHead\"><span class=\"titlemark\">9.4 <\/span> <a id=\"x1-2760004\"><\/a>Taylor Approximation<\/h3> <p class=\"noindent\">Wir erinnern daran, dass die Ableitung&nbsp;<math display=\"inline\"><msup><mrow><mi>f<\/mi><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><\/math> einer reellwertigen differenzierbaren Funktion&nbsp;<math display=\"inline\"><mi>f<\/mi><\/math> auf einem Intervall die Steigung der Tangente <\/p><table id=\"z9f621b391178\" class=\"equation-star\"><tr><td> <math class=\"equation\" display=\"block\"> <mi>y<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">+<\/mo> <msup><mrow><mi>f<\/mi><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow> <mn>0<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo> <\/math><\/td><\/tr><\/table> <p class=\"indent\">des Graphen von&nbsp;<math display=\"inline\"><mi>f<\/mi><\/math> bei&nbsp;<math display=\"inline\"><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn> <\/mrow> <\/msub> <\/math> angibt, wobei die Tangente den Graphen gut approximiert (im Sinne von&nbsp;<math display=\"inline\"><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <mi>y<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">+<\/mo> <mi>o<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><\/math> f\u00fcr&nbsp;<math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2192<\/mo> <msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><\/math>). Wir wollen hier die G\u00fcte der Approximation erh\u00f6hen indem wir statt linearen Approximationen auch noch h\u00f6here polynomiale Approximationen erlauben. <\/p><p class=\"indent\">Sei also <math display=\"inline\"><mi>f<\/mi> <mo class=\"MathClass-punc\">:<\/mo> <mo class=\"MathClass-open\">(<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>b<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">\u2192<\/mo> <mi>\u2102<\/mi><\/math> eine <math display=\"inline\"><mi>n<\/mi><\/math>-mal differenzierbare Funktion auf einem offenen, nicht-leeren (m\u00f6glicherweise unbeschr\u00e4nkten) Intervall <span class=\"maperiod\"><math display=\"inline\"><mo class=\"MathClass-open\">(<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo> <mi>b<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">\u2286<\/mo> <mi>\u211d<\/mi><\/math><\/span><span class=\"period\">.<\/span> Die <math display=\"inline\"><mstyle><mi>n<\/mi><\/mstyle><\/math><span class=\"ecbx-1095\">-te Taylor-Approximation<\/span> von <math display=\"inline\"><mi>f<\/mi><\/math> um einen Punkt <math display=\"inline\"><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub> <mo class=\"MathClass-rel\">\u2208<\/mo> <mo class=\"MathClass-open\">(<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>b<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math> ist das Polynom                                                                                                                                                                           <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><msubsup><mrow><mi>P<\/mi><\/mrow><mrow><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><mo class=\"MathClass-punc\">,<\/mo><mi>n<\/mi><\/mrow><mrow><mi>f<\/mi><\/mrow><\/msubsup> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-rel\">=<\/mo><munderover accent=\"false\" accentunder=\"false\"><mrow><mo> \u2211<\/mo> <\/mrow><mrow><mi>k<\/mi><mo class=\"MathClass-rel\">=<\/mo><mn>0<\/mn><\/mrow><mrow><mi>n<\/mi><\/mrow><\/munderover><mfrac><mrow><msup><mrow><mi>f<\/mi><\/mrow><mrow><mo class=\"MathClass-open\">(<\/mo><mi>k<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow><\/msup><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><\/mrow> <mrow><mi>k<\/mi><mo class=\"MathClass-punc\">!<\/mo><\/mrow><\/mfrac> <msup><mrow><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mi>k<\/mi><\/mrow><\/msup><mo class=\"MathClass-punc\">.<\/mo><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">Die Koeffizienten wurden dabei so gew\u00e4hlt, dass <math display=\"inline\"><msubsup><mrow><mi>P<\/mi><\/mrow><mrow><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-punc\">,<\/mo><mi>n<\/mi> <\/mrow> <mrow> <mi>f<\/mi> <\/mrow> <\/msubsup> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn> <\/mrow> <\/msub> <\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-rel\">=<\/mo> <mi>f<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><\/math> und allgemeiner <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><msup><mrow><mo class=\"MathClass-open\">(<\/mo><msubsup><mrow><mi>P<\/mi><\/mrow><mrow><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><mo class=\"MathClass-punc\">,<\/mo><mi>n<\/mi><\/mrow><mrow><mi>f<\/mi><\/mrow><\/msubsup><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mo class=\"MathClass-open\">(<\/mo><mi>k<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow><\/msup> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><msub><mrow><mi>x<\/mi><\/mrow><mrow> <mn>0<\/mn><\/mrow><\/msub><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-rel\">=<\/mo> <msup><mrow><mi>f<\/mi><\/mrow><mrow><mo class=\"MathClass-open\">(<\/mo><mi>k<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow><\/msup> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><msub><mrow><mi>x<\/mi><\/mrow><mrow> <mn>0<\/mn><\/mrow><\/msub><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">f\u00fcr <math display=\"inline\"><mi>k<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> {<\/mo><mrow><mn>0<\/mn><mo class=\"MathClass-punc\">,<\/mo> <mi class=\"MathClass-op\">\u2026<\/mi><mo> <\/mo><mo class=\"MathClass-punc\">,<\/mo><mi>n<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">}<\/mo><\/mrow><\/math> gilt (wieso?). Die vielleicht naive Hoffnung ist hierbei, dass die Taylor-Approximation, wie der Name sagt, die Funktion <math display=\"inline\"><mi>f<\/mi><\/math> approximiert. Falls <math display=\"inline\"><mi>f<\/mi><\/math> glatt ist, dann ist die <span class=\"ecbx-1095\">Taylorreihe <\/span>von <math display=\"inline\"><mi>f<\/mi><\/math> um <math display=\"inline\"><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-rel\">\u2208<\/mo> <mo class=\"MathClass-open\">(<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>b<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math> definiert als die Potenzreihe <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><msubsup><mrow><mi>T<\/mi><\/mrow><mrow><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><\/mrow><mrow><mi>f<\/mi><\/mrow><\/msubsup> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-rel\">=<\/mo><munderover accent=\"false\" accentunder=\"false\"><mrow><mo> \u2211<\/mo> <\/mrow><mrow><mi>k<\/mi><mo class=\"MathClass-rel\">=<\/mo><mn>0<\/mn><\/mrow><mrow><mi>\u221e<\/mi><\/mrow><\/munderover><mfrac><mrow><msup><mrow><mi>f<\/mi><\/mrow><mrow><mo class=\"MathClass-open\">(<\/mo><mi>k<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow><\/msup><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><\/mrow> <mrow><mi>k<\/mi><mo class=\"MathClass-punc\">!<\/mo><\/mrow><\/mfrac> <msup><mrow><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mi>k<\/mi><\/mrow><\/msup><mo class=\"MathClass-punc\">.<\/mo><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">Damit wir hier von einer Potenzreihe sprechen d\u00fcrfen, erweitern wir die Definition <a href=\"..\/..\/chapter\/potenzreihen#x1-199001r54\">7.54<\/a> wie folgt. Eine <span class=\"ecbx-1095\">Potenzreihe um einen Punkt<\/span> <math display=\"inline\"><msub><mrow><mi>z<\/mi><\/mrow><mrow><mn>0<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2102<\/mi><\/math> in der Variable <math display=\"inline\"><mi>z<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2102<\/mi><\/math> ist ein formaler Ausdruck der Form <span class=\"maperiod\"><math display=\"inline\"><msubsup><mrow><mi class=\"MathClass-op\"> \u2211<\/mi><mo> <\/mo> <\/mrow><mrow><mi>n<\/mi><mo class=\"MathClass-rel\">=<\/mo><mn>0<\/mn><\/mrow><mrow><mi>\u221e<\/mi><\/mrow><\/msubsup><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><msup><mrow><mo class=\"MathClass-open\">(<\/mo><mi>z<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <msub><mrow><mi>z<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msup><\/math><\/span><span class=\"period\">,<\/span> wobei <math display=\"inline\"><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi> <\/mrow> <\/msub> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2102<\/mi><\/math> f\u00fcr alle <math display=\"inline\"><mi>n<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2115<\/mi><\/math> die Koeffizienten der Potenzreihe sind. Der Konvergenzradius <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mi>R<\/mi> <mo class=\"MathClass-rel\">=<\/mo><msup><mrow> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><munder class=\"msub\"><mrow><mi class=\"qopname\">limsup<\/mi><mo>  <\/mo><\/mrow><mrow><mi>n<\/mi><mo class=\"MathClass-rel\">\u2192<\/mo><mi>\u221e<\/mi><\/mrow><\/munder><mroot><mrow><mo class=\"MathClass-rel\">|<\/mo><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><mo class=\"MathClass-rel\">|<\/mo><\/mrow><mrow><mi>n<\/mi><\/mrow><\/mroot><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><\/mrow><mrow><mo class=\"MathClass-bin\">\u2212<\/mo><mn>1<\/mn><\/mrow><\/msup><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">ist wie in Abschnitt <a href=\"..\/..\/chapter\/potenzreihen#x1-2000001\">7.4.1<\/a> definiert und die Potenzreihe <math display=\"inline\"><msubsup><mrow><mi class=\"MathClass-op\">\u2211<\/mi><mo> <\/mo> <\/mrow><mrow><mi>n<\/mi><mo class=\"MathClass-rel\">=<\/mo><mn>0<\/mn><\/mrow><mrow><mi>\u221e<\/mi><\/mrow><\/msubsup><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><msup><mrow><mo class=\"MathClass-open\">(<\/mo><mi>z<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <msub><mrow><mi>z<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msup><\/math> konvergiert f\u00fcr alle <math display=\"inline\"><mi>z<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2102<\/mi><\/math> mit Abstand kleiner <math display=\"inline\"><mi>R<\/mi><\/math> von <math display=\"inline\"><msub><mrow><mi>z<\/mi><\/mrow><mrow><mn>0<\/mn> <\/mrow> <\/msub> <\/math> und divergiert f\u00fcr alle <math display=\"inline\"><mi>z<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2102<\/mi><\/math> mit <math display=\"inline\"><mo class=\"MathClass-rel\">|<\/mo><mi>z<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <msub><mrow><mi>z<\/mi><\/mrow><mrow><mn>0<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-rel\">|<\/mo> <mo class=\"MathClass-rel\">&gt;<\/mo> <mi>R<\/mi><\/math> (was aus Satz <a href=\"..\/..\/chapter\/potenzreihen#x1-200002r56\">7.56<\/a> folgt, wie?). <\/p><p class=\"indent\">Soweit ist nicht klar, was der Konvergenzradius der Taylor-Reihe einer glatten Funktion <math display=\"inline\"><mi>f<\/mi><\/math> um einen Punkt <math display=\"inline\"><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn> <\/mrow> <\/msub> <\/math> im Definitionsbereich ist und ob er positiv ist. Die naive Hoffnung ist, dass die Taylor-Reihe, wo definiert, gleich der Funktion <math display=\"inline\"><mi>f<\/mi><\/math> ist, da die Taylorreihe bei <math display=\"inline\"><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn> <\/mrow> <\/msub> <\/math> die gleichen Ableitungen wie <math display=\"inline\"><mi>f<\/mi><\/math> hat. Dies ist in der Tat f\u00fcr viele Funktionen der Fall, doch, wie folgendes Beispiel zeigt, nicht immer. <\/p> <div class=\"me meexample\"> <div class=\"wp-nocaption \"><\/div><h4 id=\"zb9b320cd2ad8\"> <a id=\"x1-276001r45\"><\/a> <span class=\"ecbx-1095\">Beispiel 9.45 <\/span>(Verschwindende Taylorreihe)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Wir betrachten die Funktion<\/span> <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mi>\u03c8<\/mi> <mo class=\"MathClass-punc\">:<\/mo> <mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u211d<\/mi><mo class=\"MathClass-rel\">\u21a6<\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> {<\/mo><mrow> <mtable align=\"axis\" class=\"array\" columnlines=\"none\" equalcolumns=\"false\" equalrows=\"false\"> <mtr><mtd class=\"array\" columnalign=\"center\"><mi class=\"qopname\">exp<\/mi><mo>  <\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mo class=\"MathClass-bin\">\u2212<\/mo><mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mi>x<\/mi><\/mrow><\/mfrac> <\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><\/mtd><mtd class=\"array\" columnalign=\"center\"><mstyle class=\"text\"><mtext>falls&nbsp;<\/mtext><\/mstyle><mi>x<\/mi> <mo class=\"MathClass-rel\">&gt;<\/mo> <mn>0<\/mn><\/mtd><\/mtr> <mtr><mtd class=\"array\" columnalign=\"center\"> <mn>0<\/mn> <\/mtd> <mtd class=\"array\" columnalign=\"center\"><mstyle class=\"text\"><mtext>falls&nbsp;<\/mtext><\/mstyle> <mi>x<\/mi> <mo class=\"MathClass-rel\">\u2264<\/mo> <mn>0<\/mn><\/mtd> <\/mtr> <\/mtable> <\/mrow><mo fence=\"true\" form=\"postfix\" \/><\/mrow><mo class=\"MathClass-punc\">,<\/mo><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\"><span class=\"ecti-1095\">die nach Beispiel <\/span><a href=\"..\/..\/chapter\/die-ableitung#x1-231004r23\"><span class=\"ecti-1095\">8.23<\/span><\/a> <span class=\"ecti-1095\">glatt ist und <\/span><math display=\"inline\"><msup><mrow><mi>\u03c8<\/mi><\/mrow><mrow><mo class=\"MathClass-open\">(<\/mo><mi>n<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow><\/msup><mo class=\"MathClass-open\">(<\/mo><mn>0<\/mn><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <mn>0<\/mn><\/math> <span class=\"ecti-1095\">f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r alle <\/span><math display=\"inline\"><mi>n<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <msub><mrow><mi>\u2115<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><\/math> <span class=\"ecti-1095\">erf<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">llt. Wir<\/span> <span class=\"ecti-1095\">zeigen, dass sich <\/span><math display=\"inline\"><mi>\u03c8<\/mi><\/math> <span class=\"ecti-1095\">nicht<\/span> <span class=\"ecti-1095\">durch eine Potenzreihe um <\/span><math display=\"inline\"><mn>0<\/mn><\/math> <span class=\"ecti-1095\">darstellen l<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">sst.<\/span> <\/p><p class=\"indent\"><span class=\"ecti-1095\">Wir nehmen also indirekt an, dass es eine Potenzreihe<\/span> <math display=\"inline\"><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">=<\/mo><msubsup><mrow><mi class=\"MathClass-op\"> \u2211<\/mi><mo> <\/mo> <\/mrow><mrow><mi>n<\/mi><mo class=\"MathClass-rel\">=<\/mo><mn>0<\/mn><\/mrow><mrow><mi>\u221e<\/mi><\/mrow><\/msubsup><msub><mrow><mi>c<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><msup><mrow><mi>x<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msup><\/math> <span class=\"ecti-1095\">mit reellen Koeffizienten<\/span> <span class=\"ecti-1095\">und Konvergenzradius <\/span><math display=\"inline\"><mi>R<\/mi> <mo class=\"MathClass-rel\">&gt;<\/mo> <mn>0<\/mn><\/math> <span class=\"ecti-1095\">gibt, so dass <\/span><math display=\"inline\"><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <mi>\u03c8<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math> <span class=\"ecti-1095\">f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r alle<\/span> <math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mo class=\"MathClass-open\">(<\/mo><mo class=\"MathClass-bin\">\u2212<\/mo><mi>R<\/mi><mo class=\"MathClass-punc\">,<\/mo> <mi>R<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math><span class=\"ecti-1095\">. Nach Korollar <\/span><a href=\"..\/..