{"id":100,"date":"2021-12-15T09:53:30","date_gmt":"2021-12-15T09:53:30","guid":{"rendered":"https:\/\/wp-prd.let.ethz.ch\/analysis19\/chapter\/weitere-lernmaterialien-9\/"},"modified":"2021-12-15T09:53:30","modified_gmt":"2021-12-15T09:53:30","slug":"weitere-lernmaterialien-9","status":"publish","type":"chapter","link":"https:\/\/wp-prd.let.ethz.ch\/analysis19\/chapter\/weitere-lernmaterialien-9\/","title":{"raw":"Weitere Lernmaterialien","rendered":"Weitere Lernmaterialien"},"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=\"z9975f39a572f\" class=\"sectionHead\"><span class=\"titlemark\">9.8 <\/span> <a id=\"x1-2910008\"><\/a>Weitere Lernmaterialien<\/h3> <a id=\"x1-291001r290\"><\/a> <h4 id=\"z7f74c770ed4b\" class=\"subsectionHead\"><span class=\"titlemark\">9.8.1 <\/span> <a id=\"x1-2920001\"><\/a>Verwendung des Kapitels<\/h4> <p class=\"noindent\">Dieses Kapitel verbindet mit Hilfe des Fundamentalsatz der Analysis die zentralen Begriffe der Ableitung und des Riemann-Integrals. Damit haben wir das vollst\u00e4ndige Arsenal an Ableitungsregeln auch f\u00fcr die Berechnung von Integralen verwenden k\u00f6nnen, womit die Berechnung von Integralen mitunter deutlich einfacher wurde \u2013 auch wenn dies zus\u00e4tzliche \u00dcbung erfordert. Umgekehrt haben wir aber auch mit dem Integralrestglied im Satz zur Taylor-Approximation gesehen, dass das Riemann-Integral n\u00fctzlich sein kann um den Zusammenhang zwischen den Ableitungen und der urspr\u00fcnglichen Funktion besser zu verstehen. Falls die symbolische Integration sich als nicht machbar erweist, so ist wiederum die Taylor-Approximation n\u00fctzlich um das Riemann-Integral numerisch \u2013 zum Beispiel mit der Simpson-Methode \u2013 mit \u00fcberraschend hoher Genauigkeit zu berechnen. Ebenso ist aber der Satz zur Taylor-Approximation auch von theoretischer Wichtigkeit, da wir mit diesen asymptotische Formeln wie zum Beispiel die Sterling-Formel beweisen k\u00f6nnen. <\/p><p class=\"indent\">Zusammenfassend k\u00f6nnen wir also sagen, dass dieses Kapitel den Aufbau der eindimensionalen Analysis vollendet. Die Inhalte dieses Kapitels bilden einen zentralen Bestandteil der Analysis I\/II Vorlesung und ihrer Anwendungen. <a id=\"x1-292001r292\"><\/a> <\/p> <h4 id=\"z83b0ca29874f\" class=\"subsectionHead\"><span class=\"titlemark\">9.8.2 <\/span> <a id=\"x1-2930002\"><\/a>\u00dcbungen<\/h4> <p class=\"noindent\">Da die Schwierigkeit beim Integrieren vor allem in der Auswahl der richtigen Methode liegt, wollen wir in folgender \u00dcbung noch einige weitere Aufgaben ohne Angabe der richtigen Technik auflisten. Dazu wollen wir noch erw\u00e4hnen, dass es oft auch mehr als eine Methode gibt, die zum Erfolg f\u00fchren kann. <\/p> <div class=\"me meexample\"> <p class=\"indent\"><\/p><h4 id=\"z2fd64eb8de0a\"> <span class=\"ecbx-1095\">\u00dc<\/span><span class=\"ecbx-1095\">bung.<\/span><\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Berechnen Sie die unbestimmten Integrale<\/span> <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"> <mtable align=\"axis\" class=\"array\" columnlines=\"none\" equalcolumns=\"false\" equalrows=\"false\"> <mtr><mtd class=\"array\" columnalign=\"left\"><mo>\u222b  <\/mo> <mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><msqrt><mrow><msup><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn> <\/mrow> <\/msup> <mo class=\"MathClass-bin\">+<\/mo><mn>2<\/mn><mn>0<\/mn><mn>1<\/mn><msup><mrow><mn>7<\/mn><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><\/mrow><\/msqrt><\/mrow><\/mfrac><mspace class=\"thinspace\" width=\"0.17em\" \/> <mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi><mo class=\"MathClass-punc\">,<\/mo> <\/mtd><mtd class=\"array\" columnalign=\"left\"><mo>\u222b  <\/mo> <mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mi>x<\/mi><msqrt><mrow><mn>1<\/mn><mo class=\"MathClass-bin\">+<\/mo><msup><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><\/mrow><\/msqrt><\/mrow><\/mfrac><mspace class=\"thinspace\" width=\"0.17em\" \/> <mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi><mo class=\"MathClass-punc\">,<\/mo> <\/mtd> <\/mtr> <mtr><mtd class=\"array\" columnalign=\"left\"><mo>\u222b  <\/mo><mi class=\"qopname\">exp<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mroot><mrow><mi>x<\/mi><\/mrow><mrow><mn>3<\/mn><\/mrow><\/mroot><mo class=\"MathClass-close\">)<\/mo><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi><mo class=\"MathClass-punc\">,<\/mo> <\/mtd><mtd class=\"array\" columnalign=\"left\"><mo>\u222b  <\/mo> <mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mn>1<\/mn><mo class=\"MathClass-bin\">+<\/mo><mi class=\"qopname\">exp<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mo class=\"MathClass-bin\">\u2212<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow><\/mfrac><mspace class=\"thinspace\" width=\"0.17em\" \/> <mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi><mo class=\"MathClass-punc\">,<\/mo> <\/mtd> <\/mtr> <mtr><mtd class=\"array\" columnalign=\"left\"><mo>\u222b  <\/mo> <mfrac><mrow><mi>x<\/mi><\/mrow> <mrow><mn>1<\/mn><mo class=\"MathClass-bin\">+<\/mo><msqrt><mrow><mi>x<\/mi><\/mrow><\/msqrt><\/mrow><\/mfrac><mspace class=\"thinspace\" width=\"0.17em\" \/> <mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi><mo class=\"MathClass-punc\">,<\/mo> <\/mtd><mtd class=\"array\" columnalign=\"left\"><mo>\u222b  <\/mo><msup><mrow><mi class=\"qopname\">cos<\/mi><mo>  <\/mo><\/mrow><mrow><mn>5<\/mn><\/mrow><\/msup><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><msup><mrow><mi class=\"qopname\">sin<\/mi><mo>  <\/mo><\/mrow><mrow><mn>3<\/mn><\/mrow><\/msup><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi><mo class=\"MathClass-punc\">,<\/mo><\/mtd> <\/mtr> <mtr><mtd class=\"array\" columnalign=\"left\"><mo>\u222b  <\/mo><msup><mrow><mi class=\"qopname\">cos<\/mi><mo>  <\/mo><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><msup><mrow><mi class=\"qopname\">sin<\/mi><mo>  <\/mo><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi><mo class=\"MathClass-punc\">,<\/mo> <\/mtd><mtd class=\"array\" columnalign=\"left\"><mo>\u222b  <\/mo> <mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><msup><mrow><mi class=\"qopname\"> cos<\/mi><mo>  <\/mo> <\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><msup><mrow><mi class=\"qopname\"> sin<\/mi><mo>  <\/mo> <\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow><\/mfrac><mspace class=\"thinspace\" width=\"0.17em\" \/> <mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi><mo class=\"MathClass-punc\">,<\/mo> <\/mtd> <\/mtr> <mtr><mtd class=\"array\" columnalign=\"left\"><mo>\u222b  <\/mo><mfrac><mrow><mi class=\"qopname\"> log<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-bin\">+<\/mo><mi class=\"qopname\">sin<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi class=\"qopname\">log<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-close\">)<\/mo><msup><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><\/mrow> <mrow><msup><mrow><mi>x<\/mi><\/mrow><mrow><mn>3<\/mn><\/mrow><\/msup><\/mrow><\/mfrac> <mspace class=\"thinspace\" width=\"0.17em\" \/> <mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi><mo class=\"MathClass-punc\">,<\/mo><\/mtd><mtd class=\"array\" columnalign=\"left\"><mo>\u222b  <\/mo> <mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mi class=\"qopname\"> cos<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow><\/mfrac><mspace class=\"thinspace\" width=\"0.17em\" \/> <mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi> <\/mtd><\/mtr> <\/mtable> <\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <\/div> <div class=\"me meexample\"> <p class=\"indent\"><\/p><h4 id=\"z80ee6d103c8c\"> <span class=\"ecbx-1095\">\u00dc<\/span><span class=\"ecbx-1095\">bung.<\/span><\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Zeigen Sie, dass jede stetig differenzierbare Funktion <\/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\">auf einem kompakten Intervall <\/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\">mit Endpunkten <\/span><math display=\"inline\"><mi>a<\/mi> <mo class=\"MathClass-rel\">&lt;<\/mo> <mi>b<\/mi><\/math> <span class=\"ecti-1095\">beschr<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">nkte Variation <\/span><math display=\"inline\"><mi>V<\/mi> <mo class=\"MathClass-open\">(<\/mo><mi>f<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math> <span class=\"ecti-1095\">(siehe die entsprechende <\/span><span class=\"ecti-1095\">\u00dc<\/span><span class=\"ecti-1095\">bung in Abschnitt<\/span><span class=\"ecti-1095\">&nbsp;<\/span><a href=\"..\/..\/chapter\/weitere-lernmaterialien#x1-1270002\"><span class=\"ecti-1095\">4.8.2<\/span><\/a><span class=\"ecti-1095\">) hat und dass <\/span><math display=\"inline\"><mi>V<\/mi> <mo class=\"MathClass-open\">(<\/mo><mi>f<\/mi><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">=<\/mo><msubsup><mrow><mi class=\"MathClass-op\"> \u222b  <\/mi><mo> <\/mo><\/mrow><mrow><mi>a<\/mi><\/mrow><mrow><mi>x<\/mi><\/mrow><\/msubsup><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><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>t<\/mi><\/math> <span class=\"ecti-1095\">gilt 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> <\/p><details><summary style=\"color:#FF7F00\"><span class=\"ecti-1095\">Hinweis.<\/span><\/summary><p class=\"indent\" style=\"margin-top: 0\"><span class=\"ecti-1095\">Gegeben eine Zerlegung <\/span><span class=\"maperiod\"><math display=\"inline\"><mi>\u2128<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> {<\/mo><mrow><mi>a<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub> <mo class=\"MathClass-rel\">&lt;<\/mo> <msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub> <mo class=\"MathClass-rel\">&lt;<\/mo> <mi class=\"MathClass-op\">\u2026<\/mi><mo> <\/mo> <mo class=\"MathClass-rel\">&lt;<\/mo> <msub><mrow><mi>x<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub> <mo class=\"MathClass-rel\">=<\/mo> <mi>b<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">}<\/mo><\/mrow><\/math><\/span><span class=\"period\">,<\/span> <span class=\"ecti-1095\">wenden Sie den Mittelwertsatz auf den Ausdruck <\/span><math display=\"inline\"><msubsup><mrow><mi class=\"MathClass-op\">\u2211<\/mi><mo> <\/mo> <\/mrow><mrow><mi>i<\/mi><mo class=\"MathClass-rel\">=<\/mo><mn>1<\/mn><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msubsup><mo class=\"MathClass-rel\">|<\/mo><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mi>i<\/mi><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mi>i<\/mi><mo class=\"MathClass-bin\">\u2212<\/mo><mn>1<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-rel\">|<\/mo><\/math> <span class=\"ecti-1095\">an und betrachten Sie dann die richtige Riemann-Summe.<\/span><\/p><\/details>  <\/div> <div class=\"me meexample\"> <p class=\"indent\"><\/p><h4 id=\"za88e06db2c5c\"> <span class=\"ecbx-1095\">\u00dc<\/span><span class=\"ecbx-1095\">bung <\/span>(Abel-Summation und Partielle Integration)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Wir wollen in dieser <\/span><span class=\"ecti-1095\">\u00dc<\/span><span class=\"ecti-1095\">bung den Zusammenhang zwischen der Abel-Summation und der<\/span> <span class=\"ecti-1095\">partiellen Integration erkl<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">ren. Sei also<\/span><span class=\"ecti-1095\">&nbsp;<\/span><math display=\"inline\"><mi>a<\/mi> <mo class=\"MathClass-rel\">&lt;<\/mo> <mi>b<\/mi><\/math> <span class=\"ecti-1095\">und seien<\/span><span class=\"ecti-1095\">&nbsp;<\/span><math display=\"inline\"><mi>u<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>v<\/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\">zwei stetig differenzierbare Funktionen. Verwenden Sie die Abel-Summation von <\/span><span class=\"ecti-1095\">\u00dc<\/span><span class=\"ecti-1095\">bung <\/span><a href=\"..\/..\/chapter\/summen-und-produkte#x1-78001r3\"><span class=\"ecti-1095\">3.3<\/span><\/a><span class=\"ecti-1095\">,<\/span> <span class=\"ecti-1095\">um die partielle Integration<\/span><span class=\"ecti-1095\">&nbsp;<\/span><math display=\"inline\"><msubsup><mrow><mi class=\"MathClass-op\">\u222b  <\/mi><mo> <\/mo><\/mrow><mrow><mi>a<\/mi><\/mrow><mrow><mi>b<\/mi><\/mrow><\/msubsup><mi>u<\/mi><msup><mrow><mi>v<\/mi><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <msubsup><mrow><mo class=\"MathClass-open\">[<\/mo><mi>u<\/mi><mi>v<\/mi><mo class=\"MathClass-close\">]<\/mo><\/mrow><mrow><mi>a<\/mi><\/mrow><mrow><mi>b<\/mi><\/mrow><\/msubsup> <mo class=\"MathClass-bin\">\u2212<\/mo><msubsup><mrow><mi class=\"MathClass-op\">\u222b  <\/mi><mo> <\/mo><\/mrow><mrow><mi>a<\/mi><\/mrow><mrow><mi>b<\/mi><\/mrow><\/msubsup><msup><mrow><mi>u<\/mi><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup><mi>v<\/mi><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi><\/math> <span class=\"ecti-1095\">zu beweisen.<\/span> <\/p><p class=\"indent\"><\/p><details><summary style=\"color:#FF7F00\"><span class=\"ecti-1095\">Hinweis.