{"id":79,"date":"2021-12-15T09:53:18","date_gmt":"2021-12-15T09:53:18","guid":{"rendered":"https:\/\/wp-prd.let.ethz.ch\/analysis19\/chapter\/potenzreihen\/"},"modified":"2021-12-15T09:53:18","modified_gmt":"2021-12-15T09:53:18","slug":"potenzreihen","status":"publish","type":"chapter","link":"https:\/\/wp-prd.let.ethz.ch\/analysis19\/chapter\/potenzreihen\/","title":{"raw":"Potenzreihen","rendered":"Potenzreihen"},"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=\"zc8703823ed71\" class=\"sectionHead\"><span class=\"titlemark\">7.4 <\/span> <a id=\"x1-1990004\"><\/a>Potenzreihen<\/h3> <div class=\"me metheorem\"> <p class=\"indent\"><\/p><h4 id=\"z15eba0fbb4a0\"> <a id=\"x1-199001r54\"><\/a> <span class=\"ecbx-1095\">Definition 7.54 <\/span>(Potenzreihe)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\">F\u00fcr jedes <math display=\"inline\"><mi>n<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <msub><mrow><mi>\u2115<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><\/math> sei <span class=\"maperiod\"><math display=\"inline\"><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi> <\/mrow> <\/msub> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2102<\/mi><\/math><\/span><span class=\"period\">.<\/span> Dann ist der formale Ausdruck <\/p><math display=\"block\"><mtable class=\"align\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><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><msub><mrow><mi>a<\/mi><\/mrow><mrow> <mi>n<\/mi><\/mrow><\/msub><msup><mrow><mi>z<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msup><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"><mstyle class=\"label\" id=\"x1-199002r5\" \/><mstyle class=\"maketag\"><mtext>(7.5)<\/mtext><\/mstyle><mspace class=\"nbsp\" width=\"0.33em\" \/> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">eine <span class=\"ecbx-1095\">Potenzreihe <\/span>in der Variable <span class=\"maperiod\"><math display=\"inline\"><mi>z<\/mi><\/math><\/span><span class=\"period\">.<\/span> <\/p> <\/div> <p class=\"indent\">Es dr\u00e4ngt sich bei obiger Definition ein Vergleich zur Definition eines Polynoms in Definition <a href=\"..\/..\/chapter\/polynome#x1-81005r13\">3.13<\/a> auf. Im Gegensatz zur Diskussion von Polynomen ist aber eine Potenzreihe vorerst nur ein formaler Ausdruck. Es ist nicht klar, bei welchen komplexen Zahlen man eine Potenzreihe auswerten darf. Insbesondere wissen wir (noch) nicht, ob wir diesem formalen Ausdruck \u00fcberhaupt eine Funktion auf <math display=\"inline\"><mi>\u2102<\/mi><\/math> oder einer bestimmten Teilmenge von <math display=\"inline\"><mi>\u2102<\/mi><\/math> zuordnen k\u00f6nnen. Diese Frage h\u00e4ngt stark von den Koeffizienten <math display=\"inline\"><msub><mrow><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi> <\/mrow> <\/msub> <mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mi>n<\/mi><mo class=\"MathClass-rel\">\u2208<\/mo><msub><mrow><mi>\u2115<\/mi><\/mrow><mrow><mn>0<\/mn> <\/mrow> <\/msub> <\/mrow> <\/msub> <\/math> ab und wird in Satz <a href=\"..\/..\/chapter\/potenzreihen#x1-200002r56\">7.56<\/a> beantwortet. <\/p><p class=\"indent\">Wie schon bei Polynomen in Definition <a href=\"..\/..\/chapter\/polynome#x1-81001r11\">3.11<\/a> ist auch hier die Definition (<a href=\"..\/..\/chapter\/summen-und-produkte#x1-77003r2\">3.2<\/a>) \u00e4usserst sinnvoll, damit die Potenzreihe (<a href=\"..\/..\/chapter\/potenzreihen#x1-199002r5\">7.5<\/a>)bei <math display=\"inline\"><mi>z<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mn>0<\/mn><\/math> auf jeden Fall konvergiert und den Wert <math display=\"inline\"><msub><mrow><mi>a<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><\/math> hat. <a id=\"x1-199003r198\"><\/a> <\/p> <h4 id=\"z1dd1e58054f4\" class=\"subsectionHead\"><span class=\"titlemark\">7.4.1 <\/span> <a id=\"x1-2000001\"><\/a>Konvergenzradius<\/h4> <div class=\"me metheorem\"> <p class=\"indent\"><\/p><h4 id=\"zf0ed7d793817\"> <a id=\"x1-200001r55\"><\/a> <span class=\"ecbx-1095\">Definition 7.55.