\/chapter\/der-fundamentalsatz-der-integral--und-differentialrechnung#x1-260001r11\"><span class=\"ecti-1095\">9.11<\/span><\/a> <span class=\"ecti-1095\">(oder<\/span> <span class=\"ecti-1095\">genauer <\/span><span class=\"ecti-1095\">\u00dc<\/span><span class=\"ecti-1095\">bung <\/span><a href=\"..\/..\/chapter\/der-fundamentalsatz-der-integral--und-differentialrechnung#x1-260002r12\"><span class=\"ecti-1095\">9.12<\/span><\/a><span class=\"ecti-1095\">) ist <\/span><span class=\"maperiod\"><math display=\"inline\"><msup><mrow><mi>f<\/mi><\/mrow><mrow><mo class=\"MathClass-open\">(<\/mo><mi>n<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow><\/msup><mo class=\"MathClass-open\">(<\/mo><mn>0<\/mn><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <msub><mrow><mi>c<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><mi>n<\/mi><mo class=\"MathClass-punc\">!<\/mo><\/math><\/span><span class=\"period\">.<\/span> <span class=\"ecti-1095\">Aber da <\/span><math display=\"inline\"><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <mi>\u03c8<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math> <span class=\"ecti-1095\">f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r alle <\/span><math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mo class=\"MathClass-open\">(<\/mo><mo class=\"MathClass-bin\">\u2212<\/mo><mi>R<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>R<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math> <span class=\"ecti-1095\">angenommen wurde und da Ableiten eine lokale Operation ist, schliessen wir, dass<\/span> <math display=\"inline\"><msup><mrow><mi>f<\/mi><\/mrow><mrow><mo class=\"MathClass-open\">(<\/mo><mi>n<\/mi><mo class=\"MathClass-close\">)<\/mo> <\/mrow> <\/msup> <mo class=\"MathClass-open\">(<\/mo><mn>0<\/mn><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <mn>0<\/mn><\/math> <span class=\"ecti-1095\">und somit<\/span> <math display=\"inline\"><msub><mrow><mi>c<\/mi><\/mrow><mrow><mi>n<\/mi> <\/mrow> <\/msub> <mo class=\"MathClass-rel\">=<\/mo> <mn>0<\/mn><\/math> <span class=\"ecti-1095\">f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r alle<\/span> <math display=\"inline\"><mi>n<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <msub><mrow><mi>\u2115<\/mi><\/mrow><mrow><mn>0<\/mn> <\/mrow> <\/msub> <\/math><span class=\"ecti-1095\">. Dies widerspricht<\/span> <span class=\"ecti-1095\">jedoch <\/span><math display=\"inline\"><mi>\u03c8<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">&gt;<\/mo> <mn>0<\/mn><\/math> <span class=\"ecti-1095\">f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r<\/span> <span class=\"ecti-1095\">alle <\/span><math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">&gt;<\/mo> <mn>0<\/mn><\/math><span class=\"ecti-1095\">. Also<\/span> <span class=\"ecti-1095\">kann <\/span><math display=\"inline\"><mi>\u03c8<\/mi><\/math> <span class=\"ecti-1095\">in keiner<\/span> <span class=\"ecti-1095\">Umgebung von <\/span><math display=\"inline\"><mn>0<\/mn><\/math> <span class=\"ecti-1095\">durch eine Potenzreihe dargestellt werden.<\/span> <\/p> <\/div> <p class=\"indent\">Folgender Satz liefert nun einen direkten Vergleich zwischen einer Funktion und ihrer Taylor-Approximationen. Er impliziert f\u00fcr viele Funktionen, dass sie mit ihrer Taylorreihe \u00fcbereinstimmen. <\/p> <div class=\"me metheorem\"> <div class=\"wp-nocaption \"><\/div><h4 id=\"z2d3e739c4657\"> <a id=\"x1-276002r46\"><\/a> <span class=\"ecbx-1095\">Theorem 9.46 <\/span>(Taylor-Approximation)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Sei <\/span><math display=\"inline\"><mo class=\"MathClass-open\">(<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo> <mi>b<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">\u2286<\/mo> <mi>\u211d<\/mi><\/math> <span class=\"ecti-1095\">ein offenes, nicht-leeres<\/span> <span class=\"ecti-1095\">Intervall und sei <\/span><math display=\"inline\"><mi>f<\/mi> <mo class=\"MathClass-punc\">:<\/mo> <mo class=\"MathClass-open\">(<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>b<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">\u2192<\/mo> <mi>\u2102<\/mi><\/math> <span class=\"ecti-1095\">eine<\/span> <math display=\"inline\"><mo class=\"MathClass-open\">(<\/mo><mi>n<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mn>1<\/mn><mo class=\"MathClass-close\">)<\/mo><\/math><span class=\"ecti-1095\">-mal stetig differenzierbare<\/span> <span class=\"ecti-1095\">Funktion. Sei <\/span><span class=\"maperiod\"><math display=\"inline\"><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub> <mo class=\"MathClass-rel\">\u2208<\/mo> <mo class=\"MathClass-open\">(<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>b<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math><\/span><span class=\"period\">.<\/span> <span class=\"ecti-1095\">Dann gilt f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r alle <\/span><math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mo class=\"MathClass-open\">(<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>b<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math> <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mi>f<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-rel\">=<\/mo> <msubsup><mrow><mi>P<\/mi><\/mrow><mrow><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><mo class=\"MathClass-punc\">,<\/mo><mi>n<\/mi><\/mrow><mrow><mi>f<\/mi><\/mrow><\/msubsup> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-bin\">+<\/mo> <msubsup><mrow><mi>R<\/mi><\/mrow><mrow><msub><mrow> <mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><mo class=\"MathClass-punc\">,<\/mo><mi>n<\/mi><\/mrow><mrow><mi>f<\/mi><\/mrow><\/msubsup> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><mo class=\"MathClass-punc\">,<\/mo><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\"><span class=\"ecti-1095\">wobei <\/span><math display=\"inline\"><msubsup><mrow><mi>P<\/mi><\/mrow><mrow><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-punc\">,<\/mo><mi>n<\/mi> <\/mrow> <mrow> <mi>f<\/mi><\/mrow><\/msubsup><\/math> <span class=\"ecti-1095\">die<\/span> <math display=\"inline\"><mi>n<\/mi><\/math><span class=\"ecti-1095\">-te Taylor-Approximation<\/span> <span class=\"ecti-1095\">ist und wir den Fehlerterm <\/span><math display=\"inline\"><msubsup><mrow><mi>R<\/mi><\/mrow><mrow><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><mo class=\"MathClass-punc\">,<\/mo><mi>n<\/mi><\/mrow><mrow><mi>f<\/mi><\/mrow><\/msubsup><\/math> <span class=\"ecti-1095\">durch das sogenannte <\/span><span class=\"ecbi-1095\">Integral-Restglied<\/span> <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>b<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><mo class=\"MathClass-rel\">\u21a6<\/mo><msubsup><mrow><mi>R<\/mi><\/mrow><mrow><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><mo class=\"MathClass-punc\">,<\/mo><mi>n<\/mi><\/mrow><mrow><mi>f<\/mi><\/mrow><\/msubsup> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-rel\">=<\/mo><msubsup><mrow><mo> \u222b  <\/mo><\/mrow><mrow><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><\/mrow><mrow><mi>x<\/mi><\/mrow><\/msubsup><msup><mrow><mi>f<\/mi><\/mrow><mrow><mo class=\"MathClass-open\">(<\/mo><mi>n<\/mi><mo class=\"MathClass-bin\">+<\/mo><mn>1<\/mn><mo class=\"MathClass-close\">)<\/mo><\/mrow><\/msup> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>t<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mfrac><mrow><msup><mrow><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>t<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msup><\/mrow> <mrow><mi>n<\/mi><mo class=\"MathClass-punc\">!<\/mo><\/mrow><\/mfrac> <mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>t<\/mi><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\"><span class=\"ecti-1095\">darstellen k<\/span><span class=\"ecti-1095\">\u00f6<\/span><span class=\"ecti-1095\">nnen. Dies gilt auch f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r Funktionen auf<\/span> <math display=\"inline\"><mo class=\"MathClass-open\">[<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-punc\">,<\/mo> <mi>b<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math> <span class=\"ecti-1095\">und<\/span> <span class=\"ecti-1095\">Punkte <\/span><math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mo class=\"MathClass-open\">[<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><mo class=\"MathClass-punc\">,<\/mo><mi>b<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math> <span class=\"ecti-1095\">(beziehungsweise <\/span><math display=\"inline\"><mo class=\"MathClass-open\">(<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">]<\/mo><\/math> <span class=\"ecti-1095\">und <\/span><math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mo class=\"MathClass-open\">(<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo> <msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">]<\/mo><\/math><span class=\"ecti-1095\">).<\/span> <\/p> <\/div> <p class=\"indent\">Die Annahme im Theorem, dass <math display=\"inline\"><mi>f<\/mi><\/math> eine <math display=\"inline\"><mo class=\"MathClass-open\">(<\/mo><mi>n<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mn>1<\/mn><mo class=\"MathClass-close\">)<\/mo><\/math>-mal stetig differenzierbare Funktion ist, ist essentiell f\u00fcr obige Formulierung. In der Tat ist damit <math display=\"inline\"><msup><mrow><mi>f<\/mi><\/mrow><mrow><mo class=\"MathClass-open\">(<\/mo><mi>n<\/mi><mo class=\"MathClass-bin\">+<\/mo><mn>1<\/mn><mo class=\"MathClass-close\">)<\/mo> <\/mrow> <\/msup> <\/math> eine stetige Funktion und das Integral der stetigen Funktion <math display=\"inline\"><mi>t<\/mi><mo class=\"MathClass-rel\">\u21a6<\/mo> <msup><mrow><mi>f<\/mi><\/mrow><mrow><mo class=\"MathClass-open\">(<\/mo><mi>n<\/mi><mo class=\"MathClass-bin\">+<\/mo><mn>1<\/mn><mo class=\"MathClass-close\">)<\/mo> <\/mrow> <\/msup> <mo class=\"MathClass-open\">(<\/mo><mi>t<\/mi><mo class=\"MathClass-close\">)<\/mo><mfrac><mrow><msup><mrow><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-bin\">\u2212<\/mo><mi>t<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msup><\/mrow> <mrow><mi>n<\/mi><mo class=\"MathClass-punc\">!<\/mo><\/mrow><\/mfrac> <\/math> im Integral-Restglied existiert. <\/p><div class=\"wp-nocaption \"><\/div> <div class=\"proof\"> <p class=\"indent\"><span class=\"head\"><\/span><\/p><details open=\"open\"><summary><b>Beweis von Satz <a href=\"..\/..\/chapter\/taylor-approximation#x1-276002r46\">9.46<\/a>.<\/b><\/summary><p class=\"indent\" style=\"margin-top: 10\"> Das Theorem ergibt sich mit Induktion \u00fcber <math display=\"inline\"><mi>n<\/mi><\/math> und partieller Integration (die ebenso f\u00fcr komplexwertige Funktionen auf <math display=\"inline\"><mo class=\"MathClass-open\">(<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo> <mi>b<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math> gilt, wieso?). Ist <math display=\"inline\"><mi>n<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mn>0<\/mn><\/math> und <math display=\"inline\"><mi>f<\/mi> <mo class=\"MathClass-punc\">:<\/mo> <mo class=\"MathClass-open\">(<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo> <mi>b<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">\u2192<\/mo> <mi>\u211d<\/mi><\/math> stetig differenzierbar, so gilt nach dem Fundamentalsatz der Differential- und Integralrechnung (genauer Korollar <a href=\"..\/..\/chapter\/der-fundamentalsatz-der-integral--und-differentialrechnung#x1-259008r5\">9.5<\/a>) <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mi>f<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-rel\">=<\/mo> <mi>f<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-bin\">+<\/mo><msubsup><mrow><mo> \u222b  <\/mo><\/mrow><mrow><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><\/mrow><mrow><mi>x<\/mi><\/mrow><\/msubsup><msup><mrow><mi>f<\/mi><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup><mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>t<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>t<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <msubsup><mrow><mi>P<\/mi><\/mrow><mrow><msub><mrow> <mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><mo class=\"MathClass-punc\">,<\/mo><mn>0<\/mn><\/mrow><mrow><mi>f<\/mi><\/mrow><\/msubsup> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-bin\">+<\/mo> <msubsup><mrow><mi>R<\/mi><\/mrow><mrow><msub><mrow> <mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><mo class=\"MathClass-punc\">,<\/mo><mn>0<\/mn><\/mrow><mrow><mi>f<\/mi><\/mrow><\/msubsup> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><mo class=\"MathClass-punc\">.<\/mo><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">Ist <math display=\"inline\"><mi>f<\/mi><\/math> nun zweimal stetig differenzierbar (also <math display=\"inline\"><mi>n<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mn>1<\/mn><\/math>), so k\u00f6nnen wir auf obiges Integral partielle Integration mit <math display=\"inline\"><mi>u<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>t<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <msup><mrow><mi>f<\/mi><\/mrow><mrow><mo>\u2032<\/mo> <\/mrow> <\/msup> <mo class=\"MathClass-open\">(<\/mo><mi>t<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math> und <math display=\"inline\"><mi>v<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>t<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <mi>t<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>x<\/mi><\/math> anwenden und erhalten <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mtd> <mtd class=\"align-even\"> <mo class=\"MathClass-rel\">=<\/mo> <mi>f<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-bin\">+<\/mo><msubsup><mrow> <mrow><mo fence=\"true\" form=\"prefix\"> [<\/mo><mrow><msup><mrow><mi>f<\/mi><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup><mo class=\"MathClass-open\">(<\/mo><mi>t<\/mi><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-open\">(<\/mo><mi>t<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow><mo fence=\"true\" form=\"postfix\">]<\/mo><\/mrow><\/mrow><mrow> <mi>t<\/mi><mo class=\"MathClass-rel\">=<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><\/mrow><mrow><mi>t<\/mi><mo class=\"MathClass-rel\">=<\/mo><mi>x<\/mi><\/mrow><\/msubsup> <mo class=\"MathClass-bin\">\u2212<\/mo><msubsup><mrow><mo>\u222b  <\/mo><\/mrow><mrow><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><\/mrow><mrow><mi>x<\/mi><\/mrow><\/msubsup><msup><mrow><mi>f<\/mi><\/mrow><mrow><mo>\u2033<\/mo><\/mrow><\/msup> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>t<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>t<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>t<\/mi><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\" \/> <mtd class=\"align-label\"> <mspace width=\"2em\" \/><\/mtd><\/mtr><mtr><mtd class=\"align-odd\" columnalign=\"right\" \/> <mtd class=\"align-even\"> <mo class=\"MathClass-rel\">=<\/mo> <mi>f<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-bin\">+<\/mo> <msup><mrow><mi>f<\/mi><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup><mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><msub><mrow><mi>x<\/mi><\/mrow><mrow> <mn>0<\/mn><\/mrow><\/msub><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-bin\">+<\/mo><msubsup><mrow><mo> \u222b  <\/mo><\/mrow><mrow><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><\/mrow><mrow><mi>x<\/mi><\/mrow><\/msubsup><msup><mrow><mi>f<\/mi><\/mrow><mrow><mo>\u2033<\/mo><\/mrow><\/msup> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>t<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mfrac><mrow><msup><mrow><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>t<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msup><\/mrow> <mrow><mn>1<\/mn><mo class=\"MathClass-punc\">!