<\/span><\/summary><p class=\"indent\" style=\"margin-top: 0\"><span class=\"ecti-1095\">Erkl<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">ren Sie zuerst, wieso Sie ohne Beschr<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">nkung der Allgemeinheit<\/span> <math display=\"inline\"><mi>u<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>b<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <mi>v<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>a<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <mn>0<\/mn><\/math> <span class=\"ecti-1095\">annehmen<\/span> <span class=\"ecti-1095\">k<\/span><span class=\"ecti-1095\">\u00f6<\/span><span class=\"ecti-1095\">nnen. Sei<\/span><span class=\"ecti-1095\">&nbsp;<\/span><math display=\"inline\"><mi>\u2128<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> {<\/mo><mrow><mi>a<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub> <mo class=\"MathClass-rel\">&lt;<\/mo> <msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub> <mo class=\"MathClass-rel\">&lt;<\/mo> <mi class=\"MathClass-op\">\u2026<\/mi><mo> <\/mo> <mo class=\"MathClass-rel\">&lt;<\/mo> <msub><mrow><mi>x<\/mi><\/mrow><mrow><mi>n<\/mi><mo class=\"MathClass-bin\">\u2212<\/mo><mn>1<\/mn><\/mrow><\/msub> <mo class=\"MathClass-rel\">&lt;<\/mo> <msub><mrow><mi>x<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub> <mo class=\"MathClass-rel\">=<\/mo> <mi>b<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">}<\/mo><\/mrow><\/math> <span class=\"ecti-1095\">eine<\/span> <span class=\"ecti-1095\">Zerlegung von<\/span><span class=\"ecti-1095\">&nbsp;<\/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\">in<\/span><span class=\"ecti-1095\">&nbsp;<\/span><math display=\"inline\"><mi>n<\/mi> <mo class=\"MathClass-rel\">\u2265<\/mo> <mn>1<\/mn><\/math> <span class=\"ecti-1095\">viele Teilintervalle<\/span> <span class=\"ecti-1095\">der L<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">nge<\/span><span class=\"ecti-1095\">&nbsp;<\/span><span class=\"maperiod\"><math display=\"inline\"><mfrac><mrow><mi>b<\/mi><mo class=\"MathClass-bin\">\u2212<\/mo><mi>a<\/mi><\/mrow> <mrow><mi>n<\/mi><\/mrow><\/mfrac> <\/math><\/span><span class=\"period\">.<\/span> <span class=\"ecti-1095\">Verwenden Sie Abel-Summation auf die Summe<\/span> <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><munderover accent=\"false\" accentunder=\"false\"><mrow><mo>\u2211<\/mo> <\/mrow><mrow><mi>k<\/mi><mo class=\"MathClass-rel\">=<\/mo><mn>1<\/mn><\/mrow><mrow><mi>n<\/mi><\/mrow><\/munderover><mi>u<\/mi><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow> <mi>k<\/mi><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-open\">(<\/mo><mi>v<\/mi><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mi>k<\/mi><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>v<\/mi><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mi>k<\/mi><mo class=\"MathClass-bin\">\u2212<\/mo><mn>1<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-close\">)<\/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<\/span><span class=\"ecti-1095\">&nbsp;<\/span><math display=\"inline\"><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>k<\/mi> <\/mrow> <\/msub> <mo class=\"MathClass-rel\">=<\/mo> <mo class=\"MathClass-open\">(<\/mo><mi>v<\/mi><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mi>k<\/mi><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>v<\/mi><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mi>k<\/mi><mo class=\"MathClass-bin\">\u2212<\/mo><mn>1<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-close\">)<\/mo><\/math> <span class=\"ecti-1095\">und<\/span><span class=\"ecti-1095\">&nbsp;<\/span><math display=\"inline\"><msub><mrow><mi>b<\/mi><\/mrow><mrow><mi>k<\/mi> <\/mrow> <\/msub> <mo class=\"MathClass-rel\">=<\/mo> <mi>u<\/mi><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mi>k<\/mi><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><\/math><span class=\"ecti-1095\">. Verwenden Sie anschliessend<\/span> <span class=\"ecti-1095\">den Mittelwertsatz auf<\/span><span class=\"ecti-1095\">&nbsp;<\/span><math display=\"inline\"><mi>v<\/mi><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mi>k<\/mi><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>v<\/mi><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mi>k<\/mi><mo class=\"MathClass-bin\">\u2212<\/mo><mn>1<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><\/math> <span class=\"ecti-1095\">an und vergleichen Sie die resultierende Summen mit der Riemann-Summen f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r das Integral<\/span> <span class=\"maperiod\"><math display=\"inline\"><msubsup><mrow><mi class=\"MathClass-op\">\u222b  <\/mi><mo> <\/mo><\/mrow><mrow><mi>a<\/mi><\/mrow><mrow><mi>b<\/mi><\/mrow><\/msubsup><mi>u<\/mi><msup><mrow><mi>v<\/mi><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi><\/math><\/span><span class=\"period\">.<\/span><\/p><\/details>  <\/div> <div class=\"me meexample\"> <p class=\"indent\"><\/p><h4 id=\"zbafa692a9bd4\"> <span class=\"ecbx-1095\">\u00dc<\/span><span class=\"ecbx-1095\">bung <\/span>(Ein alternativer Beweis von Proposition <a href=\"..\/..\/chapter\/anwendungen#x1-116007r30\">4.30<\/a>)<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 hier einen Beweis von Proposition <\/span><a href=\"..\/..\/chapter\/anwendungen#x1-116007r30\"><span class=\"ecti-1095\">4.30<\/span><\/a> <span class=\"ecti-1095\">unter der Annahme, dass<\/span> <math display=\"inline\"><mi>f<\/mi><\/math> <span class=\"ecti-1095\">stetig<\/span> <span class=\"ecti-1095\">ist, durchf<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">hren. Gehen Sie wie folgt vor:<\/span> <\/p><dl class=\"enumerate\"><dt class=\"enumerate\"> <span class=\"ecti-1095\">(i)<\/span><\/dt><dd class=\"enumerate\"><span class=\"ecti-1095\">Zeigen Sie, dass die Abbildung <\/span><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><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>b<\/mi><mo class=\"MathClass-close\">]<\/mo><mo class=\"MathClass-rel\">\u21a6<\/mo><mi mathvariant=\"bold-script\">\u2110<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math> <span class=\"ecti-1095\">differenzierbar ist und dass <\/span><span class=\"maperiod\"><math display=\"inline\"><msup><mrow><mi>F<\/mi><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup> <mo class=\"MathClass-rel\">=<\/mo> <mi>f<\/mi><\/math><\/span><span class=\"period\">.<\/span> <\/dd><dt class=\"enumerate\"> <span class=\"ecti-1095\">(ii)<\/span><\/dt><dd class=\"enumerate\"><span class=\"ecti-1095\">Verwenden Sie nun den Fundamentalsatz der Integral- und Differentialrechnung, um auf<\/span> <span class=\"ecti-1095\">Proposition <\/span><a href=\"..\/..\/chapter\/anwendungen#x1-116007r30\"><span class=\"ecti-1095\">4.30<\/span><\/a> <span class=\"ecti-1095\">zu schliessen.<\/span><\/dd><\/dl> <\/div> <div class=\"me meexample\"> <p class=\"indent\"><\/p><h4 id=\"zdff40d17065c\"> <span class=\"ecbx-1095\">\u00dc<\/span><span class=\"ecbx-1095\">bung <\/span>(Volumen einer Vase)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Berechnen Sie das Volumen der Vase<\/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><mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>y<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>z<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-rel\">\u2208<\/mo> <msup><mrow><mi>\u211d<\/mi><\/mrow><mrow><mn>3<\/mn><\/mrow><\/msup><mo class=\"MathClass-rel\">\u2223<\/mo><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> [<\/mo><mrow><mo class=\"MathClass-bin\">\u2212<\/mo><mi>\u03c0<\/mi><mo class=\"MathClass-punc\">,<\/mo><mn>2<\/mn><mi>\u03c0<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">]<\/mo><\/mrow><mo class=\"MathClass-punc\">,<\/mo><mspace class=\"nbsp\" width=\"0.33em\" \/><mn>0<\/mn> <mo class=\"MathClass-rel\">\u2264<\/mo><msqrt><mrow><msup><mrow><mi>y<\/mi><\/mrow><mrow><mn>2<\/mn> <\/mrow> <\/msup> <mo class=\"MathClass-bin\">+<\/mo> <msup><mrow><mi>z<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><\/mrow><\/msqrt> <mo class=\"MathClass-rel\">\u2264<\/mo><mi class=\"qopname\"> sin<\/mi><mo>  <\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-bin\">+<\/mo> <mn>2<\/mn><\/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> <\/div> <div class=\"me meexample\"> <p class=\"indent\"><\/p><h4 id=\"z95caeeba2791\"> <span class=\"ecbx-1095\">\u00dc<\/span><span class=\"ecbx-1095\">bung <\/span>(Gleichm\u00e4ssige Konvergenz der Ableitungen)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Sei <\/span><math display=\"inline\"><msub><mrow><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>f<\/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\">eine Folge stetig  differenzierbarer  reellwertiger  Funktionen  auf  einem  kompakten  Intervall<\/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\">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> <span class=\"ecti-1095\">Angenommen die Folge <\/span><math display=\"inline\"><msub><mrow><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>f<\/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 gleichm<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">ssig gegen eine Funktion <\/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\">und die Folge der Ableitungen <\/span><math display=\"inline\"><msub><mrow><mo class=\"MathClass-open\">(<\/mo><msubsup><mrow><mi>f<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msubsup><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><\/math> <span class=\"ecti-1095\">konvergiert gleichm<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">ssig gegen eine Funktion <\/span><span class=\"maperiod\"><math display=\"inline\"><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><span class=\"period\">.<\/span> <span class=\"ecti-1095\">Zeigen Sie, dass <\/span><math display=\"inline\"><mi>f<\/mi><\/math> <span class=\"ecti-1095\">stetig differenzierbar ist mit <\/span><span class=\"maperiod\"><math display=\"inline\"><msup><mrow><mi>f<\/mi><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup> <mo class=\"MathClass-rel\">=<\/mo> <mi>g<\/mi><\/math><\/span><span class=\"period\">.<\/span> <\/p> <\/div> <div class=\"me meexample\"> <p class=\"indent\"><\/p><h4 id=\"z3e79986ba3c1\"> <span class=\"ecbx-1095\">\u00dc<\/span><span class=\"ecbx-1095\">bung.<\/span><\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Wir verwenden obige <\/span><span class=\"ecti-1095\">\u00dc<\/span><span class=\"ecti-1095\">bung, um eine glatte Funktion zu konstruieren, deren Taylorreihe um<\/span> <span class=\"ecti-1095\">einen Punkt Konvergenzradius Null hat.<\/span> <\/p><dl class=\"enumerate\"><dt class=\"enumerate\"> <span class=\"ecti-1095\">(i)<\/span><\/dt><dd class=\"enumerate\"><span class=\"ecti-1095\">Zeigen Sie, dass die Reihe <\/span><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><msup><mrow><mi class=\"qopname\">e<\/mi><mo>  <\/mo><\/mrow><mrow><mo class=\"MathClass-bin\">\u2212<\/mo><mi>n<\/mi><\/mrow><\/msup><mi class=\"qopname\"> cos<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><msup><mrow><mi>n<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><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> <mi>\u211d<\/mi><\/math> <span class=\"ecti-1095\">absolut konvergiert.<\/span> <\/dd><dt class=\"enumerate\"> <span class=\"ecti-1095\">(ii)<\/span><\/dt><dd class=\"enumerate\"><span class=\"ecti-1095\">Nach (i) definieren wir die 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>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u211d<\/mi><mo class=\"MathClass-rel\">\u21a6<\/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><msup><mrow><mi class=\"qopname\">e<\/mi><mo>  <\/mo><\/mrow><mrow><mo class=\"MathClass-bin\">\u2212<\/mo><mi>n<\/mi><\/mrow><\/msup><mi class=\"qopname\"> cos<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><msup><mrow><mi>n<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><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\">Zeigen Sie, dass <\/span><math display=\"inline\"><mi>f<\/mi><\/math> <span class=\"ecti-1095\">glatt ist.<\/span> <\/p><\/dd><dt class=\"enumerate\"> <span class=\"ecti-1095\">(iii)<\/span><\/dt><dd class=\"enumerate\"><span class=\"ecti-1095\">Berechnen Sie die Taylorreihe von <\/span><math display=\"inline\"><mi>f<\/mi><\/math> <span class=\"ecti-1095\">um Null und deren Konvergenzradius.