<\/span> <\/h4> <p class=\"indent\">Sei <math display=\"inline\"><msubsup><mrow><mi class=\"MathClass-op\"> \u2211<\/mi><mo> <\/mo> <\/mrow><mrow><mi>n<\/mi><mo class=\"MathClass-rel\">=<\/mo><mn>0<\/mn><\/mrow><mrow><mi>\u221e<\/mi><\/mrow><\/msubsup><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><msup><mrow><mi>z<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msup><\/math> eine Potenzreihe mit komplexen Koeffizienten <span class=\"maperiod\"><math display=\"inline\"><msub><mrow><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mi>n<\/mi><mo class=\"MathClass-rel\">\u2208<\/mo><msub><mrow><mi>\u2115<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><\/mrow><\/msub><\/math><\/span><span class=\"period\">.<\/span> Wir definieren den <span class=\"ecbx-1095\">Konvergenzradius <\/span>durch <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mi>R<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><munder class=\"msub\"><mrow><mi class=\"qopname\">limsup<\/mi><mo>  <\/mo><\/mrow><mrow><mi>n<\/mi><mo class=\"MathClass-rel\">\u2192<\/mo><mi>\u221e<\/mi><\/mrow><\/munder><mroot><mrow><mo class=\"MathClass-rel\">|<\/mo><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><mo class=\"MathClass-rel\">|<\/mo><\/mrow><mrow><mi>n<\/mi><\/mrow><\/mroot><\/mrow><\/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\">wobei wir <math display=\"inline\"> <mfrac> <mrow> <mn>1<\/mn><\/mrow> <mrow><mo class=\"MathClass-bin\">+<\/mo><mi>\u221e<\/mi><\/mrow><\/mfrac> <mo class=\"MathClass-rel\">=<\/mo> <mn>0<\/mn><\/math> setzen und hier (aber auch nur hier) die Vereinbarung <math display=\"inline\"><mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mn>0<\/mn><\/mrow><\/mfrac> <mo class=\"MathClass-rel\">=<\/mo> <mo class=\"MathClass-bin\">+<\/mo><mi>\u221e<\/mi><\/math> treffen. <\/p> <\/div> <div class=\"me metheorem\"> <p class=\"indent\"><\/p><h4 id=\"z481151657781\"> <a id=\"x1-200002r56\"><\/a> <span class=\"ecbx-1095\">Satz 7.56 <\/span>(\u00dcber den Konvergenzradius)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Sei <\/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><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><msup><mrow><mi>z<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msup><\/math> <span class=\"ecti-1095\">eine Potenzreihe<\/span> <span class=\"ecti-1095\">und <\/span><math display=\"inline\"><mi>R<\/mi><\/math> <span class=\"ecti-1095\">ihr Konvergenzradius.<\/span> <span class=\"ecti-1095\">Dann konvergiert 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><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><msup><mrow><mi>z<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msup><\/math> <span class=\"ecti-1095\">f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r alle <\/span><math display=\"inline\"><mi>z<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2102<\/mi><\/math> <span class=\"ecti-1095\">mit<\/span> <math display=\"inline\"><mo class=\"MathClass-rel\">|<\/mo><mi>z<\/mi><mo class=\"MathClass-rel\">|<\/mo> <mo class=\"MathClass-rel\">&lt;<\/mo> <mi>R<\/mi><\/math> <span class=\"ecti-1095\">absolut und divergiert<\/span> <span class=\"ecti-1095\">f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r alle <\/span><math display=\"inline\"><mi>z<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2102<\/mi><\/math> <span class=\"ecti-1095\">mit<\/span> <math display=\"inline\"><mo class=\"MathClass-rel\">|<\/mo><mi>z<\/mi><mo class=\"MathClass-rel\">|<\/mo> <mo class=\"MathClass-rel\">&gt;<\/mo> <mi>R<\/mi><\/math><span class=\"ecti-1095\">. Weiters konvergiert<\/span> <span class=\"ecti-1095\">die Funktionenfolge <\/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>N<\/mi><\/mrow><\/msubsup><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><msup><mrow><mi>z<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msup><\/math> <span class=\"ecti-1095\">gleichm<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">ssig gegen <\/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><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><msup><mrow><mi>z<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msup><\/math> <span class=\"ecti-1095\">auf<\/span> <span class=\"ecti-1095\">jeder Kreisscheibe der Form <\/span><math display=\"inline\"><msub><mrow><mi>B<\/mi><\/mrow><mrow><mi>S<\/mi><\/mrow><\/msub><mo