<\/mo><\/mrow><\/mfrac> <mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>t<\/mi><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\" \/> <mtd class=\"align-label\"> <mspace width=\"2em\" \/><\/mtd><\/mtr><mtr><mtd class=\"align-odd\" columnalign=\"right\" \/> <mtd class=\"align-even\"> <mo class=\"MathClass-rel\">=<\/mo> <msubsup><mrow><mi>P<\/mi><\/mrow><mrow><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><mo class=\"MathClass-punc\">,<\/mo><mn>1<\/mn><\/mrow><mrow><mi>f<\/mi><\/mrow><\/msubsup> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-bin\">+<\/mo> <msubsup><mrow><mi>R<\/mi><\/mrow><mrow><msub><mrow> <mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><mo class=\"MathClass-punc\">,<\/mo><mn>1<\/mn><\/mrow><mrow><mi>f<\/mi><\/mrow><\/msubsup> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><mo class=\"MathClass-punc\">.<\/mo><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\" \/> <mtd class=\"align-label\"> <mspace width=\"2em\" \/><\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">Wir sehen also wie sich ausgehend vom Fundamentalsatz mittels partieller Integration der n\u00e4chste Fall des Satzes ergibt. <\/p><p class=\"indent\">Angenommen die Aussage des Satzes stimmt f\u00fcr <math display=\"inline\"><mi>n<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <mn>1<\/mn> <mo class=\"MathClass-rel\">\u2265<\/mo> <mn>0<\/mn><\/math> und sei <math display=\"inline\"><mi>f<\/mi> <mo class=\"MathClass-punc\">:<\/mo> <mo class=\"MathClass-open\">(<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo> <mi>b<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">\u2192<\/mo> <mi>\u211d<\/mi><\/math> eine <math display=\"inline\"><mo class=\"MathClass-open\">(<\/mo><mi>n<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mn>1<\/mn><mo class=\"MathClass-close\">)<\/mo><\/math>-mal stetig differenzierbare Funktion. Dann gilt auf Grund der Induktionsvorraussetzung                                                                                                                                                                           <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mi>f<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-rel\">=<\/mo><munderover accent=\"false\" accentunder=\"false\"><mrow><mo> \u2211<\/mo> <\/mrow><mrow><mi>k<\/mi><mo class=\"MathClass-rel\">=<\/mo><mn>0<\/mn><\/mrow><mrow><mi>n<\/mi><mo class=\"MathClass-bin\">\u2212<\/mo><mn>1<\/mn><\/mrow><\/munderover><mfrac><mrow><msup><mrow><mi>f<\/mi><\/mrow><mrow><mo class=\"MathClass-open\">(<\/mo><mi>k<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow><\/msup><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><\/mrow> <mrow><mi>k<\/mi><mo class=\"MathClass-punc\">!<\/mo><\/mrow><\/mfrac> <msup><mrow><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mi>k<\/mi><\/mrow><\/msup> <mo class=\"MathClass-bin\">+<\/mo><msubsup><mrow><mo> \u222b  <\/mo><\/mrow><mrow><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><\/mrow><mrow><mi>x<\/mi><\/mrow><\/msubsup><msup><mrow><mi>f<\/mi><\/mrow><mrow><mo class=\"MathClass-open\">(<\/mo><mi>n<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow><\/msup> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>t<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mfrac><mrow><msup><mrow><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>t<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mi>n<\/mi><mo class=\"MathClass-bin\">\u2212<\/mo><mn>1<\/mn><\/mrow><\/msup><\/mrow> <mrow><mo class=\"MathClass-open\">(<\/mo><mi>n<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <mn>1<\/mn><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-punc\">!<\/mo><\/mrow><\/mfrac> <mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>t<\/mi><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">f\u00fcr alle <span class=\"maperiod\"><math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mo class=\"MathClass-open\">(<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>b<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math><\/span><span class=\"period\">.<\/span> <\/p><p class=\"indent\">Wir setzen <math display=\"inline\"><mi>u<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>t<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <msup><mrow><mi>f<\/mi><\/mrow><mrow><mo class=\"MathClass-open\">(<\/mo><mi>n<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow><\/msup><mo class=\"MathClass-open\">(<\/mo><mi>t<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math> und <span class=\"maperiod\"><math display=\"inline\"><mi>v<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>t<\/mi> <\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-rel\">=<\/mo> <mo class=\"MathClass-bin\">\u2212<\/mo><mfrac><mrow><msup><mrow><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-bin\">\u2212<\/mo><mi>t<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msup><\/mrow> <mrow><mi>n<\/mi><mo class=\"MathClass-punc\">!<\/mo><\/mrow><\/mfrac> <\/math><\/span><span class=\"period\">,<\/span> bemerken <math display=\"inline\"><msup><mrow><mi>v<\/mi><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup><mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>t<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-rel\">=<\/mo> <mfrac><mrow><msup><mrow><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-bin\">\u2212<\/mo><mi>t<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mi>n<\/mi><mo class=\"MathClass-bin\">\u2212<\/mo><mn>1<\/mn><\/mrow><\/msup><\/mrow> <mrow><mo class=\"MathClass-open\">(<\/mo><mi>n<\/mi><mo class=\"MathClass-bin\">\u2212<\/mo><mn>1<\/mn><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-punc\">!<\/mo><\/mrow><\/mfrac> <\/math> und wenden partielle Integration an, um <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mtd> <mtd class=\"align-even\"> <mo class=\"MathClass-rel\">=<\/mo><munderover accent=\"false\" accentunder=\"false\"><mrow><mo> \u2211<\/mo> <\/mrow><mrow><mi>k<\/mi><mo class=\"MathClass-rel\">=<\/mo><mn>0<\/mn><\/mrow><mrow><mi>n<\/mi><mo class=\"MathClass-bin\">\u2212<\/mo><mn>1<\/mn><\/mrow><\/munderover><mfrac><mrow><msup><mrow><mi>f<\/mi><\/mrow><mrow><mo class=\"MathClass-open\">(<\/mo><mi>k<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow><\/msup><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><\/mrow> <mrow><mi>k<\/mi><mo class=\"MathClass-punc\">!<\/mo><\/mrow><\/mfrac> <msup><mrow><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mi>k<\/mi><\/mrow><\/msup> <mo class=\"MathClass-bin\">\u2212<\/mo><msubsup><mrow><mrow><mo fence=\"true\" form=\"prefix\"> [<\/mo><mrow><msup><mrow><mi>f<\/mi><\/mrow><mrow><mo class=\"MathClass-open\">(<\/mo><mi>n<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow><\/msup><mo class=\"MathClass-open\">(<\/mo><mi>t<\/mi><mo class=\"MathClass-close\">)<\/mo><mfrac><mrow><msup><mrow><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>t<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msup><\/mrow> <mrow><mi>n<\/mi><mo class=\"MathClass-punc\">!<\/mo><\/mrow><\/mfrac> <\/mrow><mo fence=\"true\" form=\"postfix\">]<\/mo><\/mrow><\/mrow><mrow><mi>t<\/mi><mo class=\"MathClass-rel\">=<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><\/mrow><mrow><mi>t<\/mi><mo class=\"MathClass-rel\">=<\/mo><mi>x<\/mi><\/mrow><\/msubsup> <mo class=\"MathClass-bin\">+<\/mo><msubsup><mrow><mo> \u222b  <\/mo><\/mrow><mrow><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><\/mrow><mrow><mi>x<\/mi><\/mrow><\/msubsup><msup><mrow><mi>f<\/mi><\/mrow><mrow><mo class=\"MathClass-open\">(<\/mo><mi>n<\/mi><mo class=\"MathClass-bin\">+<\/mo><mn>1<\/mn><mo class=\"MathClass-close\">)<\/mo><\/mrow><\/msup> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>t<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mfrac><mrow><msup><mrow><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>t<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msup><\/mrow> <mrow><mi>n<\/mi><mo class=\"MathClass-punc\">!<\/mo><\/mrow><\/mfrac> <mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>t<\/mi><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\" \/> <mtd class=\"align-label\"> <mspace width=\"2em\" \/><\/mtd><\/mtr><mtr><mtd class=\"align-odd\" columnalign=\"right\" \/> <mtd class=\"align-even\"> <mo class=\"MathClass-rel\">=<\/mo><munderover accent=\"false\" accentunder=\"false\"><mrow><mo> \u2211<\/mo> <\/mrow><mrow><mi>k<\/mi><mo class=\"MathClass-rel\">=<\/mo><mn>0<\/mn><\/mrow><mrow><mi>n<\/mi><\/mrow><\/munderover><mfrac><mrow><msup><mrow><mi>f<\/mi><\/mrow><mrow><mo class=\"MathClass-open\">(<\/mo><mi>k<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow><\/msup><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><\/mrow> <mrow><mi>k<\/mi><mo class=\"MathClass-punc\">!<\/mo><\/mrow><\/mfrac> <msup><mrow><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mi>k<\/mi><\/mrow><\/msup> <mo class=\"MathClass-bin\">+<\/mo><msubsup><mrow><mo> \u222b  <\/mo><\/mrow><mrow><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><\/mrow><mrow><mi>x<\/mi><\/mrow><\/msubsup><msup><mrow><mi>f<\/mi><\/mrow><mrow><mo class=\"MathClass-open\">(<\/mo><mi>n<\/mi><mo class=\"MathClass-bin\">+<\/mo><mn>1<\/mn><mo class=\"MathClass-close\">)<\/mo><\/mrow><\/msup> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>t<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mfrac><mrow><msup><mrow><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>t<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msup><\/mrow> <mrow><mi>n<\/mi><mo class=\"MathClass-punc\">!<\/mo><\/mrow><\/mfrac> <mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>t<\/mi><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\" \/> <mtd class=\"align-label\"> <mspace width=\"2em\" \/><\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">zu erhalten. Dies beweist den Induktionsschritt und damit den Satz. <span>&nbsp;&nbsp;<\/span><\/p><div class=\"qed\">\u25a0<\/div><\/details><\/div> <p class=\"indent\">Oft werden wir das Theorem von Taylor (Theorem <a href=\"..\/..\/chapter\/taylor-approximation#x1-276002r46\">9.46<\/a>) in folgender Form verwenden. <\/p> <div class=\"me metheorem\"> <div class=\"wp-nocaption \"><\/div><h4 id=\"zad53dae02d37\"> <a id=\"x1-276003r47\"><\/a> <span class=\"ecbx-1095\">Korollar 9.47 <\/span>(Taylor-Absch\u00e4tzung)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Sei <\/span><math display=\"inline\"><mo class=\"MathClass-open\">(<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo> <mi>b<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">\u2286<\/mo> <mi>\u211d<\/mi><\/math> <span class=\"ecti-1095\">ein offenes,<\/span> <span class=\"ecti-1095\">nicht-leeres Intervall, sei <\/span><math display=\"inline\"><mi>f<\/mi> <mo class=\"MathClass-punc\">:<\/mo> <mo class=\"MathClass-open\">(<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>b<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">\u2192<\/mo> <mi>\u2102<\/mi><\/math> <span class=\"ecti-1095\">eine<\/span> <math display=\"inline\"><mo class=\"MathClass-open\">(<\/mo><mi>n<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mn>1<\/mn><mo class=\"MathClass-close\">)<\/mo><\/math><span class=\"ecti-1095\">-mal stetig differenzierbare<\/span> <span class=\"ecti-1095\">Funktion und seien <\/span><math display=\"inline\"><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><mo class=\"MathClass-punc\">,<\/mo><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mo class=\"MathClass-open\">(<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>b<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math> <span class=\"ecti-1095\">zwei<\/span> <span class=\"ecti-1095\">Punkte. Wir setzen <\/span><span class=\"maperiod\"><math display=\"inline\"><msub><mrow><mi>M<\/mi><\/mrow><mrow><mi>n<\/mi><mo class=\"MathClass-bin\">+<\/mo><mn>1<\/mn><\/mrow><\/msub> <mo class=\"MathClass-rel\">=<\/mo><mi class=\"qopname\"> max<\/mi><mo>  <\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> {<\/mo><mrow><mo class=\"MathClass-rel\">|<\/mo><msup><mrow><mi>f<\/mi><\/mrow><mrow><mo class=\"MathClass-open\">(<\/mo><mi>n<\/mi><mo class=\"MathClass-bin\">+<\/mo><mn>1<\/mn><mo class=\"MathClass-close\">)<\/mo><\/mrow><\/msup><mo class=\"MathClass-open\">(<\/mo><mi>t<\/mi><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-rel\">|<\/mo><mo class=\"MathClass-rel\">\u2223<\/mo><mi>t<\/mi><mstyle class=\"text\"><mtext>&nbsp;zwischen&nbsp;<\/mtext><\/mstyle><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><mstyle class=\"text\"><mtext>&nbsp;und&nbsp;<\/mtext><\/mstyle><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">}<\/mo><\/mrow><\/math><\/span><span class=\"period\">.<\/span> <span class=\"ecti-1095\">Dann gilt<\/span> <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"> <mrow><mo fence=\"true\" form=\"prefix\"> |<\/mo><mrow><mi>f<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-bin\">\u2212<\/mo> <msubsup><mrow><mi>P<\/mi><\/mrow><mrow><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><mo class=\"MathClass-punc\">,<\/mo><mi>n<\/mi><\/mrow><mrow><mi>f<\/mi><\/mrow><\/msubsup> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><\/mrow><mo fence=\"true\" form=\"postfix\">|<\/mo><\/mrow> <mo class=\"MathClass-rel\">=<\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> |<\/mo><mrow><msubsup><mrow><mi>R<\/mi><\/mrow><mrow><msub><mrow> <mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><mo class=\"MathClass-punc\">,<\/mo><mi>n<\/mi><\/mrow><mrow><mi>f<\/mi><\/mrow><\/msubsup> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><\/mrow><mo fence=\"true\" form=\"postfix\">|<\/mo><\/mrow> <mo class=\"MathClass-rel\">\u2264<\/mo> <mfrac><mrow><msub><mrow><mi>M<\/mi><\/mrow><mrow><mi>n<\/mi><mo class=\"MathClass-bin\">+<\/mo><mn>1<\/mn><\/mrow><\/msub><mo class=\"MathClass-rel\">|<\/mo><mi>x<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><msup><mrow><mo class=\"MathClass-rel\">|<\/mo><\/mrow><mrow><mi>n<\/mi><mo class=\"MathClass-bin\">+<\/mo><mn>1<\/mn><\/mrow><\/msup><\/mrow> <mrow><mo class=\"MathClass-open\">(<\/mo><mi>n<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mn>1<\/mn><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-punc\">!<\/mo><\/mrow><\/mfrac> <mo class=\"MathClass-punc\">.<\/mo><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\"><span class=\"ecti-1095\">Insbesondere ist <\/span><math display=\"inline\"><mi>f<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-rel\">=<\/mo> <msubsup><mrow><mi>P<\/mi><\/mrow><mrow><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><mo class=\"MathClass-punc\">,<\/mo><mi>n<\/mi><\/mrow><mrow><mi>f<\/mi><\/mrow><\/msubsup> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-bin\">+<\/mo> <mi>O<\/mi><mo class=\"MathClass-open\">(<\/mo><msup><mrow><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mi>n<\/mi><mo class=\"MathClass-bin\">+<\/mo><mn>1<\/mn><\/mrow><\/msup><mo class=\"MathClass-close\">)<\/mo><\/math> <span class=\"ecti-1095\">f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r <\/span><span class=\"maperiod\"><math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2192<\/mo> <msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn> <\/mrow> <\/msub> <\/math><\/span><span class=\"period\">.<\/span> <\/p> <\/div> <div class=\"wp-nocaption \"><\/div> <div class=\"proof\"> <p class=\"indent\"><span class=\"head\"><\/span><\/p><details open=\"open\"><summary><b>Beweis.