<\/span><\/dd><\/dl> <\/div> <div class=\"me meexample\"> <p class=\"indent\"><\/p><h4 id=\"z1d045f3b32e4\"> <span class=\"ecbx-1095\">\u00dc<\/span><span class=\"ecbx-1095\">bung <\/span>(Kr\u00fcmmung ebener Kurven)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">In dieser <\/span><span class=\"ecti-1095\">\u00dc<\/span><span class=\"ecti-1095\">bung m<\/span><span class=\"ecti-1095\">\u00f6<\/span><span class=\"ecti-1095\">chten wir die Kr<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">mmung ebener Kurven betrachten. Alle Kurven,<\/span> <span class=\"ecti-1095\">die wir dabei betrachten wollen, sollen zweimal differenzierbar, regul<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">r und einfach<\/span> <span class=\"ecti-1095\">sein, hier der Einfachheit vorerst inklusive den Endpunkten. F<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r eine solche Kurve<\/span> <math display=\"inline\"><mi>\u03b3<\/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> <msup><mrow><mi>\u211d<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><\/math> <span class=\"ecti-1095\">und ein<\/span> <math display=\"inline\"><mi>p<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <msup><mrow><mi>\u211d<\/mi><\/mrow><mrow><mn>2<\/mn> <\/mrow> <\/msup> <\/math> <span class=\"ecti-1095\">auf der Kurve sei<\/span> <math display=\"inline\"><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><\/math> <span class=\"ecti-1095\">der eindeutige<\/span> <span class=\"ecti-1095\">Zeitpunkt mit <\/span><math display=\"inline\"><mi>\u03b3<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>t<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <mi>p<\/mi><\/math><span class=\"ecti-1095\">. Dann<\/span> <span class=\"ecti-1095\">ist die Kr<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">mmung von <\/span><math display=\"inline\"><mi>\u03b3<\/mi><\/math> <span class=\"ecti-1095\">bei <\/span><math display=\"inline\"><mi>p<\/mi><\/math> <span class=\"ecti-1095\">definiert als<\/span> <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><msub><mrow><mi>\u03ba<\/mi><\/mrow><mrow><mi>\u03b3<\/mi><\/mrow><\/msub> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>p<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-rel\">=<\/mo> <mfrac><mrow><mo class=\"MathClass-open\">\u27e8<\/mo><mover accent=\"true\"><mrow><mi>\u03b3<\/mi><\/mrow><mo accent=\"true\">\u02d9<\/mo><\/mover><mo class=\"MathClass-open\">(<\/mo><mi>t<\/mi><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-punc\">,<\/mo><mi>R<\/mi><mover accent=\"true\"><mrow><mi>\u03b3<\/mi><\/mrow><mo accent=\"true\">\u00a8<\/mo><\/mover><mo class=\"MathClass-open\">(<\/mo><mi>t<\/mi><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-close\">\u27e9<\/mo><\/mrow> <mrow><mo class=\"MathClass-rel\">\u2225<\/mo><mover accent=\"true\"><mrow><mi>\u03b3<\/mi><\/mrow><mo accent=\"true\">\u02d9<\/mo><\/mover><mo class=\"MathClass-open\">(<\/mo><mi>t<\/mi><mo class=\"MathClass-close\">)<\/mo><msup><mrow><mo class=\"MathClass-rel\">\u2225<\/mo><\/mrow><mrow><mn>3<\/mn><\/mrow><\/msup><\/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\">wobei <\/span><math display=\"inline\"><mi>R<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mstyle><mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow \/><mo fence=\"true\" form=\"postfix\" \/><\/mrow><\/mstyle> <mstyle class=\"text\"><mtext \/><mstyle class=\"math\"><mtable align=\"axis\" class=\"array\" columnlines=\"none none none none none none none none none\" equalcolumns=\"false\" equalrows=\"false\"> <mtr><mtd class=\"array\" columnalign=\"center\"> <mn>0<\/mn> <\/mtd><mtd class=\"array\" columnalign=\"center\"><mn>1<\/mn><\/mtd><\/mtr> <mtr><mtd class=\"array\" columnalign=\"center\"><mo class=\"MathClass-bin\">\u2212<\/mo><mn>1<\/mn><\/mtd> <mtd class=\"array\" columnalign=\"center\"><mn>0<\/mn><\/mtd><\/mtr> <\/mtable> <\/mstyle><mtext>&nbsp;<\/mtext><\/mstyle> <mstyle><mrow><mo fence=\"true\" form=\"prefix\"> )<\/mo><mrow \/><mo fence=\"true\" form=\"postfix\" \/><\/mrow><\/mstyle> <\/math><span class=\"ecti-1095\">. Ist<\/span> <math display=\"inline\"><mi>\u03b3<\/mi><\/math> <span class=\"ecti-1095\">einfach, aber<\/span> <span class=\"ecti-1095\">erf<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">llt <\/span><math display=\"inline\"><mi>\u03b3<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>a<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <mi>\u03b3<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>b<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math><span class=\"ecti-1095\">, so reicht es<\/span> <span class=\"ecti-1095\">anzunehmen, dass <\/span><math display=\"inline\"><mover accent=\"true\"><mrow><mi>\u03b3<\/mi><\/mrow><mo accent=\"true\">\u02d9<\/mo><\/mover><mo class=\"MathClass-open\">(<\/mo><mi>a<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <mover accent=\"true\"><mrow><mi>\u03b3<\/mi><\/mrow><mo accent=\"true\">\u02d9<\/mo><\/mover><mo class=\"MathClass-open\">(<\/mo><mi>b<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math> <span class=\"ecti-1095\">und <\/span><math display=\"inline\"><mover accent=\"true\"><mrow><mi>\u03b3<\/mi><\/mrow><mo accent=\"true\">\u00a8<\/mo><\/mover> <mo class=\"MathClass-open\">(<\/mo><mi>a<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <mover accent=\"true\"><mrow><mi>\u03b3<\/mi><\/mrow><mo accent=\"true\">\u00a8<\/mo><\/mover><mo class=\"MathClass-open\">(<\/mo><mi>b<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math> <span class=\"ecti-1095\">gilt, damit obiger Ausdruck f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r die Kr<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">mmung Sinn ergibt.<\/span> <\/p><dl class=\"enumerate\"><dt class=\"enumerate\"> <span class=\"ecti-1095\">(i)<\/span><\/dt><dd class=\"enumerate\"><span class=\"ecti-1095\">Berechnen Sie die Kr<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">mmung der Kurven <\/span><math display=\"inline\"><mi>t<\/mi><mo class=\"MathClass-rel\">\u21a6<\/mo><mo class=\"MathClass-open\">(<\/mo><mi class=\"qopname\">cos<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>t<\/mi><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-punc\">,<\/mo><mi class=\"qopname\">sin<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>t<\/mi><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-close\">)<\/mo><\/math> <span class=\"ecti-1095\">und <\/span><math display=\"inline\"><mi>t<\/mi><mo class=\"MathClass-rel\">\u21a6<\/mo> <mo class=\"MathClass-open\">(<\/mo><mi class=\"qopname\">cosh<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>t<\/mi><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-punc\">,<\/mo><mi class=\"qopname\">sinh<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>t<\/mi><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-close\">)<\/mo><\/math> <span class=\"ecti-1095\">(definiert auf geeigneten Intervallen).<\/span> <\/dd><dt class=\"enumerate\"> <span class=\"ecti-1095\">(ii)<\/span><\/dt><dd class=\"enumerate\"><span class=\"ecti-1095\">Zeigen Sie, dass die Kr<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">mmung unabh<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">ngig ist von der Parametrisierung, das heisst,<\/span> <span class=\"ecti-1095\">dass f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r jede Reparametrisierung <\/span><math display=\"inline\"><mi>\u03b3<\/mi> <mo class=\"MathClass-bin\">\u2218<\/mo> <mi>\u03c8<\/mi><\/math> <span class=\"ecti-1095\">einer Kurve <\/span><math display=\"inline\"><mi>\u03b3<\/mi><\/math> <span class=\"ecti-1095\">wie oben gilt <\/span><math display=\"inline\"><msub><mrow><mi>\u03ba<\/mi><\/mrow><mrow><mi>\u03b3<\/mi><mo class=\"MathClass-bin\">\u2218<\/mo><mi>\u03c8<\/mi><\/mrow><\/msub><mo class=\"MathClass-open\">(<\/mo><mi>p<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <msub><mrow><mi>\u03ba<\/mi><\/mrow><mrow><mi>\u03b3<\/mi><\/mrow><\/msub><mo class=\"MathClass-open\">(<\/mo><mi>p<\/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 Punkte <\/span><math display=\"inline\"><mi>p<\/mi><\/math> <span class=\"ecti-1095\">auf der Kurve.<\/span><\/dd><\/dl> <\/div> <div class=\"me meexample\"> <p class=\"indent\"><\/p><h4 id=\"z8fabecc22df7\"> <span class=\"ecbx-1095\">\u00dc<\/span><span class=\"ecbx-1095\">bung <\/span>(Existenz von Kurven vorgegebener Kr\u00fcmmung)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Wie in vorheriger <\/span><span class=\"ecti-1095\">\u00dc<\/span><span class=\"ecti-1095\">bung m<\/span><span class=\"ecti-1095\">\u00f6<\/span><span class=\"ecti-1095\">chten wir hier die Kr<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">mmung ebener Kurven betrachten,<\/span> <span class=\"ecti-1095\">aber dabei zulassen, dass die betrachteten Kurven nicht einfach sind, womit f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r eine regul<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">re<\/span> <span class=\"ecti-1095\">Kurve <\/span><math display=\"inline\"><mi>\u03b3<\/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> <msup><mrow><mi>\u211d<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><\/math> <span class=\"ecti-1095\">die Kr<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">mmung <\/span><math display=\"inline\"><msub><mrow><mi>\u03ba<\/mi><\/mrow><mrow><mi>\u03b3<\/mi><\/mrow><\/msub><\/math> <span class=\"ecti-1095\">definiert ist als Funktion auf <\/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\">via <\/span><span class=\"maperiod\"><math display=\"inline\"><msub><mrow><mi>\u03ba<\/mi><\/mrow><mrow><mi>\u03b3<\/mi> <\/mrow> <\/msub> <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> <mo class=\"MathClass-open\">\u27e8<\/mo><mover accent=\"true\"><mrow><mi>\u03b3<\/mi><\/mrow><mo accent=\"true\">\u02d9<\/mo><\/mover><mo class=\"MathClass-open\">(<\/mo><mi>t<\/mi><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-punc\">,<\/mo><mi>R<\/mi><mover accent=\"true\"><mrow><mi>\u03b3<\/mi><\/mrow><mo accent=\"true\">\u00a8<\/mo><\/mover><mo class=\"MathClass-open\">(<\/mo><mi>t<\/mi><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-close\">\u27e9<\/mo><\/mrow> <mrow><mo class=\"MathClass-rel\">\u2225<\/mo><mover accent=\"true\"><mrow><mi>\u03b3<\/mi><\/mrow><mo accent=\"true\">\u02d9<\/mo><\/mover><mo class=\"MathClass-open\">(<\/mo><mi>t<\/mi><mo class=\"MathClass-close\">)<\/mo><msup><mrow><mo class=\"MathClass-rel\">\u2225<\/mo><\/mrow><mrow><mn>3<\/mn><\/mrow><\/msup><\/mrow><\/mfrac> <\/math><\/span><span class=\"period\">.<\/span> <\/p><p class=\"indent\"><span class=\"ecti-1095\">Sei nun <\/span><math display=\"inline\"><mi>\u03ba<\/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 beliebige stetige<\/span> <span class=\"ecti-1095\">Funktion auf einem Intervall <\/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\">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> <span class=\"ecti-1095\">Wir m<\/span><span class=\"ecti-1095\">\u00f6<\/span><span class=\"ecti-1095\">chten hier zeigen, dass eine zweimal stetig differenzierbare Kurve<\/span> <math display=\"inline\"><mi>\u03b3<\/mi><\/math> <span class=\"ecti-1095\">mit<\/span> <span class=\"ecti-1095\">Kr<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">mmungsfunktion <\/span><math display=\"inline\"><mi>\u03ba<\/mi><\/math> <span class=\"ecti-1095\">existiert. Dazu betrachten wir die Funktion<\/span> <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mi>\ud835\udf03<\/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> <msup><mrow><mi>\u211d<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><mo class=\"MathClass-punc\">,<\/mo><mspace class=\"nbsp\" width=\"0.33em\" \/><mi>s<\/mi><mo class=\"MathClass-rel\">\u21a6<\/mo><msubsup><mrow><mo>\u222b  <\/mo><\/mrow><mrow><mi>a<\/mi><\/mrow><mrow><mi>s<\/mi><\/mrow><\/msubsup><mi>\u03ba<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>r<\/mi><mo class=\"MathClass-close\">)<\/mo><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>r<\/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\">und setzen<\/span> <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mi>\u03b3<\/mi> <mo class=\"MathClass-punc\">:<\/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\">\u2192<\/mo> <msup><mrow><mi>\u211d<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><mo class=\"MathClass-punc\">,<\/mo><mspace class=\"nbsp\" width=\"0.33em\" \/><mi>t<\/mi><mo class=\"MathClass-rel\">\u21a6<\/mo><msubsup><mrow><mo>\u222b  <\/mo><\/mrow><mrow><mi>a<\/mi><\/mrow><mrow><mi>t<\/mi><\/mrow><\/msubsup> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi class=\"qopname\">cos<\/mi><mo>  <\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>\ud835\udf03<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>s<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><mo class=\"MathClass-punc\">,<\/mo><mi class=\"qopname\">sin<\/mi><mo>  <\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>\ud835\udf03<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>s<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>s<\/mi><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\">Zeigen Sie, dass die Komponenten von <\/span><math display=\"inline\"><mi>\u03b3<\/mi><\/math> <span class=\"ecti-1095\">jeweils zweimal stetig differenzierbar ist und dass<\/span> <math display=\"inline\"><mi>\u03b3<\/mi><\/math> <span class=\"ecti-1095\">eine<\/span> <span class=\"ecti-1095\">regul<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">re, nach Bogenl<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">nge parametrisierte Kurve ist. Verifizieren Sie anschliessend, dass<\/span> <math display=\"inline\"><msub><mrow><mi>\u03ba<\/mi><\/mrow><mrow><mi>\u03b3<\/mi> <\/mrow> <\/msub> <mo class=\"MathClass-rel\">=<\/mo> <mi>\u03ba<\/mi><\/math> <span class=\"ecti-1095\">gilt.<\/span> <\/p> <\/div> <div class=\"me meexample\"> <p class=\"indent\"><\/p><h4 id=\"zdaf019c96667\"> <span class=\"ecbx-1095\">\u00dc<\/span><span class=\"ecbx-1095\">bung <\/span>(Irrationalit\u00e4t der Kreiszahl)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">In dieser           <\/span><span class=\"ecti-1095\">\u00dc<\/span><span class=\"ecti-1095\">bung           m<\/span><span class=\"ecti-1095\">\u00f6<\/span><span class=\"ecti-1095\">chten           wir           zeigen,           dass<\/span> <math display=\"inline\"><mi>\u03c0<\/mi><\/math> <span class=\"ecti-1095\">irrational ist, wobei wir dem Beweis von Niven <\/span><span class=\"cite\"><span class=\"ecti-1095\">[<\/span><a href=\"#XpiirrationalNiven\"><span class=\"ecti-1095\">Niv47<\/span><\/a><span class=\"ecti-1095\">]<\/span><\/span> <span class=\"ecti-1095\">folgen werden.<\/span> <\/p><p class=\"indent\"><span class=\"ecti-1095\">Per Widerspruch wollen wir annehmen, dass<\/span> <math display=\"inline\"><mi>\u03c0<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mfrac> <mrow> <mi>a<\/mi><\/mrow> <mrow><mi>b<\/mi><\/mrow><\/mfrac><\/math> <span class=\"ecti-1095\">ist f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r<\/span> <math display=\"inline\"><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo> <mi>b<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2115<\/mi><\/math><span class=\"ecti-1095\">. Nun<\/span> <span class=\"ecti-1095\">betrachten wir die Polynome<\/span> <\/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> <mfrac><mrow><msup><mrow><mi>x<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msup><msup><mrow><mo class=\"MathClass-open\">(<\/mo><mi>a<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>b<\/mi><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> <mo class=\"MathClass-punc\">,<\/mo><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>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><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">\u2212<\/mo> <msup><mrow><mi>f<\/mi><\/mrow><mrow><mo class=\"MathClass-open\">(<\/mo><mn>2<\/mn><mo class=\"MathClass-close\">)<\/mo><\/mrow><\/msup><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">+<\/mo> <msup><mrow><mi>f<\/mi><\/mrow><mrow><mo class=\"MathClass-open\">(<\/mo><mn>4<\/mn><mo class=\"MathClass-close\">)<\/mo><\/mrow><\/msup><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">\u2212<\/mo><mi class=\"MathClass-op\">\u2026<\/mi><mo> <\/mo> <mo class=\"MathClass-bin\">+<\/mo> <msup><mrow><mo class=\"MathClass-open\">(<\/mo><mo class=\"MathClass-bin\">\u2212<\/mo><mn>1<\/mn><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msup><msup><mrow><mi>f<\/mi><\/mrow><mrow><mo class=\"MathClass-open\">(<\/mo><mn>2<\/mn><mi>n<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow><\/msup><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/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\">f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r ein <\/span><span class=\"maperiod\"><math display=\"inline\"><mi>n<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2115<\/mi><\/math><\/span><span class=\"period\">,<\/span> <span class=\"ecti-1095\">welches wir sp<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">ter w<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">hlen werden.<\/span> <\/p><dl class=\"enumerate\"><dt class=\"enumerate\"> <span class=\"ecti-1095\">(i)<\/span><\/dt><dd class=\"enumerate\"><span class=\"ecti-1095\">Begr<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">nden Sie, wieso <\/span><span class=\"maperiod\"><math display=\"inline\"><mi>f<\/mi><\/math><\/span><span class=\"period\">,<\/span> <span class=\"ecti-1095\">alle Ableitungen von <\/span><math display=\"inline\"><mi>f<\/mi><\/math> <span class=\"ecti-1095\">und <\/span><math display=\"inline\"><mi>F<\/mi><\/math> <span class=\"ecti-1095\">bei <\/span><math display=\"inline\"><mn>0<\/mn><\/math> <span class=\"ecti-1095\">und bei <\/span><math display=\"inline\"><mi>\u03c0<\/mi><\/math> <span class=\"ecti-1095\">ganzzahlige Werte annehmen. Verifizieren Sie des Weiteren, dass <\/span><math display=\"inline\"><mi>f<\/mi><msub><mrow><mo class=\"MathClass-rel\">|<\/mo><\/mrow><mrow><mo class=\"MathClass-open\">[<\/mo><mn>0<\/mn><mo class=\"MathClass-punc\">,<\/mo><mi>\u03c0<\/mi><mo class=\"MathClass-close\">]<\/mo><\/mrow><\/msub><\/math> <span class=\"ecti-1095\">eine nicht-negative Funktion ist, welche genau bei <\/span><math display=\"inline\"><mn>0<\/mn><\/math> <span class=\"ecti-1095\">und <\/span><math display=\"inline\"><mi>\u03c0<\/mi><\/math> <span class=\"ecti-1095\">verschwindet.<\/span> <\/dd><dt class=\"enumerate\"> <span class=\"ecti-1095\">(ii)<\/span><\/dt><dd class=\"enumerate\"><span class=\"ecti-1095\">Zeigen Sie, dass<\/span> <math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><msup><mrow><mo class=\"MathClass-open\">(<\/mo><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><mi class=\"qopname\">sin<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>F<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><mi class=\"qopname\">cos<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup> <mo class=\"MathClass-rel\">=<\/mo> <mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><mi class=\"qopname\">sin<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/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\">und <\/span><span class=\"maperiod\"><math display=\"inline\"><msubsup><mrow><mi class=\"MathClass-op\"> \u222b  <\/mi><mo> <\/mo><\/mrow><mrow><mn>0<\/mn><\/mrow><mrow><mi>\u03c0<\/mi><\/mrow><\/msubsup><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><mi class=\"qopname\">sin<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mi>F<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>\u03c0<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">+<\/mo> <mi>F<\/mi><mo class=\"MathClass-open\">(<\/mo><mn>0<\/mn><mo class=\"MathClass-close\">)<\/mo><\/math><\/span><span class=\"period\">.<\/span> <\/p><\/dd><dt class=\"enumerate\"> <span class=\"ecti-1095\">(iii)<\/span><\/dt><dd class=\"enumerate\"><span class=\"ecti-1095\">Schliessen Sie auf einen Widerspruch.<\/span><\/dd><\/dl> <\/div> <div class=\"me meexample\"> <p class=\"indent\"><\/p><h4 id=\"za3eff3dcf845\"> <span class=\"ecbx-1095\">\u00dc<\/span><span class=\"ecbx-1095\">bung <\/span>(Summe der Reziproken der Primzahlen)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">In dieser <\/span><span class=\"ecti-1095\">\u00dc<\/span><span class=\"ecti-1095\">bung m<\/span><span class=\"ecti-1095\">\u00f6<\/span><span class=\"ecti-1095\">chten wir f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r nat<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">rliche Zahlen<\/span> <math display=\"inline\"><mi>n<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2115<\/mi><\/math> <span class=\"ecti-1095\">die Summe<\/span> <math display=\"inline\"><msub><mrow><mi class=\"MathClass-op\">\u2211<\/mi><mo> <\/mo> <\/mrow><mrow><mi>p<\/mi><mo class=\"MathClass-rel\">\u2208<\/mo><mi>\u2119<\/mi><mo class=\"MathClass-punc\">:<\/mo><mi>p<\/mi><mo class=\"MathClass-rel\">\u2264<\/mo><mi>n<\/mi><\/mrow><\/msub><mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mi>p<\/mi><\/mrow><\/mfrac><\/math> <span class=\"ecti-1095\">betrachten,<\/span> <span class=\"ecti-1095\">wobei <\/span><math display=\"inline\"><mi>\u2119<\/mi> <mo class=\"MathClass-rel\">\u2286<\/mo> <mi>\u2115<\/mi><\/math> <span class=\"ecti-1095\">die Menge der Primzahlen bezeichnet. Dabei m<\/span><span class=\"ecti-1095\">\u00f6<\/span><span class=\"ecti-1095\">chten wir zeigen, dass<\/span> <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><munder class=\"msub\"><mrow><mo>\u2211<\/mo> <\/mrow><mrow><mi>p<\/mi><mo class=\"MathClass-rel\">\u2208<\/mo><mi>\u2119<\/mi><mo class=\"MathClass-punc\">:<\/mo><mi>p<\/mi><mo class=\"MathClass-rel\">\u2264<\/mo><mi>n<\/mi><\/mrow><\/munder><mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mi>p<\/mi><\/mrow><\/mfrac> <mo class=\"MathClass-rel\">\u2265<\/mo><mi class=\"qopname\"> log<\/mi><mo>  <\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi class=\"qopname\">log<\/mi><mo>  <\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>n<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-bin\">\u2212<\/mo><mi class=\"qopname\"> log<\/mi><mo>  <\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>C<\/mi><\/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\"><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\">\u2265<\/mo> <mn>3<\/mn><\/math> <span class=\"ecti-1095\">und<\/span> <span class=\"ecti-1095\">eine Konstante <\/span><span class=\"maperiod\"><math display=\"inline\"><mi>C<\/mi> <mo class=\"MathClass-rel\">&gt;<\/mo> <mn>1<\/mn><\/math><\/span><span class=\"period\">.<\/span> <span class=\"ecti-1095\">Insbesondere gibt es unendlich viele Primzahlen (wieso?). Die obige Ungleichung stellt eine (stark)<\/span> <span class=\"ecti-1095\">abgeschw<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">chte Version des zweiten Theorems von Mertens (und damit einen Vorreiter des<\/span> <span class=\"ecti-1095\">Primzahlsatzes) dar.<\/span> <\/p><dl class=\"enumerate\"><dt class=\"enumerate\"> <span class=\"ecti-1095\">(i)<\/span><\/dt><dd class=\"enumerate\"><span class=\"ecti-1095\">Verifizieren Sie 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\">die Ungleichung<\/span> <math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mi class=\"qopname\">exp<\/mi><mo>  <\/mo><mstyle><mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow \/><mo fence=\"true\" form=\"postfix\" \/><\/mrow><\/mstyle><munder class=\"msub\"><mrow><mo>\u2211<\/mo> <\/mrow><mrow><mi>p<\/mi><mo class=\"MathClass-rel\">\u2208<\/mo><mi>\u2119<\/mi><mo class=\"MathClass-punc\">:<\/mo><mi>p<\/mi><mo class=\"MathClass-rel\">\u2264<\/mo><mi>n<\/mi><\/mrow><\/munder><mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mi>p<\/mi><\/mrow><\/mfrac><mstyle><mrow><mo fence=\"true\" form=\"prefix\"> )<\/mo><mrow \/><mo fence=\"true\" form=\"postfix\" \/><\/mrow><\/mstyle> <mo class=\"MathClass-rel\">\u2265<\/mo><munder class=\"msub\"><mrow><mo>\u220f<\/mo> <\/mrow><mrow><mi>p<\/mi><mo class=\"MathClass-rel\">\u2208<\/mo><mi>\u2119<\/mi><mo class=\"MathClass-punc\">:<\/mo><mi>p<\/mi><mo class=\"MathClass-rel\">\u2264<\/mo><mi>n<\/mi><\/mrow><\/munder> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mn>1<\/mn> <mo class=\"MathClass-bin\">+<\/mo> <mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mi>p<\/mi><\/mrow><\/mfrac> <\/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> <\/dd><dt class=\"enumerate\"> <span class=\"ecti-1095\">(ii)<\/span><\/dt><dd class=\"enumerate\"><span class=\"ecti-1095\">Zeigen Sie 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> <math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><munderover accent=\"false\" accentunder=\"false\"><mrow><mo> \u2211<\/mo> <\/mrow><mrow><mi>k<\/mi><mo class=\"MathClass-rel\">=<\/mo><mn>1<\/mn><\/mrow><mrow><mi>n<\/mi><\/mrow><\/munderover><mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mi>k<\/mi><\/mrow><\/mfrac> <mo class=\"MathClass-rel\">\u2264<\/mo><mstyle><mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow \/><mo fence=\"true\" form=\"postfix\" \/><\/mrow><\/mstyle><munderover accent=\"false\" accentunder=\"false\"><mrow><mo>\u2211<\/mo> <\/mrow><mrow><mi>\u2113<\/mi><mo class=\"MathClass-rel\">=<\/mo><mn>1<\/mn><\/mrow><mrow><mi>n<\/mi><\/mrow><\/munderover> <mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><msup><mrow><mi>\u2113<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><\/mrow><\/mfrac><mstyle><mrow><mo fence=\"true\" form=\"prefix\"> )<\/mo><mrow \/><mo fence=\"true\" form=\"postfix\" \/><\/mrow><\/mstyle><munder class=\"msub\"><mrow><mo> \u220f<\/mo> <\/mrow><mrow><mi>p<\/mi><mo class=\"MathClass-rel\">\u2208<\/mo><mi>\u2119<\/mi><mo class=\"MathClass-punc\">:<\/mo><mi>p<\/mi><mo class=\"MathClass-rel\">\u2264<\/mo><mi>n<\/mi><\/mrow><\/munder> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mn>1<\/mn> <mo class=\"MathClass-bin\">+<\/mo> <mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mi>p<\/mi><\/mrow><\/mfrac> <\/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> <\/dd><dt class=\"enumerate\"> <span class=\"ecti-1095\">(iii)<\/span><\/dt><dd class=\"enumerate\"><span class=\"ecti-1095\">Schliessen Sie auf die Aussage.<\/span><\/dd><\/dl> <p class=\"indent\"><\/p><details><summary style=\"color:#FF7F00\"><span class=\"ecti-1095\">Hinweis.<\/span><\/summary><p class=\"indent\" style=\"margin-top: 0\"><span class=\"ecti-1095\">F<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r den zweiten Teil k<\/span><span class=\"ecti-1095\">\u00f6<\/span><span class=\"ecti-1095\">nnen Sie die Existenz und Eindeutigkeit einer<\/span> <span class=\"ecti-1095\">Primfaktorzerlegung verwenden, wonach sich insbesondere jede nat<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">rliche Zahl zwischen<\/span> <math display=\"inline\"><mn>1<\/mn><\/math> <span class=\"ecti-1095\">und<\/span> <math display=\"inline\"><mi>n<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2115<\/mi><\/math> <span class=\"ecti-1095\">als<\/span> <span class=\"ecti-1095\">Produkt einer Quadratzahl mit einer Zahl schreiben l<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">sst, in deren Faktorisierung jede Primzahl nur<\/span> <span class=\"ecti-1095\">einmal vorkommt. F<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r den letzten Teil k<\/span><span class=\"ecti-1095\">\u00f6<\/span><span class=\"ecti-1095\">nnen Sie <\/span><span class=\"ecti-1095\">\u00dc<\/span><span class=\"ecti-1095\">bung <\/span><a href=\"..\/..\/chapter\/das-uneigentliche-integral#x1-273008r36\"><span class=\"ecti-1095\">9.36<\/span><\/a> <span class=\"ecti-1095\">benutzen.