<\/b><\/summary><p class=\"indent\" style=\"margin-top: 10\">Angenommen <span class=\"maperiod\"><math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2265<\/mo> <msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><\/math><\/span><span class=\"period\">.<\/span> Dann gilt <\/p><math display=\"block\"> <mtable class=\"multline-star\"> <mtr><mtd class=\"multline-star\"> <mrow><mo fence=\"true\" form=\"prefix\"> |<\/mo><mrow><msubsup><mrow><mi>R<\/mi><\/mrow><mrow><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><mo class=\"MathClass-punc\">,<\/mo><mi>n<\/mi><\/mrow><mrow><mi>f<\/mi><\/mrow><\/msubsup> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><\/mrow><mo fence=\"true\" form=\"postfix\">|<\/mo><\/mrow> <mo class=\"MathClass-rel\">\u2264<\/mo><msubsup><mrow><mo>\u222b  <\/mo><\/mrow><mrow><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><\/mrow><mrow><mi>x<\/mi><\/mrow><\/msubsup> <mrow><mo fence=\"true\" form=\"prefix\"> |<\/mo><mrow><msup><mrow><mi>f<\/mi><\/mrow><mrow><mo class=\"MathClass-open\">(<\/mo><mi>n<\/mi><mo class=\"MathClass-bin\">+<\/mo><mn>1<\/mn><mo class=\"MathClass-close\">)<\/mo><\/mrow><\/msup> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>t<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><\/mrow><mo fence=\"true\" form=\"postfix\">|<\/mo><\/mrow> <mfrac><mrow><msup><mrow><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>t<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msup><\/mrow> <mrow><mi>n<\/mi><mo class=\"MathClass-punc\">!<\/mo><\/mrow><\/mfrac> <mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>t<\/mi> <mo class=\"MathClass-rel\">\u2264<\/mo> <mfrac><mrow><msub><mrow><mi>M<\/mi><\/mrow><mrow><mi>n<\/mi><mo class=\"MathClass-bin\">+<\/mo><mn>1<\/mn><\/mrow><\/msub><\/mrow> <mrow><mi>n<\/mi><mo class=\"MathClass-punc\">!<\/mo><\/mrow><\/mfrac> <msubsup><mrow><mo>\u222b  <\/mo><\/mrow><mrow><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><\/mrow><mrow><mi>x<\/mi><\/mrow><\/msubsup><msup><mrow><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>t<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msup><mspace class=\"thinspace\" width=\"0.17em\" \/> <mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>t<\/mi> <\/mtd><\/mtr><mtr><mtd class=\"multline-star\"> <mo class=\"MathClass-rel\">=<\/mo> <mfrac><mrow><msub><mrow><mi>M<\/mi><\/mrow><mrow><mi>n<\/mi><mo class=\"MathClass-bin\">+<\/mo><mn>1<\/mn><\/mrow><\/msub><\/mrow> <mrow><mi>n<\/mi><mo class=\"MathClass-punc\">!<\/mo><\/mrow><\/mfrac><msubsup><mrow> <mrow><mo fence=\"true\" form=\"prefix\"> [<\/mo><mrow><mo class=\"MathClass-bin\">\u2212<\/mo> <mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mi>n<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mn>1<\/mn><\/mrow><\/mfrac><msup><mrow><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>t<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mi>n<\/mi><mo class=\"MathClass-bin\">+<\/mo><mn>1<\/mn><\/mrow><\/msup><\/mrow><mo fence=\"true\" form=\"postfix\">]<\/mo><\/mrow> <\/mrow><mrow><msub><mrow> <mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><\/mrow><mrow><mi>x<\/mi><\/mrow><\/msubsup> <mo class=\"MathClass-rel\">=<\/mo> <mfrac><mrow><msub><mrow><mi>M<\/mi><\/mrow><mrow><mi>n<\/mi><mo class=\"MathClass-bin\">+<\/mo><mn>1<\/mn><\/mrow><\/msub><msup><mrow><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mi>n<\/mi><mo class=\"MathClass-bin\">+<\/mo><mn>1<\/mn><\/mrow><\/msup><\/mrow> <mrow><mo class=\"MathClass-open\">(<\/mo><mi>n<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mn>1<\/mn><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-punc\">!<\/mo><\/mrow><\/mfrac> <mo class=\"MathClass-punc\">.<\/mo> <\/mtd><\/mtr><\/mtable> <\/math> <p class=\"nopar\"> Der Beweis f\u00fcr <math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">&lt;<\/mo> <msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><\/math> ist analog ausgehend von                                                                                                                                                                           <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"> <mrow><mo fence=\"true\" form=\"prefix\"> |<\/mo><mrow><msubsup><mrow><mi>R<\/mi><\/mrow><mrow><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><mo class=\"MathClass-punc\">,<\/mo><mi>n<\/mi><\/mrow><mrow><mi>f<\/mi><\/mrow><\/msubsup> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><\/mrow><mo fence=\"true\" form=\"postfix\">|<\/mo><\/mrow> <mo class=\"MathClass-rel\">=<\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> |<\/mo><mrow><msubsup><mrow><mo>\u222b  <\/mo><\/mrow><mrow><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><\/mrow><mrow><mi>x<\/mi><\/mrow><\/msubsup><msup><mrow><mi>f<\/mi><\/mrow><mrow><mo class=\"MathClass-open\">(<\/mo><mi>n<\/mi><mo class=\"MathClass-bin\">+<\/mo><mn>1<\/mn><mo class=\"MathClass-close\">)<\/mo><\/mrow><\/msup> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>t<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mfrac><mrow><msup><mrow><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>t<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msup><\/mrow> <mrow><mi>n<\/mi><mo class=\"MathClass-punc\">!<\/mo><\/mrow><\/mfrac> <mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>t<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">|<\/mo><\/mrow> <mo class=\"MathClass-rel\">\u2264<\/mo><msubsup><mrow><mo>\u222b  <\/mo><\/mrow><mrow><mi>x<\/mi><\/mrow><mrow><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub> <\/mrow><\/msubsup><mo class=\"MathClass-rel\">|<\/mo><msup><mrow><mi>f<\/mi><\/mrow><mrow><mo class=\"MathClass-open\">(<\/mo><mi>n<\/mi><mo class=\"MathClass-bin\">+<\/mo><mn>1<\/mn><mo class=\"MathClass-close\">)<\/mo><\/mrow><\/msup> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>t<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><mo class=\"MathClass-rel\">|<\/mo><mfrac><mrow><msup><mrow><mo class=\"MathClass-open\">(<\/mo><mi>t<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msup><\/mrow> <mrow><mi>n<\/mi><mo class=\"MathClass-punc\">!<\/mo><\/mrow><\/mfrac> <mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>t<\/mi><mo class=\"MathClass-punc\">.<\/mo><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"><mstyle class=\"maketag\"><mtext \/><\/mstyle><mspace class=\"nbsp\" width=\"0.33em\" \/><mstyle class=\"label\" id=\"x1-276005r8\" \/> <\/mtd><\/mtr><\/mtable><\/math> <span>&nbsp;&nbsp;<\/span><div class=\"qed\">\u25a0<\/div><\/details><\/div> <div class=\"me meexample\"> <div class=\"wp-nocaption \"><\/div><h4 id=\"zeefd0afc37dd\"> <span class=\"ecti-1095\">Bemerkung.<\/span><\/h4> <p class=\"indent\">Wir h\u00e4tten andere Versionen des obigen Satz zu Taylor-Approximationen bereits in Abschnitt <a href=\"..\/..\/chapter\/zentrale-saetze-der-differentialrechnung#x1-2320002\">8.2<\/a> f\u00fcr reellwertige Funktionen beweisen k\u00f6nnen. Dies sogar unter der etwas schw\u00e4cheren Annahme an die <math display=\"inline\"><mo class=\"MathClass-open\">(<\/mo><mi>n<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mn>1<\/mn><mo class=\"MathClass-close\">)<\/mo><\/math>-te Ableitung, dass diese bloss zwischen <math display=\"inline\"><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><\/math> und <math display=\"inline\"><mi>x<\/mi><\/math> existieren soll und nicht unbedingt stetig sein muss. Unter dieser Voraussetzung gibt es ein <math display=\"inline\"><msub><mrow><mi>\u03be<\/mi><\/mrow><mrow><mi>C<\/mi> <\/mrow> <\/msub> <\/math> zwischen <math display=\"inline\"><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn> <\/mrow> <\/msub> <\/math> und <span class=\"maperiod\"><math display=\"inline\"><mi>x<\/mi><\/math><\/span><span class=\"period\">,<\/span> so dass das sogenannte <span class=\"ecbx-1095\">Restglied nach Cauchy <\/span>durch <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><msubsup><mrow><mi>R<\/mi><\/mrow><mrow><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><mo class=\"MathClass-punc\">,<\/mo><mi>n<\/mi><\/mrow><mrow><mi>f<\/mi><\/mrow><\/msubsup> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-rel\">=<\/mo> <mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mi>n<\/mi><mo class=\"MathClass-punc\">!<\/mo><\/mrow><\/mfrac><msup><mrow><mi>f<\/mi><\/mrow><mrow><mo class=\"MathClass-open\">(<\/mo><mi>n<\/mi><mo class=\"MathClass-bin\">+<\/mo><mn>1<\/mn><mo class=\"MathClass-close\">)<\/mo><\/mrow><\/msup> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><msub><mrow><mi>\u03be<\/mi><\/mrow><mrow> <mi>C<\/mi><\/mrow><\/msub><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <msup><mrow><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <msub><mrow><mi>\u03be<\/mi><\/mrow><mrow><mi>C<\/mi><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msup> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <msub><mrow><mi>x<\/mi><\/mrow><mrow> <mn>0<\/mn><\/mrow><\/msub><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">gegeben ist. Es gibt unter denselben Voraussetzungen auch ein <math display=\"inline\"><msub><mrow><mi>\u03be<\/mi><\/mrow><mrow><mi>L<\/mi> <\/mrow> <\/msub> <\/math> zwischen <math display=\"inline\"><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn> <\/mrow> <\/msub> <\/math> und <span class=\"maperiod\"><math display=\"inline\"><mi>x<\/mi><\/math><\/span><span class=\"period\">,<\/span> so dass das <span class=\"ecbx-1095\">Restglied nach Lagrange <\/span>durch <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><msubsup><mrow><mi>R<\/mi><\/mrow><mrow><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><mo class=\"MathClass-punc\">,<\/mo><mi>n<\/mi><\/mrow><mrow><mi>f<\/mi><\/mrow><\/msubsup> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-rel\">=<\/mo> <mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mo class=\"MathClass-open\">(<\/mo><mi>n<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mn>1<\/mn><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-punc\">!<\/mo><\/mrow><\/mfrac><msup><mrow><mi>f<\/mi><\/mrow><mrow><mo class=\"MathClass-open\">(<\/mo><mi>n<\/mi><mo class=\"MathClass-bin\">+<\/mo><mn>1<\/mn><mo class=\"MathClass-close\">)<\/mo><\/mrow><\/msup> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><msub><mrow><mi>\u03be<\/mi><\/mrow><mrow> <mi>L<\/mi><\/mrow><\/msub><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <msup><mrow><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mi>n<\/mi><mo class=\"MathClass-bin\">+<\/mo><mn>1<\/mn><\/mrow><\/msup><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">gegeben ist. Wir verweisen daf\u00fcr auf folgende \u00dcbung. <\/p> <\/div> <div class=\"me meexample\"> <div class=\"wp-nocaption \"><\/div><h4 id=\"z6682e7ccd91f\"> <a id=\"x1-276006r48\"><\/a> <span class=\"ecbx-1095\">\u00dc<\/span><span class=\"ecbx-1095\">bung 9.48 <\/span>(Andere Formeln f\u00fcr das Restglied)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Gegeben sei ein nicht-leeres Intervall <\/span><span class=\"maperiod\"><math display=\"inline\"><mo class=\"MathClass-open\">(<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>b<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">\u2286<\/mo> <mi>\u211d<\/mi><\/math><\/span><span class=\"period\">,<\/span> <span class=\"ecti-1095\">Zahlen <\/span><math display=\"inline\"><mi>x<\/mi><mo class=\"MathClass-punc\">,<\/mo> <msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn> <\/mrow><\/msub> <mo class=\"MathClass-rel\">\u2208<\/mo> <mo class=\"MathClass-open\">(<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>b<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math> <span class=\"ecti-1095\">und eine<\/span> <math display=\"inline\"><mo class=\"MathClass-open\">(<\/mo><mi>n<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mn>1<\/mn><mo class=\"MathClass-close\">)<\/mo><\/math><span class=\"ecti-1095\">-mal differenzierbare<\/span> <span class=\"ecti-1095\">Funktion <\/span><span class=\"maperiod\"><math display=\"inline\"><mi>f<\/mi> <mo class=\"MathClass-punc\">:<\/mo> <mo class=\"MathClass-open\">(<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>b<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">\u2192<\/mo> <mi>\u211d<\/mi><\/math><\/span><span class=\"period\">.<\/span> <\/p><dl class=\"enumerate\"><dt class=\"enumerate\"> <span class=\"ecti-1095\">(a)<\/span><\/dt><dd class=\"enumerate\"><span class=\"ecti-1095\">Zeigen Sie, dass die Ableitung der Funktion<\/span> <math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mi>F<\/mi> <mo class=\"MathClass-punc\">:<\/mo> <mi>t<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>b<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><mo class=\"MathClass-rel\">\u21a6<\/mo><munderover accent=\"false\" accentunder=\"false\"><mrow><mo>\u2211<\/mo> <\/mrow><mrow><mi>k<\/mi><mo class=\"MathClass-rel\">=<\/mo><mn>0<\/mn><\/mrow><mrow><mi>n<\/mi><\/mrow><\/munderover><mfrac><mrow><msup><mrow><mi>f<\/mi><\/mrow><mrow><mo class=\"MathClass-open\">(<\/mo><mi>k<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow><\/msup><mo class=\"MathClass-open\">(<\/mo><mi>t<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow> <mrow><mi>k<\/mi><mo class=\"MathClass-punc\">!<\/mo><\/mrow><\/mfrac> <msup><mrow><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>t<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mi>k<\/mi><\/mrow><\/msup><mo class=\"MathClass-punc\">,<\/mo><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\"><span class=\"ecti-1095\">durch <\/span><math display=\"inline\"><mi>t<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>b<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><mo class=\"MathClass-rel\">\u21a6<\/mo><mfrac><mrow><msup><mrow><mi>f<\/mi><\/mrow><mrow><mo class=\"MathClass-open\">(<\/mo><mi>n<\/mi><mo class=\"MathClass-bin\">+<\/mo><mn>1<\/mn><mo class=\"MathClass-close\">)<\/mo><\/mrow><\/msup><mo class=\"MathClass-open\">(<\/mo><mi>t<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow> <mrow><mi>n<\/mi><mo class=\"MathClass-punc\">!<\/mo><\/mrow><\/mfrac> <msup><mrow><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>t<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msup><\/math> <span class=\"ecti-1095\">gegeben ist.<\/span> <\/p><\/dd><dt class=\"enumerate\"> <span class=\"ecti-1095\">(b)<\/span><\/dt><dd class=\"enumerate\"><span class=\"ecti-1095\">Wenden Sie den Mittelwertsatz (Theorem <\/span><a href=\"..\/..\/chapter\/zentrale-saetze-der-differentialrechnung#x1-233002r29\"><span class=\"ecti-1095\">8.29<\/span><\/a><span class=\"ecti-1095\">) auf die obige Funktion<\/span> <math display=\"inline\"><mi>F<\/mi><\/math> <span class=\"ecti-1095\">an, um die<\/span> <span class=\"ecti-1095\">Formel <\/span><math display=\"inline\"><msubsup><mrow><mi>R<\/mi><\/mrow><mrow><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><mo class=\"MathClass-punc\">,<\/mo><mi>n<\/mi><\/mrow><mrow><mi>f<\/mi><\/mrow><\/msubsup> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-rel\">=<\/mo> <mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mi>n<\/mi><mo class=\"MathClass-punc\">!<\/mo><\/mrow><\/mfrac><msup><mrow><mi>f<\/mi><\/mrow><mrow><mo class=\"MathClass-open\">(<\/mo><mi>n<\/mi><mo class=\"MathClass-bin\">+<\/mo><mn>1<\/mn><mo class=\"MathClass-close\">)<\/mo><\/mrow><\/msup> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><msub><mrow><mi>\u03be<\/mi><\/mrow><mrow><mi>C<\/mi><\/mrow><\/msub><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><msup><mrow> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <msub><mrow><mi>\u03be<\/mi><\/mrow><mrow><mi>C<\/mi><\/mrow><\/msub><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <\/mrow><mrow><mi>n<\/mi><\/mrow><\/msup> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><\/math> <span class=\"ecti-1095\">f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r das Restglied nach Cauchy zu beweisen.