<\/span><\/p><\/details>  <\/div> <p class=\"indent\"> <a id=\"x1-293014r258\"><\/a> <\/p> \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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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=\"z9975f39a572f\" class=\"sectionHead\"><span class=\"titlemark\">9.8 <\/span> <a id=\"x1-2910008\"><\/a>Weitere Lernmaterialien<\/h3> <a id=\"x1-291001r290\"><\/a> <h4 id=\"z7f74c770ed4b\" class=\"subsectionHead\"><span class=\"titlemark\">9.8.1 <\/span> <a id=\"x1-2920001\"><\/a>Verwendung des Kapitels<\/h4> <p class=\"noindent\">Dieses Kapitel verbindet mit Hilfe des Fundamentalsatz der Analysis die zentralen Begriffe der Ableitung und des Riemann-Integrals. Damit haben wir das vollst\u00e4ndige Arsenal an Ableitungsregeln auch f\u00fcr die Berechnung von Integralen verwenden k\u00f6nnen, womit die Berechnung von Integralen mitunter deutlich einfacher wurde \u2013 auch wenn dies zus\u00e4tzliche \u00dcbung erfordert. Umgekehrt haben wir aber auch mit dem Integralrestglied im Satz zur Taylor-Approximation gesehen, dass das Riemann-Integral n\u00fctzlich sein kann um den Zusammenhang zwischen den Ableitungen und der urspr\u00fcnglichen Funktion besser zu verstehen. Falls die symbolische Integration sich als nicht machbar erweist, so ist wiederum die Taylor-Approximation n\u00fctzlich um das Riemann-Integral numerisch \u2013 zum Beispiel mit der Simpson-Methode \u2013 mit \u00fcberraschend hoher Genauigkeit zu berechnen. Ebenso ist aber der Satz zur Taylor-Approximation auch von theoretischer Wichtigkeit, da wir mit diesen asymptotische Formeln wie zum Beispiel die Sterling-Formel beweisen k\u00f6nnen. <\/p><p class=\"indent\">Zusammenfassend k\u00f6nnen wir also sagen, dass dieses Kapitel den Aufbau der eindimensionalen Analysis vollendet. Die Inhalte dieses Kapitels bilden einen zentralen Bestandteil der Analysis I\/II Vorlesung und ihrer Anwendungen. <a id=\"x1-292001r292\"><\/a> <\/p> <h4 id=\"z83b0ca29874f\" class=\"subsectionHead\"><span class=\"titlemark\">9.8.2 <\/span> <a id=\"x1-2930002\"><\/a>\u00dcbungen<\/h4> <p class=\"noindent\">Da die Schwierigkeit beim Integrieren vor allem in der Auswahl der richtigen Methode liegt, wollen wir in folgender \u00dcbung noch einige weitere Aufgaben ohne Angabe der richtigen Technik auflisten. Dazu wollen wir noch erw\u00e4hnen, dass es oft auch mehr als eine Methode gibt, die zum Erfolg f\u00fchren kann. <\/p> <div class=\"me meexample\"> <div class=\"wp-nocaption \"><\/div><h4 id=\"z2fd64eb8de0a\"> <span class=\"ecbx-1095\">\u00dc<\/span><span class=\"ecbx-1095\">bung.<\/span><\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Berechnen Sie die unbestimmten Integrale<\/span> <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"> <mtable align=\"axis\" class=\"array\" columnlines=\"none\" equalcolumns=\"false\" equalrows=\"false\"> <mtr><mtd class=\"array\" columnalign=\"left\"><mo>\u222b  <\/mo> <mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><msqrt><mrow><msup><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn> <\/mrow> <\/msup> <mo class=\"MathClass-bin\">+<\/mo><mn>2<\/mn><mn>0<\/mn><mn>1<\/mn><msup><mrow><mn>7<\/mn><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><\/mrow><\/msqrt><\/mrow><\/mfrac><mspace class=\"thinspace\" width=\"0.17em\" \/> <mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi><mo class=\"MathClass-punc\">,<\/mo> <\/mtd><mtd class=\"array\" columnalign=\"left\"><mo>\u222b  <\/mo> <mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mi>x<\/mi><msqrt><mrow><mn>1<\/mn><mo class=\"MathClass-bin\">+<\/mo><msup><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><\/mrow><\/msqrt><\/mrow><\/mfrac><mspace class=\"thinspace\" width=\"0.17em\" \/> <mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi><mo class=\"MathClass-punc\">,<\/mo> <\/mtd> <\/mtr> <mtr><mtd class=\"array\" columnalign=\"left\"><mo>\u222b  <\/mo><mi class=\"qopname\">exp<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mroot><mrow><mi>x<\/mi><\/mrow><mrow><mn>3<\/mn><\/mrow><\/mroot><mo class=\"MathClass-close\">)<\/mo><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi><mo class=\"MathClass-punc\">,<\/mo> <\/mtd><mtd class=\"array\" columnalign=\"left\"><mo>\u222b  <\/mo> <mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mn>1<\/mn><mo class=\"MathClass-bin\">+<\/mo><mi class=\"qopname\">exp<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mo class=\"MathClass-bin\">\u2212<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow><\/mfrac><mspace class=\"thinspace\" width=\"0.17em\" \/> <mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi><mo class=\"MathClass-punc\">,<\/mo> <\/mtd> <\/mtr> <mtr><mtd class=\"array\" columnalign=\"left\"><mo>\u222b  <\/mo> <mfrac><mrow><mi>x<\/mi><\/mrow> <mrow><mn>1<\/mn><mo class=\"MathClass-bin\">+<\/mo><msqrt><mrow><mi>x<\/mi><\/mrow><\/msqrt><\/mrow><\/mfrac><mspace class=\"thinspace\" width=\"0.17em\" \/> <mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi><mo class=\"MathClass-punc\">,<\/mo> <\/mtd><mtd class=\"array\" columnalign=\"left\"><mo>\u222b  <\/mo><msup><mrow><mi class=\"qopname\">cos<\/mi><mo>  <\/mo><\/mrow><mrow><mn>5<\/mn><\/mrow><\/msup><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><msup><mrow><mi class=\"qopname\">sin<\/mi><mo>  <\/mo><\/mrow><mrow><mn>3<\/mn><\/mrow><\/msup><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi><mo class=\"MathClass-punc\">,<\/mo><\/mtd> <\/mtr> <mtr><mtd class=\"array\" columnalign=\"left\"><mo>\u222b  <\/mo><msup><mrow><mi class=\"qopname\">cos<\/mi><mo>  <\/mo><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><msup><mrow><mi class=\"qopname\">sin<\/mi><mo>  <\/mo><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi><mo class=\"MathClass-punc\">,<\/mo> <\/mtd><mtd class=\"array\" columnalign=\"left\"><mo>\u222b  <\/mo> <mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><msup><mrow><mi class=\"qopname\"> cos<\/mi><mo>  <\/mo> <\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><msup><mrow><mi class=\"qopname\"> sin<\/mi><mo>  <\/mo> <\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow><\/mfrac><mspace class=\"thinspace\" width=\"0.17em\" \/> <mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi><mo class=\"MathClass-punc\">,<\/mo> <\/mtd> <\/mtr> <mtr><mtd class=\"array\" columnalign=\"left\"><mo>\u222b  <\/mo><mfrac><mrow><mi class=\"qopname\"> log<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-bin\">+<\/mo><mi class=\"qopname\">sin<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi class=\"qopname\">log<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-close\">)<\/mo><msup><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><\/mrow> <mrow><msup><mrow><mi>x<\/mi><\/mrow><mrow><mn>3<\/mn><\/mrow><\/msup><\/mrow><\/mfrac> <mspace class=\"thinspace\" width=\"0.17em\" \/> <mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi><mo class=\"MathClass-punc\">,<\/mo><\/mtd><mtd class=\"array\" columnalign=\"left\"><mo>\u222b  <\/mo> <mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mi class=\"qopname\"> cos<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow><\/mfrac><mspace class=\"thinspace\" width=\"0.17em\" \/> <mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi> <\/mtd><\/mtr> <\/mtable> <\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <\/div> <div class=\"me meexample\"> <div class=\"wp-nocaption \"><\/div><h4 id=\"z80ee6d103c8c\"> <span class=\"ecbx-1095\">\u00dc<\/span><span class=\"ecbx-1095\">bung.<\/span><\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Zeigen Sie, dass jede stetig differenzierbare Funktion <\/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\">auf einem kompakten Intervall <\/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\">mit Endpunkten <\/span><math display=\"inline\"><mi>a<\/mi> <mo class=\"MathClass-rel\">&lt;<\/mo> <mi>b<\/mi><\/math> <span class=\"ecti-1095\">beschr<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">nkte Variation <\/span><math display=\"inline\"><mi>V<\/mi> <mo class=\"MathClass-open\">(<\/mo><mi>f<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math> <span class=\"ecti-1095\">(siehe die entsprechende <\/span><span class=\"ecti-1095\">\u00dc<\/span><span class=\"ecti-1095\">bung in Abschnitt<\/span><span class=\"ecti-1095\">&nbsp;<\/span><a href=\"..\/..\/chapter\/weitere-lernmaterialien#x1-1270002\"><span class=\"ecti-1095\">4.8.2<\/span><\/a><span class=\"ecti-1095\">) hat und dass <\/span><math display=\"inline\"><mi>V<\/mi> <mo class=\"MathClass-open\">(<\/mo><mi>f<\/mi><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">=<\/mo><msubsup><mrow><mi class=\"MathClass-op\"> \u222b  <\/mi><mo> <\/mo><\/mrow><mrow><mi>a<\/mi><\/mrow><mrow><mi>x<\/mi><\/mrow><\/msubsup><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><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>t<\/mi><\/math> <span class=\"ecti-1095\">gilt 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> <\/p><details><summary style=\"color:#FF7F00\"><span class=\"ecti-1095\">Hinweis.<\/span><\/summary><p class=\"indent\" style=\"margin-top: 0\"><span class=\"ecti-1095\">Gegeben eine Zerlegung <\/span><span class=\"maperiod\"><math display=\"inline\"><mi>\u2128<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> {<\/mo><mrow><mi>a<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub> <mo class=\"MathClass-rel\">&lt;<\/mo> <msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub> <mo class=\"MathClass-rel\">&lt;<\/mo> <mi class=\"MathClass-op\">\u2026<\/mi><mo> <\/mo> <mo class=\"MathClass-rel\">&lt;<\/mo> <msub><mrow><mi>x<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub> <mo class=\"MathClass-rel\">=<\/mo> <mi>b<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">}<\/mo><\/mrow><\/math><\/span><span class=\"period\">,<\/span> <span class=\"ecti-1095\">wenden Sie den Mittelwertsatz auf den Ausdruck <\/span><math display=\"inline\"><msubsup><mrow><mi class=\"MathClass-op\">\u2211<\/mi><mo> <\/mo> <\/mrow><mrow><mi>i<\/mi><mo class=\"MathClass-rel\">=<\/mo><mn>1<\/mn><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msubsup><mo class=\"MathClass-rel\">|<\/mo><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mi>i<\/mi><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mi>i<\/mi><mo class=\"MathClass-bin\">\u2212<\/mo><mn>1<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-rel\">|<\/mo><\/math> <span class=\"ecti-1095\">an und betrachten Sie dann die richtige Riemann-Summe.<\/span><\/p><\/details>  <\/div> <div class=\"me meexample\"> <div class=\"wp-nocaption \"><\/div><h4 id=\"za88e06db2c5c\"> <span class=\"ecbx-1095\">\u00dc<\/span><span class=\"ecbx-1095\">bung <\/span>(Abel-Summation und Partielle Integration)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Wir wollen in dieser <\/span><span class=\"ecti-1095\">\u00dc<\/span><span class=\"ecti-1095\">bung den Zusammenhang zwischen der Abel-Summation und der<\/span> <span class=\"ecti-1095\">partiellen Integration erkl<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">ren. Sei also<\/span><span class=\"ecti-1095\">&nbsp;<\/span><math display=\"inline\"><mi>a<\/mi> <mo class=\"MathClass-rel\">&lt;<\/mo> <mi>b<\/mi><\/math> <span class=\"ecti-1095\">und seien<\/span><span class=\"ecti-1095\">&nbsp;<\/span><math display=\"inline\"><mi>u<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>v<\/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\">zwei stetig differenzierbare Funktionen. Verwenden Sie die Abel-Summation von <\/span><span class=\"ecti-1095\">\u00dc<\/span><span class=\"ecti-1095\">bung <\/span><a href=\"..\/..\/chapter\/summen-und-produkte#x1-78001r3\"><span class=\"ecti-1095\">3.3<\/span><\/a><span class=\"ecti-1095\">,<\/span> <span class=\"ecti-1095\">um die partielle Integration<\/span><span class=\"ecti-1095\">&nbsp;<\/span><math display=\"inline\"><msubsup><mrow><mi class=\"MathClass-op\">\u222b  <\/mi><mo> <\/mo><\/mrow><mrow><mi>a<\/mi><\/mrow><mrow><mi>b<\/mi><\/mrow><\/msubsup><mi>u<\/mi><msup><mrow><mi>v<\/mi><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <msubsup><mrow><mo class=\"MathClass-open\">[<\/mo><mi>u<\/mi><mi>v<\/mi><mo class=\"MathClass-close\">]<\/mo><\/mrow><mrow><mi>a<\/mi><\/mrow><mrow><mi>b<\/mi><\/mrow><\/msubsup> <mo class=\"MathClass-bin\">\u2212<\/mo><msubsup><mrow><mi class=\"MathClass-op\">\u222b  <\/mi><mo> <\/mo><\/mrow><mrow><mi>a<\/mi><\/mrow><mrow><mi>b<\/mi><\/mrow><\/msubsup><msup><mrow><mi>u<\/mi><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup><mi>v<\/mi><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi><\/math> <span class=\"ecti-1095\">zu beweisen.<\/span> <\/p><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\">Erkl<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">ren Sie zuerst, wieso Sie ohne Beschr<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">nkung der Allgemeinheit<\/span> <math display=\"inline\"><mi>u<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>b<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <mi>v<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>a<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <mn>0<\/mn><\/math> <span class=\"ecti-1095\">annehmen<\/span> <span class=\"ecti-1095\">k<\/span><span class=\"ecti-1095\">\u00f6<\/span><span class=\"ecti-1095\">nnen. Sei<\/span><span class=\"ecti-1095\">&nbsp;<\/span><math display=\"inline\"><mi>\u2128<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> {<\/mo><mrow><mi>a<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub> <mo class=\"MathClass-rel\">&lt;<\/mo> <msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub> <mo class=\"MathClass-rel\">&lt;<\/mo> <mi class=\"MathClass-op\">\u2026<\/mi><mo> <\/mo> <mo class=\"MathClass-rel\">&lt;<\/mo> <msub><mrow><mi>x<\/mi><\/mrow><mrow><mi>n<\/mi><mo class=\"MathClass-bin\">\u2212<\/mo><mn>1<\/mn><\/mrow><\/msub> <mo class=\"MathClass-rel\">&lt;<\/mo> <msub><mrow><mi>x<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub> <mo class=\"MathClass-rel\">=<\/mo> <mi>b<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">}<\/mo><\/mrow><\/math> <span class=\"ecti-1095\">eine<\/span> <span class=\"ecti-1095\">Zerlegung von<\/span><span class=\"ecti-1095\">&nbsp;<\/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\">in<\/span><span class=\"ecti-1095\">&nbsp;<\/span><math display=\"inline\"><mi>n<\/mi> <mo class=\"MathClass-rel\">\u2265<\/mo> <mn>1<\/mn><\/math> <span class=\"ecti-1095\">viele Teilintervalle<\/span> <span class=\"ecti-1095\">der L<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">nge<\/span><span class=\"ecti-1095\">&nbsp;<\/span><span class=\"maperiod\"><math display=\"inline\"><mfrac><mrow><mi>b<\/mi><mo class=\"MathClass-bin\">\u2212<\/mo><mi>a<\/mi><\/mrow> <mrow><mi>n<\/mi><\/mrow><\/mfrac> <\/math><\/span><span class=\"period\">.<\/span> <span class=\"ecti-1095\">Verwenden Sie Abel-Summation auf die Summe<\/span> <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><munderover accent=\"false\" accentunder=\"false\"><mrow><mo>\u2211<\/mo> <\/mrow><mrow><mi>k<\/mi><mo class=\"MathClass-rel\">=<\/mo><mn>1<\/mn><\/mrow><mrow><mi>n<\/mi><\/mrow><\/munderover><mi>u<\/mi><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow> <mi>k<\/mi><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-open\">(<\/mo><mi>v<\/mi><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mi>k<\/mi><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>v<\/mi><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mi>k<\/mi><mo class=\"MathClass-bin\">\u2212<\/mo><mn>1<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-close\">)<\/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<\/span><span class=\"ecti-1095\">&nbsp;<\/span><math display=\"inline\"><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>k<\/mi> <\/mrow> <\/msub> <mo class=\"MathClass-rel\">=<\/mo> <mo class=\"MathClass-open\">(<\/mo><mi>v<\/mi><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mi>k<\/mi><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>v<\/mi><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mi>k<\/mi><mo class=\"MathClass-bin\">\u2212<\/mo><mn>1<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-close\">)<\/mo><\/math> <span class=\"ecti-1095\">und<\/span><span class=\"ecti-1095\">&nbsp;<\/span><math display=\"inline\"><msub><mrow><mi>b<\/mi><\/mrow><mrow><mi>k<\/mi> <\/mrow> <\/msub> <mo class=\"MathClass-rel\">=<\/mo> <mi>u<\/mi><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mi>k<\/mi><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><\/math><span class=\"ecti-1095\">. Verwenden Sie anschliessend<\/span> <span class=\"ecti-1095\">den Mittelwertsatz auf<\/span><span class=\"ecti-1095\">&nbsp;<\/span><math display=\"inline\"><mi>v<\/mi><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mi>k<\/mi><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>v<\/mi><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mi>k<\/mi><mo class=\"MathClass-bin\">\u2212<\/mo><mn>1<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><\/math> <span class=\"ecti-1095\">an und vergleichen Sie die resultierende Summen mit der Riemann-Summen f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r das Integral<\/span> <span class=\"maperiod\"><math display=\"inline\"><msubsup><mrow><mi class=\"MathClass-op\">\u222b  <\/mi><mo> <\/mo><\/mrow><mrow><mi>a<\/mi><\/mrow><mrow><mi>b<\/mi><\/mrow><\/msubsup><mi>u<\/mi><msup><mrow><mi>v<\/mi><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi><\/math><\/span><span class=\"period\">.<\/span><\/p><\/details>  <\/div> <div class=\"me meexample\"> <div class=\"wp-nocaption \"><\/div><h4 id=\"zbafa692a9bd4\"> <span class=\"ecbx-1095\">\u00dc<\/span><span class=\"ecbx-1095\">bung <\/span>(Ein alternativer Beweis von Proposition <a href=\"..\/..\/chapter\/anwendungen#x1-116007r30\">4.30<\/a>)<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 hier einen Beweis von Proposition <\/span><a href=\"..\/..\/chapter\/anwendungen#x1-116007r30\"><span class=\"ecti-1095\">4.30<\/span><\/a> <span class=\"ecti-1095\">unter der Annahme, dass<\/span> <math display=\"inline\"><mi>f<\/mi><\/math> <span class=\"ecti-1095\">stetig<\/span> <span class=\"ecti-1095\">ist, durchf<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">hren. Gehen Sie wie folgt vor:<\/span> <\/p><dl class=\"enumerate\"><dt class=\"enumerate\"> <span class=\"ecti-1095\">(i)<\/span><\/dt><dd class=\"enumerate\"><span class=\"ecti-1095\">Zeigen Sie, dass die Abbildung <\/span><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><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>b<\/mi><mo class=\"MathClass-close\">]<\/mo><mo class=\"MathClass-rel\">\u21a6<\/mo><mi mathvariant=\"bold-script\">\u2110<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math> <span class=\"ecti-1095\">differenzierbar ist und dass <\/span><span class=\"maperiod\"><math display=\"inline\"><msup><mrow><mi>F<\/mi><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup> <mo class=\"MathClass-rel\">=<\/mo> <mi>f<\/mi><\/math><\/span><span class=\"period\">.<\/span> <\/dd><dt class=\"enumerate\"> <span class=\"ecti-1095\">(ii)<\/span><\/dt><dd class=\"enumerate\"><span class=\"ecti-1095\">Verwenden Sie nun den Fundamentalsatz der Integral- und Differentialrechnung, um auf<\/span> <span class=\"ecti-1095\">Proposition <\/span><a href=\"..\/..\/chapter\/anwendungen#x1-116007r30\"><span class=\"ecti-1095\">4.30<\/span><\/a> <span class=\"ecti-1095\">zu schliessen.<\/span><\/dd><\/dl> <\/div> <div class=\"me meexample\"> <div class=\"wp-nocaption \"><\/div><h4 id=\"zdff40d17065c\"> <span class=\"ecbx-1095\">\u00dc<\/span><span class=\"ecbx-1095\">bung <\/span>(Volumen einer Vase)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Berechnen Sie das Volumen der Vase<\/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><mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>y<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>z<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-rel\">\u2208<\/mo> <msup><mrow><mi>\u211d<\/mi><\/mrow><mrow><mn>3<\/mn><\/mrow><\/msup><mo class=\"MathClass-rel\">\u2223<\/mo><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> [<\/mo><mrow><mo class=\"MathClass-bin\">\u2212<\/mo><mi>\u03c0<\/mi><mo class=\"MathClass-punc\">,<\/mo><mn>2<\/mn><mi>\u03c0<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">]<\/mo><\/mrow><mo class=\"MathClass-punc\">,<\/mo><mspace class=\"nbsp\" width=\"0.33em\" \/><mn>0<\/mn> <mo class=\"MathClass-rel\">\u2264<\/mo><msqrt><mrow><msup><mrow><mi>y<\/mi><\/mrow><mrow><mn>2<\/mn> <\/mrow> <\/msup> <mo class=\"MathClass-bin\">+<\/mo> <msup><mrow><mi>z<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><\/mrow><\/msqrt> <mo class=\"MathClass-rel\">\u2264<\/mo><mi class=\"qopname\"> sin<\/mi><mo>  <\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-bin\">+<\/mo> <mn>2<\/mn><\/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> <\/div> <div class=\"me meexample\"> <div class=\"wp-nocaption \"><\/div><h4 id=\"z95caeeba2791\"> <span class=\"ecbx-1095\">\u00dc<\/span><span class=\"ecbx-1095\">bung <\/span>(Gleichm\u00e4ssige Konvergenz der Ableitungen)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Sei <\/span><math display=\"inline\"><msub><mrow><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>f<\/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\">eine Folge stetig  differenzierbarer  reellwertiger  Funktionen  auf  einem  kompakten  Intervall<\/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\">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> <span class=\"ecti-1095\">Angenommen die Folge <\/span><math display=\"inline\"><msub><mrow><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>f<\/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 gleichm<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">ssig gegen eine Funktion <\/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\">und die Folge der Ableitungen <\/span><math display=\"inline\"><msub><mrow><mo class=\"MathClass-open\">(<\/mo><msubsup><mrow><mi>f<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msubsup><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><\/math> <span class=\"ecti-1095\">konvergiert gleichm<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">ssig gegen eine Funktion <\/span><span class=\"maperiod\"><math display=\"inline\"><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><span class=\"period\">.<\/span> <span class=\"ecti-1095\">Zeigen Sie, dass <\/span><math display=\"inline\"><mi>f<\/mi><\/math> <span class=\"ecti-1095\">stetig differenzierbar ist mit <\/span><span class=\"maperiod\"><math display=\"inline\"><msup><mrow><mi>f<\/mi><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup> <mo class=\"MathClass-rel\">=<\/mo> <mi>g<\/mi><\/math><\/span><span class=\"period\">.<\/span> <\/p> <\/div> <div class=\"me meexample\"> <div class=\"wp-nocaption \"><\/div><h4 id=\"z3e79986ba3c1\"> <span class=\"ecbx-1095\">\u00dc<\/span><span class=\"ecbx-1095\">bung.<\/span><\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Wir verwenden obige <\/span><span class=\"ecti-1095\">\u00dc<\/span><span class=\"ecti-1095\">bung, um eine glatte Funktion zu konstruieren, deren Taylorreihe um<\/span> <span class=\"ecti-1095\">einen Punkt Konvergenzradius Null hat.<\/span> <\/p><dl class=\"enumerate\"><dt class=\"enumerate\"> <span class=\"ecti-1095\">(i)<\/span><\/dt><dd class=\"enumerate\"><span class=\"ecti-1095\">Zeigen Sie, dass die Reihe <\/span><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><msup><mrow><mi class=\"qopname\">e<\/mi><mo>  <\/mo><\/mrow><mrow><mo class=\"MathClass-bin\">\u2212<\/mo><mi>n<\/mi><\/mrow><\/msup><mi class=\"qopname\"> cos<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><msup><mrow><mi>n<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><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> <mi>\u211d<\/mi><\/math> <span class=\"ecti-1095\">absolut konvergiert.<\/span> <\/dd><dt class=\"enumerate\"> <span class=\"ecti-1095\">(ii)<\/span><\/dt><dd class=\"enumerate\"><span class=\"ecti-1095\">Nach (i) definieren wir die 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>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u211d<\/mi><mo class=\"MathClass-rel\">\u21a6<\/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><msup><mrow><mi class=\"qopname\">e<\/mi><mo>  <\/mo><\/mrow><mrow><mo class=\"MathClass-bin\">\u2212<\/mo><mi>n<\/mi><\/mrow><\/msup><mi class=\"qopname\"> cos<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><msup><mrow><mi>n<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><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\">Zeigen Sie, dass <\/span><math display=\"inline\"><mi>f<\/mi><\/math> <span class=\"ecti-1095\">glatt ist.<\/span> <\/p><\/dd><dt class=\"enumerate\"> <span class=\"ecti-1095\">(iii)<\/span><\/dt><dd class=\"enumerate\"><span class=\"ecti-1095\">Berechnen Sie die Taylorreihe von <\/span><math display=\"inline\"><mi>f<\/mi><\/math> <span class=\"ecti-1095\">um Null und deren Konvergenzradius.