<\/span> <\/dd><dt class=\"enumerate\"> <span class=\"ecti-1095\">(c)<\/span><\/dt><dd class=\"enumerate\"><span class=\"ecti-1095\">Verwenden Sie den verallgemeinerten Mittelwertsatz (Satz <\/span><a href=\"..\/..\/chapter\/zentrale-saetze-der-differentialrechnung#x1-236001r48\"><span class=\"ecti-1095\">8.48<\/span><\/a><span class=\"ecti-1095\">) f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r obiges<\/span> <math display=\"inline\"><mi>F<\/mi><\/math> <span class=\"ecti-1095\">und die<\/span> <span class=\"ecti-1095\">Funktion <\/span><math display=\"inline\"><mi>g<\/mi> <mo class=\"MathClass-punc\">:<\/mo> <mi>t<\/mi><mo class=\"MathClass-rel\">\u21a6<\/mo><msup><mrow><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>t<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mi>n<\/mi><mo class=\"MathClass-bin\">+<\/mo><mn>1<\/mn><\/mrow><\/msup><\/math><span class=\"ecti-1095\">, um<\/span> <span class=\"ecti-1095\">die Formel <\/span><math display=\"inline\"><msubsup><mrow><mi>R<\/mi><\/mrow><mrow><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><mo class=\"MathClass-punc\">,<\/mo><mi>n<\/mi><\/mrow><mrow><mi>f<\/mi><\/mrow><\/msubsup> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-rel\">=<\/mo> <mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mo class=\"MathClass-open\">(<\/mo><mi>n<\/mi><mo class=\"MathClass-bin\">+<\/mo><mn>1<\/mn><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-punc\">!<\/mo><\/mrow><\/mfrac><msup><mrow><mi>f<\/mi><\/mrow><mrow><mo class=\"MathClass-open\">(<\/mo><mi>n<\/mi><mo class=\"MathClass-bin\">+<\/mo><mn>1<\/mn><mo class=\"MathClass-close\">)<\/mo><\/mrow><\/msup> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><msub><mrow><mi>\u03be<\/mi><\/mrow><mrow><mi>L<\/mi><\/mrow><\/msub><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <msup><mrow><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mi>n<\/mi><mo class=\"MathClass-bin\">+<\/mo><mn>1<\/mn><\/mrow><\/msup><\/math> <span class=\"ecti-1095\">f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r das Restglied nach Lagrange zu beweisen.<\/span><\/dd><\/dl> <\/div> <div class=\"me meexample\"> <div class=\"wp-nocaption \"><\/div><h4 id=\"zb051224d12d3\"> <a id=\"x1-276010r49\"><\/a> <span class=\"ecbx-1095\">Beispiel 9.49 <\/span>(Extremwerte)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Wir wollen die Taylor-Approximation verwenden um die Diskussion in Abschnitt <\/span><a href=\"..\/..\/chapter\/die-ableitung#x1-2290003\"><span class=\"ecti-1095\">8.1.3<\/span><\/a> <span class=\"ecti-1095\">und Korollar <\/span><a href=\"..\/..\/chapter\/zentrale-saetze-der-differentialrechnung#x1-234005r37\"><span class=\"ecti-1095\">8.37<\/span><\/a> <span class=\"ecti-1095\">zu verfeinern.<\/span> <span class=\"ecti-1095\">Sei <\/span><math display=\"inline\"><mo class=\"MathClass-open\">(<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo> <mi>b<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">\u2286<\/mo> <mi>\u211d<\/mi><\/math> <span class=\"ecti-1095\">ein offenes, nicht-leeres<\/span> <span class=\"ecti-1095\">Intervall und sei <\/span><math display=\"inline\"><mi>f<\/mi> <mo class=\"MathClass-punc\">:<\/mo> <mo class=\"MathClass-open\">(<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>b<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">\u2192<\/mo> <mi>\u2102<\/mi><\/math> <span class=\"ecti-1095\">eine<\/span> <math display=\"inline\"><mi>n<\/mi><\/math><span class=\"ecti-1095\">-mal stetig differenzierbare<\/span> <span class=\"ecti-1095\">Funktion. Angenommen <\/span><math display=\"inline\"><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub> <mo class=\"MathClass-rel\">\u2208<\/mo> <mo class=\"MathClass-open\">(<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>b<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math> <span class=\"ecti-1095\">erf<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">llt<\/span> <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><msup><mrow><mi>f<\/mi><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow> <mn>0<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <mi class=\"MathClass-op\">\u2026<\/mi><mo> <\/mo> <mo class=\"MathClass-rel\">=<\/mo> <msup><mrow><mi>f<\/mi><\/mrow><mrow><mo class=\"MathClass-open\">(<\/mo><mi>n<\/mi><mo class=\"MathClass-bin\">\u2212<\/mo><mn>1<\/mn><mo class=\"MathClass-close\">)<\/mo><\/mrow><\/msup><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow> <mn>0<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <mn>0<\/mn><mo class=\"MathClass-punc\">.<\/mo><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\"><span class=\"ecti-1095\">Dann gelten folgende Implikationen.<\/span> <\/p> <div class=\"custom-itemize\"><div class=\"item-head\"> <span class=\"tcrm-1095\">\u2022<\/span><\/div><div class=\"item-content\"><span class=\"ecti-1095\">Falls <\/span><math display=\"inline\"><msup><mrow><mi>f<\/mi><\/mrow><mrow><mo class=\"MathClass-open\">(<\/mo><mi>n<\/mi><mo class=\"MathClass-close\">)<\/mo> <\/mrow> <\/msup> <mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">&lt;<\/mo> <mn>0<\/mn><\/math> <span class=\"ecti-1095\">ist und <\/span><math display=\"inline\"><mi>n<\/mi><\/math> <span class=\"ecti-1095\">gerade ist, so nimmt <\/span><math display=\"inline\"><mi>f<\/mi><\/math> <span class=\"ecti-1095\">in <\/span><math display=\"inline\"><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn> <\/mrow> <\/msub> <\/math> <span class=\"ecti-1095\">ein lokales Maximum an.<\/span> <\/div><div class=\"item-head\"> <span class=\"tcrm-1095\">\u2022<\/span><\/div><div class=\"item-content\"><span class=\"ecti-1095\">Falls <\/span><math display=\"inline\"><msup><mrow><mi>f<\/mi><\/mrow><mrow><mo class=\"MathClass-open\">(<\/mo><mi>n<\/mi><mo class=\"MathClass-close\">)<\/mo> <\/mrow> <\/msup> <mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">&gt;<\/mo> <mn>0<\/mn><\/math> <span class=\"ecti-1095\">ist und <\/span><math display=\"inline\"><mi>n<\/mi><\/math> <span class=\"ecti-1095\">gerade ist, so nimmt <\/span><math display=\"inline\"><mi>f<\/mi><\/math> <span class=\"ecti-1095\">in <\/span><math display=\"inline\"><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn> <\/mrow> <\/msub> <\/math> <span class=\"ecti-1095\">ein lokales Minimum an.<\/span> <\/div><div class=\"item-head\"> <span class=\"tcrm-1095\">\u2022<\/span><\/div><div class=\"item-content\"><span class=\"ecti-1095\">Falls <\/span><math display=\"inline\"><msup><mrow><mi>f<\/mi><\/mrow><mrow><mo class=\"MathClass-open\">(<\/mo><mi>n<\/mi><mo class=\"MathClass-close\">)<\/mo> <\/mrow> <\/msup> <mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-rel\">\u2260<\/mo><mn>0<\/mn><\/math> <span class=\"ecti-1095\">und <\/span><math display=\"inline\"><mi>n<\/mi><\/math> <span class=\"ecti-1095\">ungerade ist, so nimmt <\/span><math display=\"inline\"><mi>f<\/mi><\/math> <span class=\"ecti-1095\">in <\/span><math display=\"inline\"><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn> <\/mrow> <\/msub> <\/math> <span class=\"ecti-1095\">kein lokales Extremum an.<\/span><\/div><\/div> <p class=\"indent\"><span class=\"ecti-1095\">Alle drei Aussagen folgen aus der Taylor-Approximation in Theorem <\/span><a href=\"..\/..\/chapter\/taylor-approximation#x1-276002r46\"><span class=\"ecti-1095\">9.46<\/span><\/a><span class=\"ecti-1095\">, die in diesem Fall f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r<\/span> <math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mo class=\"MathClass-open\">(<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo> <mi>b<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math> <span class=\"ecti-1095\">die<\/span> <span class=\"ecti-1095\">Form<\/span> <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mi>f<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-rel\">=<\/mo> <mi>f<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-bin\">+<\/mo><msubsup><mrow><mo> \u222b  <\/mo><\/mrow><mrow><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><\/mrow><mrow><mi>x<\/mi><\/mrow><\/msubsup><msup><mrow><mi>f<\/mi><\/mrow><mrow><mo class=\"MathClass-open\">(<\/mo><mi>n<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow><\/msup> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>t<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mfrac><mrow><msup><mrow><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>t<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mi>n<\/mi><mo class=\"MathClass-bin\">\u2212<\/mo><mn>1<\/mn><\/mrow><\/msup><\/mrow> <mrow><mo class=\"MathClass-open\">(<\/mo><mi>n<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <mn>1<\/mn><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-punc\">!<\/mo><\/mrow><\/mfrac> <mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>t<\/mi><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\"><span class=\"ecti-1095\">annimmt. Falls <\/span><math display=\"inline\"><msup><mrow><mi>f<\/mi><\/mrow><mrow><mo class=\"MathClass-open\">(<\/mo><mi>n<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow><\/msup><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">&gt;<\/mo> <mn>0<\/mn><\/math> <span class=\"ecti-1095\">ist, existiert ein <\/span><math display=\"inline\"><mi>\u03b4<\/mi> <mo class=\"MathClass-rel\">&gt;<\/mo> <mn>0<\/mn><\/math> <span class=\"ecti-1095\">mit <\/span><math display=\"inline\"><msup><mrow><mi>f<\/mi><\/mrow><mrow><mo class=\"MathClass-open\">(<\/mo><mi>n<\/mi><mo class=\"MathClass-close\">)<\/mo> <\/mrow> <\/msup> <mo class=\"MathClass-open\">(<\/mo><mi>t<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">&gt;<\/mo> <mn>0<\/mn><\/math> <span class=\"ecti-1095\">f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r alle<\/span> <math display=\"inline\"><mi>t<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>\u03b4<\/mi><mo class=\"MathClass-punc\">,<\/mo> <msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub> <mo class=\"MathClass-bin\">+<\/mo> <mi>\u03b4<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math><span class=\"ecti-1095\">. Wir betrachten nun mehrere<\/span> <span class=\"ecti-1095\">F<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">lle. Ist <\/span><math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><mo class=\"MathClass-punc\">,<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub> <mo class=\"MathClass-bin\">+<\/mo> <mi>\u03b4<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math><span class=\"ecti-1095\">, so ist obiges<\/span> <span class=\"ecti-1095\">Integral positiv und damit <\/span><span class=\"maperiod\"><math display=\"inline\"><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">&gt;<\/mo> <mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><\/math><\/span><span class=\"period\">.<\/span> <span class=\"ecti-1095\">Ist <\/span><math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>\u03b4<\/mi><mo class=\"MathClass-punc\">,<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><\/math> <span class=\"ecti-1095\">und<\/span> <math display=\"inline\"><mi>n<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <mn>1<\/mn><\/math> <span class=\"ecti-1095\">gerade, so<\/span> <span class=\"ecti-1095\">ist obiges Integral (auf Grund der umgekehrten Reihenfolge der Integrationsgrenzen) negativ, womit<\/span> <math display=\"inline\"><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">&lt;<\/mo> <mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><\/math> <span class=\"ecti-1095\">gilt<\/span> <span class=\"ecti-1095\">und <\/span><math display=\"inline\"><mi>f<\/mi><\/math> <span class=\"ecti-1095\">in<\/span> <math display=\"inline\"><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn> <\/mrow> <\/msub> <\/math> <span class=\"ecti-1095\">kein lokales Extremum<\/span> <span class=\"ecti-1095\">annimmt. Ist hingegen <\/span><math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>\u03b4<\/mi><mo class=\"MathClass-punc\">,<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><\/math> <span class=\"ecti-1095\">und <\/span><math display=\"inline\"><mi>n<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <mn>1<\/mn><\/math> <span class=\"ecti-1095\">ungerade, so ergibt<\/span> <span class=\"ecti-1095\">sich auf dieselbe Weise <\/span><span class=\"maperiod\"><math display=\"inline\"><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">&gt;<\/mo> <mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><\/math><\/span><span class=\"period\">,<\/span> <span class=\"ecti-1095\">womit <\/span><math display=\"inline\"><mi>f<\/mi><\/math> <span class=\"ecti-1095\">ein lokales<\/span> <span class=\"ecti-1095\">Minimum in <\/span><math display=\"inline\"><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><\/math> <span class=\"ecti-1095\">annimmt.<\/span> <span class=\"ecti-1095\">F<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r <\/span><math display=\"inline\"><msup><mrow><mi>f<\/mi><\/mrow><mrow><mo class=\"MathClass-open\">(<\/mo><mi>n<\/mi><mo class=\"MathClass-close\">)<\/mo> <\/mrow> <\/msup> <mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">&lt;<\/mo> <mn>0<\/mn><\/math> <span class=\"ecti-1095\">k<\/span><span class=\"ecti-1095\">\u00f6<\/span><span class=\"ecti-1095\">nnen wir<\/span> <span class=\"ecti-1095\">obige Diskussion auf <\/span><math display=\"inline\"> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>f<\/mi><\/math> <span class=\"ecti-1095\">anwenden.<\/span> <\/p> <\/div> <a id=\"x1-276011r275\"><\/a> <h4 id=\"z2f16eff2c7c4\" class=\"subsectionHead\"><span class=\"titlemark\">9.4.1 <\/span> <a id=\"x1-2770001\"><\/a>Analytische Funktionen<\/h4> <div class=\"me meexample\"> <div class=\"wp-nocaption \"><\/div><h4 id=\"zf64cf6a02dcc\"> <a id=\"x1-277001r50\"><\/a> <span class=\"ecbx-1095\">Applet 9.50 <\/span>(Taylor-Approximationen)<span class=\"ecbx-1095\">.<\/span> <\/h4> <div class=\"wp-nocaption \"><\/div><div class=\"geoapplet\" style=\"width: 688px\"><iframe height=\"416px\" scrolling=\"no\" src=\"https:\/\/www.geogebra.org\/material\/iframe\/id\/Vh93DD4U\/width\/688\/height\/416\/border\/888888\/rc\/false\/ai\/false\/sdz\/true\/smb\/false\/stb\/false\/stbh\/false\/ld\/false\/sri\/false\" style=\"border:0px\"><\/iframe><\/div><p class=\"indent\"><span class=\"ecti-1095\">Wir stellen einige Taylor-Approximationen bei verschiebbaren Fusspunkten zu bekannten<\/span> <span class=\"ecti-1095\">Funktionen dar.<\/span> <\/p> <\/div> <p class=\"indent\">Betrachtet man eine glatte Funktion <math display=\"inline\"><mi>f<\/mi><\/math> in Theorem <a href=\"..\/..\/chapter\/taylor-approximation#x1-276002r46\">9.46<\/a>, f\u00fcr die der Wert der Ableitung von <math display=\"inline\"><mi>f<\/mi><\/math> nicht zu wild wird f\u00fcr hohe Ableitungen, dann geht das Restglied zu einem fest gew\u00e4hlten <math display=\"inline\"><mi>x<\/mi><\/math> f\u00fcr                                                                                                                                                                           <math display=\"inline\"><mi>n<\/mi> <mo class=\"MathClass-rel\">\u2192<\/mo> <mi>\u221e<\/mi><\/math> gegen Null und die Funktion <math display=\"inline\"><mi>f<\/mi><\/math> wird durch ihre Taylorreihe beschrieben. Wie bereits gesehen, muss dies jedoch nicht unbedingt der Fall sein. Eine m\u00f6gliche Absch\u00e4tzung, die ausreichen w\u00fcrde, ist <\/p><table id=\"zeb0349350474\" class=\"equation-star\"><tr><td> <math class=\"equation\" display=\"block\"> <munder class=\"msub\"><mrow><mi class=\"qopname\">max<\/mi><mo>  <\/mo><\/mrow><mrow><mi>t<\/mi><mo class=\"MathClass-rel\">\u2208<\/mo><mo class=\"MathClass-open\">[<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><mo class=\"MathClass-punc\">,<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">]<\/mo><\/mrow><\/munder> <mrow><mo fence=\"true\" form=\"prefix\"> |<\/mo><mrow><msup><mrow><mi>f<\/mi><\/mrow><mrow><mo class=\"MathClass-open\">(<\/mo><mi>n<\/mi><mo class=\"MathClass-bin\">+<\/mo><mn>1<\/mn><mo class=\"MathClass-close\">)<\/mo><\/mrow><\/msup> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>t<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><\/mrow><mo fence=\"true\" form=\"postfix\">|<\/mo><\/mrow> <mo class=\"MathClass-rel\">\u2264<\/mo> <mi>c<\/mi><msup><mrow><mi>A<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msup> <\/math><\/td><\/tr><\/table> <p class=\"indent\">f\u00fcr alle&nbsp;<math display=\"inline\"><mi>n<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2115<\/mi><\/math> und zwei Konstanten&nbsp;<span class=\"maperiod\"><math display=\"inline\"><mi>c<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>A<\/mi> <mo class=\"MathClass-rel\">\u2265<\/mo> <mn>1<\/mn><\/math><\/span><span class=\"period\">.<\/span> Eine Absch\u00e4tzung dieser Art findet man beispielsweise f\u00fcr <math display=\"inline\"><mi class=\"qopname\">exp<\/mi><mo>  <\/mo><mo class=\"MathClass-punc\">,<\/mo><mi class=\"qopname\"> sin<\/mi><mo>  <\/mo> <mo class=\"MathClass-punc\">,<\/mo><mi class=\"qopname\"> cos<\/mi><mo>  <\/mo> <\/math> und Kombinationen dieser Funktionen. <\/p> <div class=\"me meexample\"> <div class=\"wp-nocaption \"><\/div><h4 id=\"ze538c7206e48\"> <a id=\"x1-277002r51\"><\/a> <span class=\"ecbx-1095\">\u00dc<\/span><span class=\"ecbx-1095\">bung 9.51.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Sei <\/span><math display=\"inline\"><mo class=\"MathClass-open\">[<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo> <mi>b<\/mi><mo class=\"MathClass-close\">]<\/mo> <mo class=\"MathClass-rel\">\u2286<\/mo> <mi>\u211d<\/mi><\/math> <span class=\"ecti-1095\">ein kompaktes<\/span> <span class=\"ecti-1095\">Intervall mit Endpunkten <\/span><span class=\"maperiod\"><math display=\"inline\"><mi>a<\/mi> <mo class=\"MathClass-rel\">&lt;<\/mo> <mi>b<\/mi><\/math><\/span><span class=\"period\">.<\/span> <\/p><dl class=\"enumerate\"><dt class=\"enumerate\"> <span class=\"ecti-1095\">(a)<\/span><\/dt><dd class=\"enumerate\"><span class=\"ecti-1095\">Sei <\/span><math display=\"inline\"><mi>f<\/mi> <mo class=\"MathClass-punc\">:<\/mo> <mo class=\"MathClass-open\">[<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo> <mi>b<\/mi><mo class=\"MathClass-close\">]<\/mo> <mo class=\"MathClass-rel\">\u2192<\/mo> <mi>\u211d<\/mi><\/math> <span class=\"ecti-1095\">ein Polynom oder eine<\/span> <span class=\"ecti-1095\">der Funktionen <\/span><math display=\"inline\"><mi class=\"qopname\">exp<\/mi><mo>  <\/mo><mo class=\"MathClass-punc\">,<\/mo><mi class=\"qopname\">sin<\/mi><mo>  <\/mo><mo class=\"MathClass-punc\">,<\/mo><mi class=\"qopname\">cos<\/mi><mo>  <\/mo><\/math><span class=\"ecti-1095\">. Zeigen<\/span> <span class=\"ecti-1095\">Sie, dass Konstanten <\/span><math display=\"inline\"><mi>c<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>A<\/mi> <mo class=\"MathClass-rel\">\u2265<\/mo> <mn>1<\/mn><\/math> <span class=\"ecti-1095\">mit<\/span> <math display=\"block\"><mtable class=\"align\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><munder class=\"msub\"><mrow><mi class=\"qopname\">max<\/mi><mo>  <\/mo><\/mrow><mrow><mi>t<\/mi><mo class=\"MathClass-rel\">\u2208<\/mo><mo class=\"MathClass-open\">[<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>b<\/mi><mo class=\"MathClass-close\">]<\/mo><\/mrow><\/munder> <mrow><mo fence=\"true\" form=\"prefix\"> |<\/mo><mrow><msup><mrow><mi>f<\/mi><\/mrow><mrow><mo class=\"MathClass-open\">(<\/mo><mi>n<\/mi><mo class=\"MathClass-bin\">+<\/mo><mn>1<\/mn><mo class=\"MathClass-close\">)<\/mo><\/mrow><\/msup> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>t<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><\/mrow><mo fence=\"true\" form=\"postfix\">|<\/mo><\/mrow> <mo class=\"MathClass-rel\">\u2264<\/mo> <mi>c<\/mi><msup><mrow><mi>A<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msup><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"><mstyle class=\"label\" id=\"x1-277004r9\" \/><mstyle class=\"maketag\"><mtext>(9.9)<\/mtext><\/mstyle><mspace class=\"nbsp\" width=\"0.33em\" \/> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\"><span class=\"ecti-1095\">existieren.<\/span> <\/p><\/dd><dt class=\"enumerate\"> <span class=\"ecti-1095\">(b)<\/span><\/dt><dd class=\"enumerate\"><span class=\"ecti-1095\">Zeigen Sie, dass falls <\/span><math display=\"inline\"><mi>f<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>g<\/mi> <mo class=\"MathClass-punc\">:<\/mo> <mo class=\"MathClass-open\">[<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>b<\/mi><mo class=\"MathClass-close\">]<\/mo> <mo class=\"MathClass-rel\">\u2192<\/mo> <mi>\u211d<\/mi><\/math> <span class=\"ecti-1095\">Funktionen<\/span> <span class=\"ecti-1095\">mit der Eigenschaft<\/span> (<a href=\"..\/..\/chapter\/taylor-approximation#x1-277004r9\">9.9<\/a>) <span class=\"ecti-1095\">auch <\/span><math display=\"inline\"><mi>f<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mi>g<\/mi><\/math> <span class=\"ecti-1095\">und <\/span><math display=\"inline\"><mi>f<\/mi> <mo class=\"MathClass-bin\">\u22c5<\/mo> <mi>g<\/mi><\/math> <span class=\"ecti-1095\">diese Eigenschaften besitzen.<\/span><\/dd><\/dl> <\/div> <p class=\"indent\">Eine Funktion, die sich um jeden Punkt im Definitionsbereich als Potenzreihe darstellen l\u00e4sst, nennen wir analytisch. Genauer sei <math display=\"inline\"><mo class=\"MathClass-open\">(<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo> <mi>b<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">\u2286<\/mo> <mi>\u211d<\/mi><\/math> ein offenes, nicht-leeres Intervall und <math display=\"inline\"><mi>f<\/mi> <mo class=\"MathClass-punc\">:<\/mo> <mo class=\"MathClass-open\">(<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo> <mi>b<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">\u2192<\/mo> <mi>\u2102<\/mi><\/math> eine glatte Funktion. Die Funktion <math display=\"inline\"><mi>f<\/mi><\/math> heisst <span class=\"ecbx-1095\">analytisch<\/span>, falls die Taylorreihe von <math display=\"inline\"><mi>f<\/mi><\/math> um jeden Punkt in <math display=\"inline\"><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub> <mo class=\"MathClass-rel\">\u2208<\/mo> <mo class=\"MathClass-open\">(<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>b<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math> positiven Konvergenzradius&nbsp;<math display=\"inline\"><mi>R<\/mi><\/math> hat und auf <math display=\"inline\"><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>R<\/mi><mo class=\"MathClass-punc\">,<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub> <mo class=\"MathClass-bin\">+<\/mo> <mi>R<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">\u2229<\/mo> <mo class=\"MathClass-open\">(<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>b<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math> mit <math display=\"inline\"><mi>f<\/mi><\/math> \u00fcbereinstimmt. <\/p><p class=\"indent\">Wir wollen ein weiteres Beispiel, wo sich eine glatte Funktion mit ihrer Taylorreihe vergleichen l\u00e4sst, Interessierten als \u00dcbung \u00fcberlassen (vergleichen Sie dies mit \u00dcbung <a href=\"..\/..\/chapter\/der-fundamentalsatz-der-integral--und-differentialrechnung#x1-260004r14\">9.14<\/a>). <\/p> <div class=\"me meexample\"> <div class=\"wp-nocaption \"><\/div><h4 id=\"zb2696af92b1c\"> <a id=\"x1-277006r52\"><\/a> <span class=\"ecbx-1095\">\u00dc<\/span><span class=\"ecbx-1095\">bung 9.52.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Sei <\/span><span class=\"maperiod\"><math display=\"inline\"><mi>\u03b1<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2102<\/mi><\/math><\/span><span class=\"period\">.<\/span> <span class=\"ecti-1095\">Zeigen Sie, dass<\/span> <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><msup><mrow><mo class=\"MathClass-open\">(<\/mo><mn>1<\/mn> <mo class=\"MathClass-bin\">+<\/mo> <mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mi>\u03b1<\/mi><\/mrow><\/msup> <mo class=\"MathClass-rel\">=<\/mo><munderover accent=\"false\" accentunder=\"false\"><mrow><mo> \u2211<\/mo> <\/mrow><mrow><mi>n<\/mi><mo class=\"MathClass-rel\">=<\/mo><mn>0<\/mn><\/mrow><mrow><mi>\u221e<\/mi><\/mrow><\/munderover><mfenced close=\")\" open=\"(\" separators><mfrac linethickness=\"0.0pt\"><mrow><mi>\u03b1<\/mi><\/mrow> <mrow><mi>n<\/mi><\/mrow><\/mfrac><\/mfenced><msup><mrow><mi>x<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msup><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\"><span class=\"ecti-1095\">f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r alle <\/span><span class=\"maperiod\"><math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mo class=\"MathClass-open\">(<\/mo><mo class=\"MathClass-bin\">\u2212<\/mo><mn>1<\/mn><mo class=\"MathClass-punc\">,<\/mo><mn>1<\/mn><mo class=\"MathClass-close\">)<\/mo><\/math><\/span><span class=\"period\">,<\/span> <span class=\"ecti-1095\">wobei f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r <\/span><math display=\"inline\"><mi>n<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <msub><mrow><mi>\u2115<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><\/math> <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mfenced close=\")\" open=\"(\" separators><mfrac linethickness=\"0.0pt\"><mrow><mi>\u03b1<\/mi><\/mrow> <mrow><mi>n<\/mi><\/mrow><\/mfrac><\/mfenced> <mo class=\"MathClass-rel\">=<\/mo> <mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mi>n<\/mi><mo class=\"MathClass-punc\">!<\/mo><\/mrow><\/mfrac><munderover accent=\"false\" accentunder=\"false\"><mrow><mo>\u220f<\/mo> <\/mrow><mrow><mi>k<\/mi><mo class=\"MathClass-rel\">=<\/mo><mn>0<\/mn><\/mrow><mrow><mi>n<\/mi><mo class=\"MathClass-bin\">\u2212<\/mo><mn>1<\/mn><\/mrow><\/munderover> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>\u03b1<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>k<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-rel\">=<\/mo> <mfrac><mrow><mi>\u03b1<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>\u03b1<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <mn>1<\/mn><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-rel\">\u22ef<\/mo><mo class=\"MathClass-open\">(<\/mo><mi>\u03b1<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>n<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mn>1<\/mn><mo class=\"MathClass-close\">)<\/mo><\/mrow> <mrow><mi>n<\/mi><mo class=\"MathClass-punc\">!<\/mo><\/mrow><\/mfrac> <mo class=\"MathClass-punc\">.<\/mo><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <div class=\"wp-nocaption \"><\/div><details><summary style=\"color:#FF7F00\"><span class=\"ecti-1095\">Hinweis.<\/span><\/summary><p class=\"indent\" style=\"margin-top: 0\"><span class=\"ecti-1095\">Sei <\/span><span class=\"maperiod\"><math display=\"inline\"><mi>f<\/mi> <mo class=\"MathClass-punc\">:<\/mo> <mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mo class=\"MathClass-open\">(<\/mo><mo class=\"MathClass-bin\">\u2212<\/mo><mn>1<\/mn><mo class=\"MathClass-punc\">,<\/mo><mn>1<\/mn><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-rel\">\u21a6<\/mo><msup><mrow><mo class=\"MathClass-open\">(<\/mo><mn>1<\/mn> <mo class=\"MathClass-bin\">+<\/mo> <mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mi>\u03b1<\/mi><\/mrow><\/msup><\/math><\/span><span class=\"period\">.<\/span> <span class=\"ecti-1095\">Berechnen Sie zuerst <\/span><math display=\"inline\"><msup><mrow><mi>f<\/mi><\/mrow><mrow><mo class=\"MathClass-open\">(<\/mo><mi>n<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow><\/msup><\/math> <span class=\"ecti-1095\">f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r alle <\/span><math display=\"inline\"><mi>n<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2115<\/mi><\/math> <span class=\"ecti-1095\">und geben Sie<\/span> <span class=\"ecti-1095\">die Taylorreihe von <\/span><math display=\"inline\"><mi>f<\/mi><\/math> <span class=\"ecti-1095\">um Null an. Zeigen Sie dann, dass das Restglied<\/span> <span class=\"ecti-1095\">\u201e<\/span><span class=\"ecti-1095\">klein<\/span><span class=\"ecti-1095\">\u201c<\/span> <span class=\"ecti-1095\">wird.<\/span><\/p><\/details>  <\/div> <div class=\"me meexample\"> <div class=\"wp-nocaption \"><\/div><h4 id=\"zdbec45531e83\"> <span class=\"ecti-1095\">Bemerkung.<\/span><\/h4> <p class=\"indent\">Analytische  Funktionen  haben  im  Zusammenhang  mit  \u201eholomorphen  Funktionen\u201c  auf offenen Teilmengen von <math display=\"inline\"><mi>\u2102<\/mi><\/math> nach <math display=\"inline\"><mi>\u2102<\/mi><\/math> eine sch\u00f6ne, geschlossene Behandlung, die in der Vorlesung \u201eFunktionentheorie\u201c im zweiten Studienjahr thematisiert wird. Der komplexe Blickwinkel erkl\u00e4rt zum Beispiel, warum die analytische Funktion&nbsp;<math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u211d<\/mi> <mo class=\"MathClass-rel\">\u2192<\/mo> <mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mn>1<\/mn><mo class=\"MathClass-bin\">+<\/mo><msup><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><\/mrow><\/mfrac><\/math> bei&nbsp;<math display=\"inline\"><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-rel\">=<\/mo> <mn>0<\/mn><\/math> eine Taylorreihe mit Konvergenzradius&nbsp;<math display=\"inline\"><mn>1<\/mn><\/math> besitzt, obwohl die Funktion auf ganz&nbsp;<math display=\"inline\"><mi>\u211d<\/mi><\/math>                                                                                                                                                                           definiert ist. <\/p> <\/div> <a id=\"x1-277007r277\"><\/a> <h4 id=\"zc3d5550b5f20\" class=\"subsectionHead\"><span class=\"titlemark\">9.4.2 <\/span> <a id=\"x1-2780002\"><\/a>Konvergenzgeschwindigkeit des Newton-Verfahrens*<\/h4> <p class=\"noindent\">Als Anwendung des Satzes von Taylor (Theorem <a href=\"..\/..\/chapter\/taylor-approximation#x1-276002r46\">9.46<\/a>) m\u00f6chten wir in diesem Teilabschnitt das Newton-Verfahren zur approximativen Berechnung einer Nullstelle einer gegebenen Funktion diskutieren. Weitere Anwendung werden in den n\u00e4chsten Abschnitten dieses Kapitels folgen. <\/p> <div class=\"me meexample\"> <div class=\"wp-nocaption \"><\/div><h4 id=\"z4801a04426fa\"> <a id=\"x1-278001r53\"><\/a> <span class=\"ecbx-1095\">Beispiel 9.53.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Sei <\/span><math display=\"inline\"><mo class=\"MathClass-open\">[<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo> <mi>b<\/mi><mo class=\"MathClass-close\">]<\/mo> <mo class=\"MathClass-rel\">\u2286<\/mo> <mi>\u211d<\/mi><\/math> <span class=\"ecti-1095\">ein kompaktes<\/span> <span class=\"ecti-1095\">Intervall mit Endpunkten <\/span><math display=\"inline\"><mi>a<\/mi> <mo class=\"MathClass-rel\">&lt;<\/mo> <mi>b<\/mi><\/math> <span class=\"ecti-1095\">und <\/span><math display=\"inline\"><mi>f<\/mi> <mo class=\"MathClass-punc\">:<\/mo> <mo class=\"MathClass-open\">[<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo> <mi>b<\/mi><mo class=\"MathClass-close\">]<\/mo> <mo class=\"MathClass-rel\">\u2192<\/mo> <mi>\u211d<\/mi><\/math> <span class=\"ecti-1095\">eine stetig<\/span> <span class=\"ecti-1095\">differenzierbare Funktion mit <\/span><math display=\"inline\"><msup><mrow><mi>f<\/mi><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">&gt;<\/mo> <mn>0<\/mn><\/math> <span class=\"ecti-1095\">f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r alle <\/span><span class=\"maperiod\"><math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mo class=\"MathClass-open\">[<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>b<\/mi><mo class=\"MathClass-close\">]<\/mo><\/math><\/span><span class=\"period\">.