<\/span><\/dd><\/dl> <\/div> <div class=\"me meexample\"> <div class=\"wp-nocaption \"><\/div><h4 id=\"z1d045f3b32e4\"> <span class=\"ecbx-1095\">\u00dc<\/span><span class=\"ecbx-1095\">bung <\/span>(Kr\u00fcmmung ebener Kurven)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">In dieser <\/span><span class=\"ecti-1095\">\u00dc<\/span><span class=\"ecti-1095\">bung m<\/span><span class=\"ecti-1095\">\u00f6<\/span><span class=\"ecti-1095\">chten wir die Kr<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">mmung ebener Kurven betrachten. Alle Kurven,<\/span> <span class=\"ecti-1095\">die wir dabei betrachten wollen, sollen zweimal differenzierbar, regul<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">r und einfach<\/span> <span class=\"ecti-1095\">sein, hier der Einfachheit vorerst inklusive den Endpunkten. F<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r eine solche Kurve<\/span> <math display=\"inline\"><mi>\u03b3<\/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> <msup><mrow><mi>\u211d<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><\/math> <span class=\"ecti-1095\">und ein<\/span> <math display=\"inline\"><mi>p<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <msup><mrow><mi>\u211d<\/mi><\/mrow><mrow><mn>2<\/mn> <\/mrow> <\/msup> <\/math> <span class=\"ecti-1095\">auf der Kurve sei<\/span> <math display=\"inline\"><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><\/math> <span class=\"ecti-1095\">der eindeutige<\/span> <span class=\"ecti-1095\">Zeitpunkt mit <\/span><math display=\"inline\"><mi>\u03b3<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>t<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <mi>p<\/mi><\/math><span class=\"ecti-1095\">. Dann<\/span> <span class=\"ecti-1095\">ist die Kr<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">mmung von <\/span><math display=\"inline\"><mi>\u03b3<\/mi><\/math> <span class=\"ecti-1095\">bei <\/span><math display=\"inline\"><mi>p<\/mi><\/math> <span class=\"ecti-1095\">definiert als<\/span> <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><msub><mrow><mi>\u03ba<\/mi><\/mrow><mrow><mi>\u03b3<\/mi><\/mrow><\/msub> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>p<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-rel\">=<\/mo> <mfrac><mrow><mo class=\"MathClass-open\">\u27e8<\/mo><mover accent=\"true\"><mrow><mi>\u03b3<\/mi><\/mrow><mo accent=\"true\">\u02d9<\/mo><\/mover><mo class=\"MathClass-open\">(<\/mo><mi>t<\/mi><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-punc\">,<\/mo><mi>R<\/mi><mover accent=\"true\"><mrow><mi>\u03b3<\/mi><\/mrow><mo accent=\"true\">\u00a8<\/mo><\/mover><mo class=\"MathClass-open\">(<\/mo><mi>t<\/mi><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-close\">\u27e9<\/mo><\/mrow> <mrow><mo class=\"MathClass-rel\">\u2225<\/mo><mover accent=\"true\"><mrow><mi>\u03b3<\/mi><\/mrow><mo accent=\"true\">\u02d9<\/mo><\/mover><mo class=\"MathClass-open\">(<\/mo><mi>t<\/mi><mo class=\"MathClass-close\">)<\/mo><msup><mrow><mo class=\"MathClass-rel\">\u2225<\/mo><\/mrow><mrow><mn>3<\/mn><\/mrow><\/msup><\/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\">wobei <\/span><math display=\"inline\"><mi>R<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mstyle><mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow \/><mo fence=\"true\" form=\"postfix\" \/><\/mrow><\/mstyle> <mstyle class=\"text\"><mtext \/><mstyle class=\"math\"><mtable align=\"axis\" class=\"array\" columnlines=\"none none none none none none none none none\" equalcolumns=\"false\" equalrows=\"false\"> <mtr><mtd class=\"array\" columnalign=\"center\"> <mn>0<\/mn> <\/mtd><mtd class=\"array\" columnalign=\"center\"><mn>1<\/mn><\/mtd><\/mtr> <mtr><mtd class=\"array\" columnalign=\"center\"><mo class=\"MathClass-bin\">\u2212<\/mo><mn>1<\/mn><\/mtd> <mtd class=\"array\" columnalign=\"center\"><mn>0<\/mn><\/mtd><\/mtr> <\/mtable> <\/mstyle><mtext>&nbsp;<\/mtext><\/mstyle> <mstyle><mrow><mo fence=\"true\" form=\"prefix\"> )<\/mo><mrow \/><mo fence=\"true\" form=\"postfix\" \/><\/mrow><\/mstyle> <\/math><span class=\"ecti-1095\">. Ist<\/span> <math display=\"inline\"><mi>\u03b3<\/mi><\/math> <span class=\"ecti-1095\">einfach, aber<\/span> <span class=\"ecti-1095\">erf<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">llt <\/span><math display=\"inline\"><mi>\u03b3<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>a<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <mi>\u03b3<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>b<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math><span class=\"ecti-1095\">, so reicht es<\/span> <span class=\"ecti-1095\">anzunehmen, dass <\/span><math display=\"inline\"><mover accent=\"true\"><mrow><mi>\u03b3<\/mi><\/mrow><mo accent=\"true\">\u02d9<\/mo><\/mover><mo class=\"MathClass-open\">(<\/mo><mi>a<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <mover accent=\"true\"><mrow><mi>\u03b3<\/mi><\/mrow><mo accent=\"true\">\u02d9<\/mo><\/mover><mo class=\"MathClass-open\">(<\/mo><mi>b<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math> <span class=\"ecti-1095\">und <\/span><math display=\"inline\"><mover accent=\"true\"><mrow><mi>\u03b3<\/mi><\/mrow><mo accent=\"true\">\u00a8<\/mo><\/mover> <mo class=\"MathClass-open\">(<\/mo><mi>a<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <mover accent=\"true\"><mrow><mi>\u03b3<\/mi><\/mrow><mo accent=\"true\">\u00a8<\/mo><\/mover><mo class=\"MathClass-open\">(<\/mo><mi>b<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math> <span class=\"ecti-1095\">gilt, damit obiger Ausdruck f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r die Kr<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">mmung Sinn ergibt.<\/span> <\/p><dl class=\"enumerate\"><dt class=\"enumerate\"> <span class=\"ecti-1095\">(i)<\/span><\/dt><dd class=\"enumerate\"><span class=\"ecti-1095\">Berechnen Sie die Kr<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">mmung der Kurven <\/span><math display=\"inline\"><mi>t<\/mi><mo class=\"MathClass-rel\">\u21a6<\/mo><mo class=\"MathClass-open\">(<\/mo><mi class=\"qopname\">cos<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>t<\/mi><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-punc\">,<\/mo><mi class=\"qopname\">sin<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>t<\/mi><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-close\">)<\/mo><\/math> <span class=\"ecti-1095\">und <\/span><math display=\"inline\"><mi>t<\/mi><mo class=\"MathClass-rel\">\u21a6<\/mo> <mo class=\"MathClass-open\">(<\/mo><mi class=\"qopname\">cosh<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>t<\/mi><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-punc\">,<\/mo><mi class=\"qopname\">sinh<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>t<\/mi><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-close\">)<\/mo><\/math> <span class=\"ecti-1095\">(definiert auf geeigneten Intervallen).<\/span> <\/dd><dt class=\"enumerate\"> <span class=\"ecti-1095\">(ii)<\/span><\/dt><dd class=\"enumerate\"><span class=\"ecti-1095\">Zeigen Sie, dass die Kr<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">mmung unabh<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">ngig ist von der Parametrisierung, das heisst,<\/span> <span class=\"ecti-1095\">dass f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r jede Reparametrisierung <\/span><math display=\"inline\"><mi>\u03b3<\/mi> <mo class=\"MathClass-bin\">\u2218<\/mo> <mi>\u03c8<\/mi><\/math> <span class=\"ecti-1095\">einer Kurve <\/span><math display=\"inline\"><mi>\u03b3<\/mi><\/math> <span class=\"ecti-1095\">wie oben gilt <\/span><math display=\"inline\"><msub><mrow><mi>\u03ba<\/mi><\/mrow><mrow><mi>\u03b3<\/mi><mo class=\"MathClass-bin\">\u2218<\/mo><mi>\u03c8<\/mi><\/mrow><\/msub><mo class=\"MathClass-open\">(<\/mo><mi>p<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <msub><mrow><mi>\u03ba<\/mi><\/mrow><mrow><mi>\u03b3<\/mi><\/mrow><\/msub><mo class=\"MathClass-open\">(<\/mo><mi>p<\/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 Punkte <\/span><math display=\"inline\"><mi>p<\/mi><\/math> <span class=\"ecti-1095\">auf der Kurve.<\/span><\/dd><\/dl> <\/div> <div class=\"me meexample\"> <div class=\"wp-nocaption \"><\/div><h4 id=\"z8fabecc22df7\"> <span class=\"ecbx-1095\">\u00dc<\/span><span class=\"ecbx-1095\">bung <\/span>(Existenz von Kurven vorgegebener Kr\u00fcmmung)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Wie in vorheriger <\/span><span class=\"ecti-1095\">\u00dc<\/span><span class=\"ecti-1095\">bung m<\/span><span class=\"ecti-1095\">\u00f6<\/span><span class=\"ecti-1095\">chten wir hier die Kr<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">mmung ebener Kurven betrachten,<\/span> <span class=\"ecti-1095\">aber dabei zulassen, dass die betrachteten Kurven nicht einfach sind, womit f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r eine regul<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">re<\/span> <span class=\"ecti-1095\">Kurve <\/span><math display=\"inline\"><mi>\u03b3<\/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> <msup><mrow><mi>\u211d<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><\/math> <span class=\"ecti-1095\">die Kr<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">mmung <\/span><math display=\"inline\"><msub><mrow><mi>\u03ba<\/mi><\/mrow><mrow><mi>\u03b3<\/mi><\/mrow><\/msub><\/math> <span class=\"ecti-1095\">definiert ist als Funktion auf <\/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\">via <\/span><span class=\"maperiod\"><math display=\"inline\"><msub><mrow><mi>\u03ba<\/mi><\/mrow><mrow><mi>\u03b3<\/mi> <\/mrow> <\/msub> <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> <mo class=\"MathClass-open\">\u27e8<\/mo><mover accent=\"true\"><mrow><mi>\u03b3<\/mi><\/mrow><mo accent=\"true\">\u02d9<\/mo><\/mover><mo class=\"MathClass-open\">(<\/mo><mi>t<\/mi><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-punc\">,<\/mo><mi>R<\/mi><mover accent=\"true\"><mrow><mi>\u03b3<\/mi><\/mrow><mo accent=\"true\">\u00a8<\/mo><\/mover><mo class=\"MathClass-open\">(<\/mo><mi>t<\/mi><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-close\">\u27e9<\/mo><\/mrow> <mrow><mo class=\"MathClass-rel\">\u2225<\/mo><mover accent=\"true\"><mrow><mi>\u03b3<\/mi><\/mrow><mo accent=\"true\">\u02d9<\/mo><\/mover><mo class=\"MathClass-open\">(<\/mo><mi>t<\/mi><mo class=\"MathClass-close\">)<\/mo><msup><mrow><mo class=\"MathClass-rel\">\u2225<\/mo><\/mrow><mrow><mn>3<\/mn><\/mrow><\/msup><\/mrow><\/mfrac> <\/math><\/span><span class=\"period\">.<\/span> <\/p><p class=\"indent\"><span class=\"ecti-1095\">Sei nun <\/span><math display=\"inline\"><mi>\u03ba<\/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 beliebige stetige<\/span> <span class=\"ecti-1095\">Funktion auf einem Intervall <\/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\">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> <span class=\"ecti-1095\">Wir m<\/span><span class=\"ecti-1095\">\u00f6<\/span><span class=\"ecti-1095\">chten hier zeigen, dass eine zweimal stetig differenzierbare Kurve<\/span> <math display=\"inline\"><mi>\u03b3<\/mi><\/math> <span class=\"ecti-1095\">mit<\/span> <span class=\"ecti-1095\">Kr<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">mmungsfunktion <\/span><math display=\"inline\"><mi>\u03ba<\/mi><\/math> <span class=\"ecti-1095\">existiert. Dazu betrachten wir die Funktion<\/span> <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mi>\ud835\udf03<\/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> <msup><mrow><mi>\u211d<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><mo class=\"MathClass-punc\">,<\/mo><mspace class=\"nbsp\" width=\"0.33em\" \/><mi>s<\/mi><mo class=\"MathClass-rel\">\u21a6<\/mo><msubsup><mrow><mo>\u222b  <\/mo><\/mrow><mrow><mi>a<\/mi><\/mrow><mrow><mi>s<\/mi><\/mrow><\/msubsup><mi>\u03ba<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>r<\/mi><mo class=\"MathClass-close\">)<\/mo><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>r<\/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\">und setzen<\/span> <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mi>\u03b3<\/mi> <mo class=\"MathClass-punc\">:<\/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\">\u2192<\/mo> <msup><mrow><mi>\u211d<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><mo class=\"MathClass-punc\">,<\/mo><mspace class=\"nbsp\" width=\"0.33em\" \/><mi>t<\/mi><mo class=\"MathClass-rel\">\u21a6<\/mo><msubsup><mrow><mo>\u222b  <\/mo><\/mrow><mrow><mi>a<\/mi><\/mrow><mrow><mi>t<\/mi><\/mrow><\/msubsup> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi class=\"qopname\">cos<\/mi><mo>  <\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>\ud835\udf03<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>s<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><mo class=\"MathClass-punc\">,<\/mo><mi class=\"qopname\">sin<\/mi><mo>  <\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>\ud835\udf03<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>s<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>s<\/mi><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\">Zeigen Sie, dass die Komponenten von <\/span><math display=\"inline\"><mi>\u03b3<\/mi><\/math> <span class=\"ecti-1095\">jeweils zweimal stetig differenzierbar ist und dass<\/span> <math display=\"inline\"><mi>\u03b3<\/mi><\/math> <span class=\"ecti-1095\">eine<\/span> <span class=\"ecti-1095\">regul<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">re, nach Bogenl<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">nge parametrisierte Kurve ist. Verifizieren Sie anschliessend, dass<\/span> <math display=\"inline\"><msub><mrow><mi>\u03ba<\/mi><\/mrow><mrow><mi>\u03b3<\/mi> <\/mrow> <\/msub> <mo class=\"MathClass-rel\">=<\/mo> <mi>\u03ba<\/mi><\/math> <span class=\"ecti-1095\">gilt.<\/span> <\/p> <\/div> <div class=\"me meexample\"> <div class=\"wp-nocaption \"><\/div><h4 id=\"zdaf019c96667\"> <span class=\"ecbx-1095\">\u00dc<\/span><span class=\"ecbx-1095\">bung <\/span>(Irrationalit\u00e4t der Kreiszahl)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">In dieser           <\/span><span class=\"ecti-1095\">\u00dc<\/span><span class=\"ecti-1095\">bung           m<\/span><span class=\"ecti-1095\">\u00f6<\/span><span class=\"ecti-1095\">chten           wir           zeigen,           dass<\/span> <math display=\"inline\"><mi>\u03c0<\/mi><\/math> <span class=\"ecti-1095\">irrational ist, wobei wir dem Beweis von Niven <\/span><span class=\"cite\"><span class=\"ecti-1095\">[<\/span><a href=\"#XpiirrationalNiven\"><span class=\"ecti-1095\">Niv47<\/span><\/a><span class=\"ecti-1095\">]<\/span><\/span> <span class=\"ecti-1095\">folgen werden.