<\/span> <span class=\"ecti-1095\">(Wir erinnern daran, dass dies im Allgemeinen nicht dasselbe wie strenge Monotonie von<\/span> <math display=\"inline\"><mi>f<\/mi><\/math> <span class=\"ecti-1095\">ist, aber strenge Monotonie impliziert.) Angenommen es gilt<\/span> <math display=\"inline\"><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>a<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">&lt;<\/mo> <mn>0<\/mn><\/math> <span class=\"ecti-1095\">und<\/span> <span class=\"maperiod\"><math display=\"inline\"><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>b<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">&gt;<\/mo> <mn>0<\/mn><\/math><\/span><span class=\"period\">.<\/span> <span class=\"ecti-1095\">Dann gibt es auf Grund des Zwischenwertsatzes (Satz <\/span><a href=\"..\/..\/chapter\/der-zwischenwertsatz#x1-96001r58\"><span class=\"ecti-1095\">3.58<\/span><\/a><span class=\"ecti-1095\">) ein<\/span> <math display=\"inline\"><mi>z<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mo class=\"MathClass-open\">[<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo> <mi>b<\/mi><mo class=\"MathClass-close\">]<\/mo><\/math> <span class=\"ecti-1095\">mit<\/span> <math display=\"inline\"><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>z<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <mn>0<\/mn><\/math><span class=\"ecti-1095\">, das wegen der<\/span> <span class=\"ecti-1095\">strengen Monotonie von <\/span><math display=\"inline\"><mi>f<\/mi><\/math> <span class=\"ecti-1095\">eindeutig bestimmt ist. Beginnend mit einem gew<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">hlten Punkt<\/span> <math display=\"inline\"><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-rel\">\u2208<\/mo> <mo class=\"MathClass-open\">[<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo> <mi>b<\/mi><mo class=\"MathClass-close\">]<\/mo><\/math> <span class=\"ecti-1095\">definieren wir rekursiv<\/span> <\/p><math display=\"block\"><mtable class=\"align\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><msub><mrow><mi>x<\/mi><\/mrow><mrow><mi>n<\/mi><mo class=\"MathClass-bin\">+<\/mo><mn>1<\/mn><\/mrow><\/msub> <mo class=\"MathClass-rel\">=<\/mo> <msub><mrow><mi>x<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub> <mo class=\"MathClass-bin\">\u2212<\/mo> <mfrac><mrow><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><\/mrow> <mrow><msup><mrow><mi>f<\/mi><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><\/mrow><\/mfrac><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"><mstyle class=\"label\" id=\"x1-278002r10\" \/><mstyle class=\"maketag\"><mtext>(9.10)<\/mtext><\/mstyle><mspace class=\"nbsp\" width=\"0.33em\" \/> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\"><span class=\"ecti-1095\">f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r alle <\/span><math display=\"inline\"><mi>n<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <msub><mrow><mi>\u2115<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><\/math> <span class=\"ecti-1095\">(falls <\/span><math display=\"inline\"><msub><mrow><mi>x<\/mi><\/mrow><mrow><mi>n<\/mi> <\/mrow> <\/msub> <mo class=\"MathClass-rel\">\u2208<\/mo> <mo class=\"MathClass-open\">[<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>b<\/mi><mo class=\"MathClass-close\">]<\/mo><\/math><span class=\"ecti-1095\">).<\/span> <span class=\"ecti-1095\">Die Idee hinter diesem Verfahren m<\/span><span class=\"ecti-1095\">\u00f6<\/span><span class=\"ecti-1095\">chten wir in folgendem Bild erkl<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">ren.<\/span> <\/p> <div class=\"center\"> <div class=\"wp-nocaption \"><\/div><div class=\"wp-nocaption \"><\/div><div class=\"mefigcentered\" id=\"wpsize=634&amp;url=Pictures\/fundsatz\/newtonverfahren.pdf\"><img decoding=\"async\" id=\"zf983508b420b\" alt=\"PIC\" src=\"https:\/\/people.math.ethz.ch\/~einsiedl\/Pictures\/fundsatz\/newtonverfahren.svg\" width=\"634\" \/><\/div> <a id=\"x1-278003r2\"><\/a> <a id=\"x1-278004\"><\/a> <br \/><div class=\"caption\"><span class=\"id\">&nbsp;&nbsp;&nbsp;&nbsp;              Figur&nbsp;9.2:     <\/span><span class=\"content\">In     einem     ersten     Schritt     approximiert     man               <math display=\"inline\"><mi>f<\/mi><\/math>             mit der Tangenten <math display=\"inline\"><msub><mrow><mi>g<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub> <mo class=\"MathClass-punc\">:<\/mo> <mi>t<\/mi><mo class=\"MathClass-rel\">\u21a6<\/mo><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">+<\/mo> <msup><mrow><mi>f<\/mi><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-open\">(<\/mo><mi>t<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><\/math>               bei <span class=\"maperiod\"><math display=\"inline\"><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><\/math><\/span><span class=\"period\">.<\/span>               Von    dieser    berechnet    man    nun    die    eindeutige    Nullstelle               <span class=\"maperiod\"><math display=\"inline\"><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub> <mo class=\"MathClass-rel\">=<\/mo> <msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub> <mo class=\"MathClass-bin\">\u2212<\/mo> <mfrac><mrow><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><\/mrow> <mrow><msup><mrow><mi>f<\/mi><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><\/mrow><\/mfrac><\/math><\/span><span class=\"period\">.<\/span>               In  einem  zweiten  Schritt  verwendet  man  nun  diesen  neuen  Punkt               <math display=\"inline\"><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><\/math>                   und            berechnet            die            eindeutige            Nullstelle               <math display=\"inline\"><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><\/math>                   der Tangenten <math display=\"inline\"><msub><mrow><mi>g<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><\/math>                   bei <span class=\"maperiod\"><math display=\"inline\"><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><\/math><\/span><span class=\"period\">.<\/span>               Dies f\u00fchrt man iterativ so fort.                                                   &nbsp;&nbsp;&nbsp;&nbsp; <\/span><\/div> <\/div> <p class=\"indent\"><span class=\"ecti-1095\">Falls die Folge <\/span><math display=\"inline\"><msub><mrow><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><\/math> <span class=\"ecti-1095\">definiert ist (das heisst, <\/span><math display=\"inline\"><msub><mrow><mi>x<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub> <mo class=\"MathClass-rel\">\u2208<\/mo> <mo class=\"MathClass-open\">[<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>b<\/mi><mo class=\"MathClass-close\">]<\/mo><\/math> <span class=\"ecti-1095\">f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r alle <\/span><math display=\"inline\"><mi>n<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2115<\/mi><\/math><span class=\"ecti-1095\">)<\/span> <span class=\"ecti-1095\">und konvergiert, dann ist ihr Grenzwert eine Nullstelle von<\/span> <math display=\"inline\"><mi>f<\/mi><\/math> <span class=\"ecti-1095\">und somit unter unseren<\/span> <span class=\"ecti-1095\">Annahmen gleich der Nullstelle <\/span><span class=\"maperiod\"><math display=\"inline\"><mi>z<\/mi><\/math><\/span><span class=\"period\">.<\/span> <span class=\"ecti-1095\">In der Tat gilt f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r <\/span><math display=\"inline\"><msup><mrow><mi>z<\/mi><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup> <mo class=\"MathClass-rel\">=<\/mo><munder class=\"msub\"><mrow><mi class=\"qopname\"> lim<\/mi><mo>  <\/mo><\/mrow><mrow><mi>n<\/mi><mo class=\"MathClass-rel\">\u2192<\/mo><mi>\u221e<\/mi><\/mrow><\/munder><msub><mrow><mi>x<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><\/math> <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><msup><mrow><mi>z<\/mi><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup> <mo class=\"MathClass-rel\">=<\/mo><munder class=\"msub\"><mrow><mi class=\"qopname\"> lim<\/mi><mo>  <\/mo><\/mrow><mrow> <mi>n<\/mi><mo class=\"MathClass-rel\">\u2192<\/mo><mi>\u221e<\/mi><\/mrow><\/munder><msub><mrow><mi>x<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub> <mo class=\"MathClass-rel\">=<\/mo><munder class=\"msub\"><mrow><mi class=\"qopname\"> lim<\/mi><mo>  <\/mo><\/mrow><mrow><mi>n<\/mi><mo class=\"MathClass-rel\">\u2192<\/mo><mi>\u221e<\/mi><\/mrow><\/munder><msub><mrow><mi>x<\/mi><\/mrow><mrow><mi>n<\/mi><mo class=\"MathClass-bin\">+<\/mo><mn>1<\/mn><\/mrow><\/msub> <mo class=\"MathClass-rel\">=<\/mo><munder class=\"msub\"><mrow><mi class=\"qopname\"> lim<\/mi><mo>  <\/mo><\/mrow><mrow><mi>n<\/mi><mo class=\"MathClass-rel\">\u2192<\/mo><mi>\u221e<\/mi><\/mrow><\/munder><mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><msub><mrow><mi>x<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub> <mo class=\"MathClass-bin\">\u2212<\/mo> <mfrac><mrow><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><\/mrow> <mrow><msup><mrow><mi>f<\/mi><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><\/mrow><\/mfrac> <\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-rel\">=<\/mo> <msup><mrow><mi>z<\/mi><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup><mo class=\"MathClass-bin\">\u2212<\/mo> <mfrac><mrow><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><msup><mrow><mi>z<\/mi><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup><mo class=\"MathClass-close\">)<\/mo><\/mrow> <mrow><msup><mrow><mi>f<\/mi><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup><mo class=\"MathClass-open\">(<\/mo><msup><mrow><mi>z<\/mi><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup><mo class=\"MathClass-close\">)<\/mo><\/mrow><\/mfrac><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\"><span class=\"ecti-1095\">und somit <\/span><math display=\"inline\"><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><msup><mrow><mi>z<\/mi><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <mn>0<\/mn><\/math> <span class=\"ecti-1095\">und <\/span><span class=\"maperiod\"><math display=\"inline\"><msup><mrow><mi>z<\/mi><\/mrow><mrow><mo>\u2032<\/mo> <\/mrow> <\/msup> <mo class=\"MathClass-rel\">=<\/mo> <mi>z<\/mi><\/math><\/span><span class=\"period\">.<\/span> <\/p><p class=\"indent\"><span class=\"ecti-1095\">Das Konvergenzverhalten der Folge <\/span><math display=\"inline\"><msub><mrow><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><\/math> <span class=\"ecti-1095\">ist im Allgemeinen<\/span> <span class=\"ecti-1095\">\u201e<\/span><span class=\"ecti-1095\">sehr chaotisch<\/span><span class=\"ecti-1095\">\u201c<\/span><span class=\"ecti-1095\">. Unter etwas st<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">rkeren Annahmen m<\/span><span class=\"ecti-1095\">\u00f6<\/span><span class=\"ecti-1095\">chten wir nun aber zeigen, dass<\/span> <span class=\"ecti-1095\">die Folge <\/span><math display=\"inline\"><msub><mrow><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><\/math> <span class=\"ecti-1095\">konvergiert und die Konvergenzgeschwindigkeit untersuchen.<\/span> <\/p><p class=\"indent\"><span class=\"ecti-1095\">Wir nehmen zus<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">tzlich an, dass <\/span><math display=\"inline\"><mi>f<\/mi><\/math> <span class=\"ecti-1095\">zweimal stetig differenzierbar ist und definieren<\/span> <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mi>M<\/mi><\/mtd> <mtd class=\"align-even\"> <mo class=\"MathClass-rel\">=<\/mo><mi class=\"qopname\"> max<\/mi><mo>  <\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> {<\/mo><mrow><mo class=\"MathClass-rel\">|<\/mo><msup><mrow><mi>f<\/mi><\/mrow><mrow><mi class=\"qopname\">\u2033<\/mi><mo>  <\/mo><\/mrow><\/msup><mo class=\"MathClass-open\">(<\/mo><mi>t<\/mi><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-rel\">|<\/mo><mo class=\"MathClass-rel\">\u2223<\/mo><mi>t<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mo class=\"MathClass-open\">[<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>b<\/mi><mo class=\"MathClass-close\">]<\/mo><\/mrow><mo fence=\"true\" form=\"postfix\">}<\/mo><\/mrow><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\" \/> <mtd class=\"align-label\"> <mspace width=\"2em\" \/><\/mtd><\/mtr><mtr><mtd class=\"align-odd\" columnalign=\"right\"><mi>m<\/mi><\/mtd> <mtd class=\"align-even\"> <mo class=\"MathClass-rel\">=<\/mo><mi class=\"qopname\"> min<\/mi><mo>  <\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> {<\/mo><mrow><mo class=\"MathClass-rel\">|<\/mo><msup><mrow><mi>f<\/mi><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup><mo class=\"MathClass-open\">(<\/mo><mi>t<\/mi><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-rel\">|<\/mo><mo class=\"MathClass-rel\">\u2223<\/mo><mi>t<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mo class=\"MathClass-open\">[<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>b<\/mi><mo class=\"MathClass-close\">]<\/mo><\/mrow><mo fence=\"true\" form=\"postfix\">}<\/mo><\/mrow><mo class=\"MathClass-punc\">.<\/mo><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\" \/> <mtd class=\"align-label\"> <mspace width=\"2em\" \/><\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\"><span class=\"ecti-1095\">Nun wenden wir Korollar <\/span><a href=\"..\/..\/chapter\/taylor-approximation#x1-276003r47\"><span class=\"ecti-1095\">9.47<\/span><\/a> <span class=\"ecti-1095\">an, um den Fehlerterm<\/span> <math display=\"inline\"><msubsup><mrow><mi>R<\/mi><\/mrow><mrow><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-punc\">,<\/mo><mn>1<\/mn> <\/mrow> <mrow> <mi>f<\/mi> <\/mrow> <\/msubsup><\/math> <span class=\"ecti-1095\">bei der Approximation<\/span> <span class=\"ecti-1095\">von <\/span><math display=\"inline\"><mi>f<\/mi><\/math> <span class=\"ecti-1095\">durch die Tangente<\/span> <span class=\"ecti-1095\">um <\/span><math display=\"inline\"><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn> <\/mrow> <\/msub> <\/math> <span class=\"ecti-1095\">abzusch<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">tzen.<\/span> <span class=\"ecti-1095\">Bei der Nullstelle <\/span><math display=\"inline\"><mi>z<\/mi><\/math> <span class=\"ecti-1095\">ergibt dies<\/span> <\/p><math display=\"block\"><mtable class=\"align\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"> <mrow><mo fence=\"true\" form=\"prefix\"> |<\/mo><mrow><msubsup><mrow><mi>R<\/mi><\/mrow><mrow><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><mo class=\"MathClass-punc\">,<\/mo><mn>1<\/mn><\/mrow><mrow><mi>f<\/mi><\/mrow><\/msubsup> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>z<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><\/mrow><mo fence=\"true\" form=\"postfix\">|<\/mo><\/mrow> <mo class=\"MathClass-rel\">\u2264<\/mo> <mfrac><mrow><mi>M<\/mi><mo class=\"MathClass-rel\">|<\/mo><mi>z<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><msup><mrow><mo class=\"MathClass-rel\">|<\/mo><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><\/mrow> <mrow><mn>2<\/mn><\/mrow><\/mfrac> <\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"><mstyle class=\"label\" id=\"x1-278005r11\" \/><mstyle class=\"maketag\"><mtext>(9.11)<\/mtext><\/mstyle><mspace class=\"nbsp\" width=\"0.33em\" \/> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\"><span class=\"ecti-1095\">Des Weiteren gilt per Definition<\/span> <\/p><math display=\"block\"><mtable class=\"align\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mn>0<\/mn> <mo class=\"MathClass-rel\">=<\/mo> <mi>f<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>z<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-rel\">=<\/mo> <mi>f<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-bin\">+<\/mo> <msup><mrow><mi>f<\/mi><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup><mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><msub><mrow><mi>x<\/mi><\/mrow><mrow> <mn>0<\/mn><\/mrow><\/msub><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>z<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-bin\">+<\/mo> <msubsup><mrow><mi>R<\/mi><\/mrow><mrow><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><mo class=\"MathClass-punc\">,<\/mo><mn>1<\/mn><\/mrow><mrow><mi>f<\/mi><\/mrow><\/msubsup> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>z<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><mo class=\"MathClass-punc\">.<\/mo><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"><mstyle class=\"label\" id=\"x1-278006r12\" \/><mstyle class=\"maketag\"><mtext>(9.12)<\/mtext><\/mstyle><mspace class=\"nbsp\" width=\"0.33em\" \/> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\"><span class=\"ecti-1095\">Wir dividieren Gleichung<\/span> (<a href=\"..\/..\/chapter\/taylor-approximation#x1-278006r12\">9.12<\/a>) <span class=\"ecti-1095\">durch <\/span><math display=\"inline\"><msup><mrow><mi>f<\/mi><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><\/math> <span class=\"ecti-1095\">und erhalten<\/span> <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mn>0<\/mn> <mo class=\"MathClass-rel\">=<\/mo> <mi>z<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo><munder><mrow><munder accentunder=\"false\"><mrow><mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub> <mo class=\"MathClass-bin\">\u2212<\/mo> <mfrac><mrow><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><\/mrow> <mrow><msup><mrow><mi>f<\/mi><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><\/mrow><\/mfrac> <\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><\/mrow><mo>\ufe38<\/mo><\/munder><\/mrow><mrow> <mo class=\"MathClass-rel\">=<\/mo> <msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><\/mrow><\/munder> <mo class=\"MathClass-bin\">+<\/mo> <mfrac><mrow><msubsup><mrow><mi>R<\/mi><\/mrow><mrow><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><mo class=\"MathClass-punc\">,<\/mo><mn>1<\/mn><\/mrow><mrow><mi>f<\/mi><\/mrow><\/msubsup><mo class=\"MathClass-open\">(<\/mo><mi>z<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow> <mrow><msup><mrow><mi>f<\/mi><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><\/mrow><\/mfrac> <mo class=\"MathClass-punc\">.<\/mo><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\"><span class=\"ecti-1095\">Mit der Absch<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">tzung<\/span> (<a href=\"..\/..\/chapter\/taylor-approximation#x1-278005r11\">9.11<\/a>) <span class=\"ecti-1095\">ergibt dies<\/span> <\/p><math display=\"block\"><mtable class=\"align\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"> <mrow><mo fence=\"true\" form=\"prefix\"> |<\/mo><mrow><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>z<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">|<\/mo><\/mrow> <mo class=\"MathClass-rel\">=<\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> |<\/mo><mrow><mfrac><mrow><msubsup><mrow><mi>R<\/mi><\/mrow><mrow><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><mo class=\"MathClass-punc\">,<\/mo><mn>1<\/mn><\/mrow><mrow><mi>f<\/mi><\/mrow><\/msubsup><mo class=\"MathClass-open\">(<\/mo><mi>z<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow> <mrow><msup><mrow><mi>f<\/mi><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><\/mrow><\/mfrac> <\/mrow><mo fence=\"true\" form=\"postfix\">|<\/mo><\/mrow> <mo class=\"MathClass-rel\">\u2264<\/mo><mfrac><mrow> <mi>M<\/mi><\/mrow> <mrow><mn>2<\/mn><mi>m<\/mi><\/mrow><\/mfrac><mspace class=\"thinspace\" width=\"0.17em\" \/><msup><mrow> <mrow><mo fence=\"true\" form=\"prefix\"> |<\/mo><mrow><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><mo class=\"MathClass-bin\">\u2212<\/mo><\/mrow><mo fence=\"true\" form=\"postfix\">|<\/mo><\/mrow><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"><mstyle class=\"label\" id=\"x1-278007r13\" \/><mstyle class=\"maketag\"><mtext>(9.13)<\/mtext><\/mstyle><mspace class=\"nbsp\" width=\"0.33em\" \/> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\"><span class=\"ecti-1095\">Falls nun <\/span><math display=\"inline\"><mi>\ud835\udf00<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mo class=\"MathClass-open\">(<\/mo><mn>0<\/mn><mo class=\"MathClass-punc\">,<\/mo> <mfrac><mrow><mi>m<\/mi><\/mrow> <mrow><mi>M<\/mi><\/mrow><\/mfrac><mo class=\"MathClass-close\">)<\/mo><\/math> <span class=\"ecti-1095\">klein<\/span> <span class=\"ecti-1095\">genug ist, so dass <\/span><math display=\"inline\"><msub><mrow><mi>B<\/mi><\/mrow><mrow><mi>\ud835\udf00<\/mi><\/mrow><\/msub><mo class=\"MathClass-open\">(<\/mo><mi>z<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">\u2286<\/mo> <mo class=\"MathClass-open\">[<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>b<\/mi><mo class=\"MathClass-close\">]<\/mo><\/math> <span class=\"ecti-1095\">gilt, und falls <\/span><math display=\"inline\"><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub> <mo class=\"MathClass-rel\">\u2208<\/mo> <msub><mrow><mi>B<\/mi><\/mrow><mrow><mi>\ud835\udf00<\/mi><\/mrow><\/msub><mo class=\"MathClass-open\">(<\/mo><mi>z<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math> <span class=\"ecti-1095\">ist, dann folgt aus<\/span> (<a href=\"..\/..\/chapter\/taylor-approximation#x1-278007r13\">9.13<\/a>)<span class=\"ecti-1095\">, dass<\/span> <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mo class=\"MathClass-rel\">|<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>z<\/mi><mo class=\"MathClass-rel\">|<\/mo><mo class=\"MathClass-rel\">\u2264<\/mo><mfrac><mrow> <mi>M<\/mi><\/mrow> <mrow><mn>2<\/mn><mi>m<\/mi><\/mrow><\/mfrac><mspace class=\"thinspace\" width=\"0.17em\" \/><mo class=\"MathClass-rel\">|<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>z<\/mi><msup><mrow><mo class=\"MathClass-rel\">|<\/mo><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup> <mo class=\"MathClass-rel\">\u2264<\/mo><mfrac><mrow> <mi>M<\/mi><\/mrow> <mrow><mn>2<\/mn><mi>m<\/mi><\/mrow><\/mfrac><mspace class=\"thinspace\" width=\"0.17em\" \/><mo class=\"MathClass-rel\">|<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>z<\/mi><mo class=\"MathClass-rel\">|<\/mo><mi>\ud835\udf00<\/mi> <mo class=\"MathClass-rel\">\u2264<\/mo><mfrac><mrow> <mn>1<\/mn><\/mrow> <mrow><mn>2<\/mn><\/mrow><\/mfrac><mo class=\"MathClass-rel\">|<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>z<\/mi><mo class=\"MathClass-rel\">|<\/mo> <mo class=\"MathClass-rel\">&lt;<\/mo> <mfrac><mrow><mi>\ud835\udf00<\/mi><\/mrow> <mrow><mn>2<\/mn><\/mrow><\/mfrac><mo class=\"MathClass-punc\">.<\/mo><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\"><span class=\"ecti-1095\">Daher liegt <\/span><math display=\"inline\"><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><\/math> <span class=\"ecti-1095\">auch<\/span> <span class=\"ecti-1095\">in <\/span><math display=\"inline\"><mo class=\"MathClass-open\">[<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo> <mi>b<\/mi><mo class=\"MathClass-close\">]<\/mo><\/math> <span class=\"ecti-1095\">und ist bereits<\/span> <span class=\"ecti-1095\">n<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">her an <\/span><math display=\"inline\"><mi>z<\/mi><\/math> <span class=\"ecti-1095\">als<\/span> <math display=\"inline\"><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn> <\/mrow> <\/msub> <\/math><span class=\"ecti-1095\">. Komplett analog beweist<\/span> <span class=\"ecti-1095\">man, dass f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r alle <\/span><math display=\"inline\"><mi>n<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <msub><mrow><mi>\u2115<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><\/math> <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mo class=\"MathClass-rel\">|<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mi>n<\/mi><mo class=\"MathClass-bin\">+<\/mo><mn>1<\/mn><\/mrow><\/msub> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>z<\/mi><mo class=\"MathClass-rel\">|<\/mo><mo class=\"MathClass-rel\">\u2264<\/mo><mfrac><mrow> <mn>1<\/mn><\/mrow> <mrow><mn>2<\/mn><\/mrow><\/mfrac><mspace class=\"thinspace\" width=\"0.17em\" \/><mo class=\"MathClass-rel\">|<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>z<\/mi><mo class=\"MathClass-rel\">|<\/mo><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\"><span class=\"ecti-1095\">gilt, was mit vollst<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">ndiger Induktion auch<\/span> <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mo class=\"MathClass-rel\">|<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>z<\/mi><mo class=\"MathClass-rel\">|<\/mo><mo class=\"MathClass-rel\">\u2264<\/mo> <msup><mrow><mn>2<\/mn><\/mrow><mrow><mo class=\"MathClass-bin\">\u2212<\/mo><mi>n<\/mi><\/mrow><\/msup><mo class=\"MathClass-rel\">|<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow> <mn>0<\/mn><\/mrow><\/msub> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>z<\/mi><mo class=\"MathClass-rel\">|<\/mo><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\"><span class=\"ecti-1095\">f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r alle <\/span><math display=\"inline\"><mi>n<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <msub><mrow><mi>\u2115<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><\/math> <span class=\"ecti-1095\">zeigt. Insbesondere<\/span> <span class=\"ecti-1095\">konvergiert die Folge <\/span><math display=\"inline\"><msub><mrow><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><\/math> <span class=\"ecti-1095\">und wir k<\/span><span class=\"ecti-1095\">\u00f6<\/span><span class=\"ecti-1095\">nnen das Newton-Verfahren (unter geeigneter Wahl eines Startpunktes<\/span> <math display=\"inline\"><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn> <\/mrow> <\/msub> <\/math><span class=\"ecti-1095\">)<\/span> <span class=\"ecti-1095\">verwenden, um die Nullstelle mit hoher Genauigkeit zu approximieren. Die Konvergenz ist noch<\/span> <span class=\"ecti-1095\">schneller als diese Absch<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">tzung beweist; man spricht von quadratischer Konvergenz.<\/span> <\/p> <\/div> <div class=\"me meexample\"> <div class=\"wp-nocaption \"><\/div><h4 id=\"z56f6c93a40bd\"> <a id=\"x1-278008r54\"><\/a> <span class=\"ecbx-1095\">\u00dc<\/span><span class=\"ecbx-1095\">bung 9.54 <\/span>(Quadratische Konvergenz)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Wir m<\/span><span class=\"ecti-1095\">\u00f6<\/span><span class=\"ecti-1095\">chten die Konvergenzgeschwindigkeit des Newton-Verfahrens hier genauer analysieren. In<\/span> <span class=\"ecti-1095\">obigem Beispiel wurde gezeigt, dass sich der Fehler in jedem Schritt mindestens halbiert. Wir<\/span> <span class=\"ecti-1095\">betrachten nun das Argument etwas genauer (und behalten dementsprechend die Notation). Sei<\/span> <math display=\"inline\"><mi>\u03b2<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mfrac> <mrow> <mi>M<\/mi><\/mrow> <mrow><mn>2<\/mn><mi>m<\/mi><\/mrow><\/mfrac><\/math><span class=\"ecti-1095\">. Zeigen Sie f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r den<\/span> <span class=\"ecti-1095\">mit <\/span><math display=\"inline\"><mi>\u03b2<\/mi><\/math> <span class=\"ecti-1095\">gewichteten<\/span> <span class=\"ecti-1095\">Abstand<\/span><span class=\"ecti-1095\">&nbsp;<\/span><math display=\"inline\"><mi>\u03b2<\/mi><mo class=\"MathClass-rel\">|<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>z<\/mi><mo class=\"MathClass-rel\">|<\/mo><\/math> <span class=\"ecti-1095\">die Absch<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">tzung<\/span> <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mi>\u03b2<\/mi><mo class=\"MathClass-rel\">|<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>z<\/mi><mo class=\"MathClass-rel\">|<\/mo><mo class=\"MathClass-rel\">\u2264<\/mo> <msup><mrow><mo class=\"MathClass-open\">(<\/mo><mi>\u03b2<\/mi><mo class=\"MathClass-rel\">|<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>z<\/mi><mo class=\"MathClass-rel\">|<\/mo><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><msup><mrow><mn>2<\/mn><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msup> <\/mrow><\/msup><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\"><span class=\"ecti-1095\">f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r jedes <\/span><math display=\"inline\"><mi>n<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2115<\/mi><\/math> <span class=\"ecti-1095\">gilt.<\/span> <\/p> <\/div> <div class=\"me meexample\"> <div class=\"wp-nocaption \"><\/div><h4 id=\"z7bc48d837598\"> <a id=\"x1-278009r55\"><\/a> <span class=\"ecbx-1095\">\u00dc<\/span><span class=\"ecbx-1095\">bung 9.55.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Verwenden                Sie                das                Newton-Verfahren,                um<\/span> <math display=\"inline\"><msqrt><mrow> <mn>2<\/mn><\/mrow><\/msqrt><\/math> <span class=\"ecti-1095\">(oder alternativ eine andere irrationale, reelle algebraische Zahl) auf drei Dezimalstellen genau<\/span> <span class=\"ecti-1095\">zu berechnen. F<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r die genaue, numerische Berechnung des Resultats d<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">rfen Sie einen Rechner<\/span> <span class=\"ecti-1095\">oder ein Computer-Programm benutzen.<\/span> <\/p> <\/div> <a id=\"x1-278010r276\"><\/a> \n","protected":false},"author":1089,"menu_order":4,"template":"","meta":{"pb_show_title":"","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-96","chapter","type-chapter","status-publish","hentry"],"part":92,"_links":{"self":[{"href":"https:\/\/wp-prd.let.ethz.ch\/analysis19\/wp-json\/pressbooks\/v2\/chapters\/96","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/wp-prd.let.ethz.ch\/analysis19\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/wp-prd.let.ethz.ch\/analysis19\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/wp-prd.let.ethz.ch\/analysis19\/wp-json\/wp\/v2\/users\/1089"}],"version-history":[{"count":0,"href":"https:\/\/wp-prd.let.ethz.ch\/analysis19\/wp-json\/pressbooks\/v2\/chapters\/96\/revisions"}],"part":[{"href":"https:\/\/wp-prd.let.ethz.ch\/analysis19\/wp-json\/pressbooks\/v2\/parts\/92"}],"metadata":[{"href":"https:\/\/wp-prd.let.ethz.ch\/analysis19\/wp-json\/pressbooks\/v2\/chapters\/96\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/wp-prd.let.ethz.ch\/analysis19\/wp-json\/wp\/v2\/media?parent=96"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/wp-prd.let.ethz.ch\/analysis19\/wp-json\/pressbooks\/v2\/chapter-type?post=96"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/wp-prd.let.ethz.ch\/analysis19\/wp-json\/wp\/v2\/contributor?post=96"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/wp-prd.let.ethz.ch\/analysis19\/wp-json\/wp\/v2\/license?post=96"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}