<\/span> <\/p><p class=\"indent\"><span class=\"ecti-1095\">Per Widerspruch wollen wir annehmen, dass<\/span> <math display=\"inline\"><mi>\u03c0<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mfrac> <mrow> <mi>a<\/mi><\/mrow> <mrow><mi>b<\/mi><\/mrow><\/mfrac><\/math> <span class=\"ecti-1095\">ist f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r<\/span> <math display=\"inline\"><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo> <mi>b<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2115<\/mi><\/math><span class=\"ecti-1095\">. Nun<\/span> <span class=\"ecti-1095\">betrachten wir die Polynome<\/span> <\/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> <mfrac><mrow><msup><mrow><mi>x<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msup><msup><mrow><mo class=\"MathClass-open\">(<\/mo><mi>a<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>b<\/mi><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> <mo class=\"MathClass-punc\">,<\/mo><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>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><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">\u2212<\/mo> <msup><mrow><mi>f<\/mi><\/mrow><mrow><mo class=\"MathClass-open\">(<\/mo><mn>2<\/mn><mo class=\"MathClass-close\">)<\/mo><\/mrow><\/msup><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">+<\/mo> <msup><mrow><mi>f<\/mi><\/mrow><mrow><mo class=\"MathClass-open\">(<\/mo><mn>4<\/mn><mo class=\"MathClass-close\">)<\/mo><\/mrow><\/msup><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">\u2212<\/mo><mi class=\"MathClass-op\">\u2026<\/mi><mo> <\/mo> <mo class=\"MathClass-bin\">+<\/mo> <msup><mrow><mo class=\"MathClass-open\">(<\/mo><mo class=\"MathClass-bin\">\u2212<\/mo><mn>1<\/mn><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msup><msup><mrow><mi>f<\/mi><\/mrow><mrow><mo class=\"MathClass-open\">(<\/mo><mn>2<\/mn><mi>n<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow><\/msup><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/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\">f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r ein <\/span><span class=\"maperiod\"><math display=\"inline\"><mi>n<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2115<\/mi><\/math><\/span><span class=\"period\">,<\/span> <span class=\"ecti-1095\">welches wir sp<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">ter w<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">hlen werden.<\/span> <\/p><dl class=\"enumerate\"><dt class=\"enumerate\"> <span class=\"ecti-1095\">(i)<\/span><\/dt><dd class=\"enumerate\"><span class=\"ecti-1095\">Begr<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">nden Sie, wieso <\/span><span class=\"maperiod\"><math display=\"inline\"><mi>f<\/mi><\/math><\/span><span class=\"period\">,<\/span> <span class=\"ecti-1095\">alle Ableitungen von <\/span><math display=\"inline\"><mi>f<\/mi><\/math> <span class=\"ecti-1095\">und <\/span><math display=\"inline\"><mi>F<\/mi><\/math> <span class=\"ecti-1095\">bei <\/span><math display=\"inline\"><mn>0<\/mn><\/math> <span class=\"ecti-1095\">und bei <\/span><math display=\"inline\"><mi>\u03c0<\/mi><\/math> <span class=\"ecti-1095\">ganzzahlige Werte annehmen. Verifizieren Sie des Weiteren, dass <\/span><math display=\"inline\"><mi>f<\/mi><msub><mrow><mo class=\"MathClass-rel\">|<\/mo><\/mrow><mrow><mo class=\"MathClass-open\">[<\/mo><mn>0<\/mn><mo class=\"MathClass-punc\">,<\/mo><mi>\u03c0<\/mi><mo class=\"MathClass-close\">]<\/mo><\/mrow><\/msub><\/math> <span class=\"ecti-1095\">eine nicht-negative Funktion ist, welche genau bei <\/span><math display=\"inline\"><mn>0<\/mn><\/math> <span class=\"ecti-1095\">und <\/span><math display=\"inline\"><mi>\u03c0<\/mi><\/math> <span class=\"ecti-1095\">verschwindet.<\/span> <\/dd><dt class=\"enumerate\"> <span class=\"ecti-1095\">(ii)<\/span><\/dt><dd class=\"enumerate\"><span class=\"ecti-1095\">Zeigen Sie, dass<\/span> <math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><msup><mrow><mo class=\"MathClass-open\">(<\/mo><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><mi class=\"qopname\">sin<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>F<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><mi class=\"qopname\">cos<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup> <mo class=\"MathClass-rel\">=<\/mo> <mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><mi class=\"qopname\">sin<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/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\">und <\/span><span class=\"maperiod\"><math display=\"inline\"><msubsup><mrow><mi class=\"MathClass-op\"> \u222b  <\/mi><mo> <\/mo><\/mrow><mrow><mn>0<\/mn><\/mrow><mrow><mi>\u03c0<\/mi><\/mrow><\/msubsup><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><mi class=\"qopname\">sin<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mi>F<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>\u03c0<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">+<\/mo> <mi>F<\/mi><mo class=\"MathClass-open\">(<\/mo><mn>0<\/mn><mo class=\"MathClass-close\">)<\/mo><\/math><\/span><span class=\"period\">.<\/span> <\/p><\/dd><dt class=\"enumerate\"> <span class=\"ecti-1095\">(iii)<\/span><\/dt><dd class=\"enumerate\"><span class=\"ecti-1095\">Schliessen Sie auf einen Widerspruch.<\/span><\/dd><\/dl> <\/div> <div class=\"me meexample\"> <div class=\"wp-nocaption \"><\/div><h4 id=\"za3eff3dcf845\"> <span class=\"ecbx-1095\">\u00dc<\/span><span class=\"ecbx-1095\">bung <\/span>(Summe der Reziproken der Primzahlen)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">In dieser <\/span><span class=\"ecti-1095\">\u00dc<\/span><span class=\"ecti-1095\">bung m<\/span><span class=\"ecti-1095\">\u00f6<\/span><span class=\"ecti-1095\">chten wir f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r nat<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">rliche Zahlen<\/span> <math display=\"inline\"><mi>n<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2115<\/mi><\/math> <span class=\"ecti-1095\">die Summe<\/span> <math display=\"inline\"><msub><mrow><mi class=\"MathClass-op\">\u2211<\/mi><mo> <\/mo> <\/mrow><mrow><mi>p<\/mi><mo class=\"MathClass-rel\">\u2208<\/mo><mi>\u2119<\/mi><mo class=\"MathClass-punc\">:<\/mo><mi>p<\/mi><mo class=\"MathClass-rel\">\u2264<\/mo><mi>n<\/mi><\/mrow><\/msub><mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mi>p<\/mi><\/mrow><\/mfrac><\/math> <span class=\"ecti-1095\">betrachten,<\/span> <span class=\"ecti-1095\">wobei <\/span><math display=\"inline\"><mi>\u2119<\/mi> <mo class=\"MathClass-rel\">\u2286<\/mo> <mi>\u2115<\/mi><\/math> <span class=\"ecti-1095\">die Menge der Primzahlen bezeichnet. Dabei m<\/span><span class=\"ecti-1095\">\u00f6<\/span><span class=\"ecti-1095\">chten wir zeigen, dass<\/span> <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><munder class=\"msub\"><mrow><mo>\u2211<\/mo> <\/mrow><mrow><mi>p<\/mi><mo class=\"MathClass-rel\">\u2208<\/mo><mi>\u2119<\/mi><mo class=\"MathClass-punc\">:<\/mo><mi>p<\/mi><mo class=\"MathClass-rel\">\u2264<\/mo><mi>n<\/mi><\/mrow><\/munder><mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mi>p<\/mi><\/mrow><\/mfrac> <mo class=\"MathClass-rel\">\u2265<\/mo><mi class=\"qopname\"> log<\/mi><mo>  <\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi class=\"qopname\">log<\/mi><mo>  <\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>n<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-bin\">\u2212<\/mo><mi class=\"qopname\"> log<\/mi><mo>  <\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>C<\/mi><\/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\"><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\">\u2265<\/mo> <mn>3<\/mn><\/math> <span class=\"ecti-1095\">und<\/span> <span class=\"ecti-1095\">eine Konstante <\/span><span class=\"maperiod\"><math display=\"inline\"><mi>C<\/mi> <mo class=\"MathClass-rel\">&gt;<\/mo> <mn>1<\/mn><\/math><\/span><span class=\"period\">.<\/span> <span class=\"ecti-1095\">Insbesondere gibt es unendlich viele Primzahlen (wieso?). Die obige Ungleichung stellt eine (stark)<\/span> <span class=\"ecti-1095\">abgeschw<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">chte Version des zweiten Theorems von Mertens (und damit einen Vorreiter des<\/span> <span class=\"ecti-1095\">Primzahlsatzes) dar.<\/span> <\/p><dl class=\"enumerate\"><dt class=\"enumerate\"> <span class=\"ecti-1095\">(i)<\/span><\/dt><dd class=\"enumerate\"><span class=\"ecti-1095\">Verifizieren Sie 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\">die Ungleichung<\/span> <math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mi class=\"qopname\">exp<\/mi><mo>  <\/mo><mstyle><mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow \/><mo fence=\"true\" form=\"postfix\" \/><\/mrow><\/mstyle><munder class=\"msub\"><mrow><mo>\u2211<\/mo> <\/mrow><mrow><mi>p<\/mi><mo class=\"MathClass-rel\">\u2208<\/mo><mi>\u2119<\/mi><mo class=\"MathClass-punc\">:<\/mo><mi>p<\/mi><mo class=\"MathClass-rel\">\u2264<\/mo><mi>n<\/mi><\/mrow><\/munder><mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mi>p<\/mi><\/mrow><\/mfrac><mstyle><mrow><mo fence=\"true\" form=\"prefix\"> )<\/mo><mrow \/><mo fence=\"true\" form=\"postfix\" \/><\/mrow><\/mstyle> <mo class=\"MathClass-rel\">\u2265<\/mo><munder class=\"msub\"><mrow><mo>\u220f<\/mo> <\/mrow><mrow><mi>p<\/mi><mo class=\"MathClass-rel\">\u2208<\/mo><mi>\u2119<\/mi><mo class=\"MathClass-punc\">:<\/mo><mi>p<\/mi><mo class=\"MathClass-rel\">\u2264<\/mo><mi>n<\/mi><\/mrow><\/munder> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mn>1<\/mn> <mo class=\"MathClass-bin\">+<\/mo> <mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mi>p<\/mi><\/mrow><\/mfrac> <\/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> <\/dd><dt class=\"enumerate\"> <span class=\"ecti-1095\">(ii)<\/span><\/dt><dd class=\"enumerate\"><span class=\"ecti-1095\">Zeigen Sie 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> <math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><munderover accent=\"false\" accentunder=\"false\"><mrow><mo> \u2211<\/mo> <\/mrow><mrow><mi>k<\/mi><mo class=\"MathClass-rel\">=<\/mo><mn>1<\/mn><\/mrow><mrow><mi>n<\/mi><\/mrow><\/munderover><mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mi>k<\/mi><\/mrow><\/mfrac> <mo class=\"MathClass-rel\">\u2264<\/mo><mstyle><mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow \/><mo fence=\"true\" form=\"postfix\" \/><\/mrow><\/mstyle><munderover accent=\"false\" accentunder=\"false\"><mrow><mo>\u2211<\/mo> <\/mrow><mrow><mi>\u2113<\/mi><mo class=\"MathClass-rel\">=<\/mo><mn>1<\/mn><\/mrow><mrow><mi>n<\/mi><\/mrow><\/munderover> <mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><msup><mrow><mi>\u2113<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><\/mrow><\/mfrac><mstyle><mrow><mo fence=\"true\" form=\"prefix\"> )<\/mo><mrow \/><mo fence=\"true\" form=\"postfix\" \/><\/mrow><\/mstyle><munder class=\"msub\"><mrow><mo> \u220f<\/mo> <\/mrow><mrow><mi>p<\/mi><mo class=\"MathClass-rel\">\u2208<\/mo><mi>\u2119<\/mi><mo class=\"MathClass-punc\">:<\/mo><mi>p<\/mi><mo class=\"MathClass-rel\">\u2264<\/mo><mi>n<\/mi><\/mrow><\/munder> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mn>1<\/mn> <mo class=\"MathClass-bin\">+<\/mo> <mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mi>p<\/mi><\/mrow><\/mfrac> <\/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> <\/dd><dt class=\"enumerate\"> <span class=\"ecti-1095\">(iii)<\/span><\/dt><dd class=\"enumerate\"><span class=\"ecti-1095\">Schliessen Sie auf die Aussage.<\/span><\/dd><\/dl> <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\">F<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r den zweiten Teil k<\/span><span class=\"ecti-1095\">\u00f6<\/span><span class=\"ecti-1095\">nnen Sie die Existenz und Eindeutigkeit einer<\/span> <span class=\"ecti-1095\">Primfaktorzerlegung verwenden, wonach sich insbesondere jede nat<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">rliche Zahl zwischen<\/span> <math display=\"inline\"><mn>1<\/mn><\/math> <span class=\"ecti-1095\">und<\/span> <math display=\"inline\"><mi>n<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2115<\/mi><\/math> <span class=\"ecti-1095\">als<\/span> <span class=\"ecti-1095\">Produkt einer Quadratzahl mit einer Zahl schreiben l<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">sst, in deren Faktorisierung jede Primzahl nur<\/span> <span class=\"ecti-1095\">einmal vorkommt. F<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r den letzten Teil k<\/span><span class=\"ecti-1095\">\u00f6<\/span><span class=\"ecti-1095\">nnen Sie <\/span><span class=\"ecti-1095\">\u00dc<\/span><span class=\"ecti-1095\">bung <\/span><a href=\"..\/..\/chapter\/das-uneigentliche-integral#x1-273008r36\"><span class=\"ecti-1095\">9.36<\/span><\/a> <span class=\"ecti-1095\">benutzen.<\/span><\/p><\/details>  <\/div> <p class=\"indent\"> <a id=\"x1-293014r258\"><\/a> <\/p> \n","protected":false},"author":1089,"menu_order":8,"template":"","meta":{"pb_show_title":"","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-100","chapter","type-chapter","status-publish","hentry"],"part":92,"_links":{"self":[{"href":"https:\/\/wp-prd.let.ethz.ch\/analysis19\/wp-json\/pressbooks\/v2\/chapters\/100","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\/100\/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\/100\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/wp-prd.let.ethz.ch\/analysis19\/wp-json\/wp\/v2\/media?parent=100"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/wp-prd.let.ethz.ch\/analysis19\/wp-json\/pressbooks\/v2\/chapter-type?post=100"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/wp-prd.let.ethz.ch\/analysis19\/wp-json\/wp\/v2\/contributor?post=100"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/wp-prd.let.ethz.ch\/analysis19\/wp-json\/wp\/v2\/license?post=100"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}