class=\"MathClass-open\">(<\/mo><mn>0<\/mn><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> {<\/mo><mrow><mi>z<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2102<\/mi><mo class=\"MathClass-rel\">\u2223<\/mo><mo class=\"MathClass-rel\">|<\/mo><mi>z<\/mi><mo class=\"MathClass-rel\">|<\/mo> <mo class=\"MathClass-rel\">&lt;<\/mo> <mi>S<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">}<\/mo><\/mrow><\/math> <span class=\"ecti-1095\">f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r jedes <\/span><span class=\"maperiod\"><math display=\"inline\"><mi>S<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mo class=\"MathClass-open\">(<\/mo><mn>0<\/mn><mo class=\"MathClass-punc\">,<\/mo><mi>R<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math><\/span><span class=\"period\">.<\/span> <span class=\"ecti-1095\">Insbesondere definiert die Potenzreihe die stetige Abbildung<\/span> <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mi>z<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <msub><mrow><mi>B<\/mi><\/mrow><mrow><mi>R<\/mi><\/mrow><\/msub><mo class=\"MathClass-open\">(<\/mo><mn>0<\/mn><mo class=\"MathClass-close\">)<\/mo><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><msub><mrow><mi>a<\/mi><\/mrow><mrow> <mi>n<\/mi><\/mrow><\/msub><msup><mrow><mi>z<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msup> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2102<\/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> <\/div> <div class=\"center\"> <p class=\"noindent\"> <\/p><p class=\"noindent\"><\/p><div class=\"mefigcentered\" id=\"wpsize=467&amp;url=Pictures\/reihen\/konvradius.pdf\"><img id=\"z81fb2cb33b47\" alt=\"PIC\" src=\"https:\/\/people.math.ethz.ch\/~einsiedl\/Pictures\/reihen\/konvradius.svg\" width=\"467\"><\/div>  <\/div> <p class=\"indent\"> <\/p> <div class=\"proof\"> <p class=\"indent\"><span class=\"head\"><\/span><\/p><details open><summary><b>Beweis.<\/b><\/summary><p class=\"indent\" style=\"margin-top: 10\">Wir verwenden das Wurzelkriterium aus Korollar <a href=\"..\/..\/chapter\/absolute-konvergenz#x1-193002r30\">7.30<\/a> f\u00fcr ein beliebiges <math display=\"inline\"><mi>z<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2102<\/mi><\/math> und die Reihe <math display=\"inline\"><msubsup><mrow><mi class=\"MathClass-op\"> \u2211<\/mi><mo> <\/mo> <\/mrow><mrow><mi>n<\/mi><mo class=\"MathClass-rel\">=<\/mo><mn>0<\/mn><\/mrow><mrow><mi>\u221e<\/mi><\/mrow><\/msubsup><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><msup><mrow><mi>z<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msup><\/math> und berechnen deswegen <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><munder class=\"msub\"><mrow><mi class=\"qopname\"> limsup<\/mi><mo>  <\/mo><\/mrow><mrow><mi>n<\/mi><mo class=\"MathClass-rel\">\u2192<\/mo><mi>\u221e<\/mi><\/mrow><\/munder><mroot><mrow><mo class=\"MathClass-rel\">|<\/mo><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><msup><mrow><mi>z<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msup><mo class=\"MathClass-rel\">|<\/mo><\/mrow><mrow><mi>n<\/mi><\/mrow><\/mroot> <mo class=\"MathClass-rel\">=<\/mo><munder class=\"msub\"><mrow><mi class=\"qopname\"> limsup<\/mi><mo>  <\/mo><\/mrow><mrow> <mi>n<\/mi><mo class=\"MathClass-rel\">\u2192<\/mo><mi>\u221e<\/mi><\/mrow><\/munder><mroot><mrow><mo class=\"MathClass-rel\">|<\/mo><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><mo class=\"MathClass-rel\">|<\/mo><\/mrow><mrow><mi>n<\/mi><\/mrow><\/mroot> <mrow><mo fence=\"true\" form=\"prefix\"> |<\/mo><mrow><mi>z<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">|<\/mo><\/mrow> <mo class=\"MathClass-rel\">=<\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> |<\/mo><mrow><mi>z<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">|<\/mo><\/mrow><munder class=\"msub\"><mrow><mi class=\"qopname\">limsup<\/mi><mo>  <\/mo><\/mrow><mrow><mi>n<\/mi><mo class=\"MathClass-rel\">\u2192<\/mo><mi>\u221e<\/mi><\/mrow><\/munder><mroot><mrow><mo class=\"MathClass-rel\">|<\/mo><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><mo class=\"MathClass-rel\">|<\/mo><\/mrow><mrow><mi>n<\/mi><\/mrow><\/mroot> <mo class=\"MathClass-rel\">=<\/mo> <mfrac><mrow><mo class=\"MathClass-rel\">|<\/mo><mi>z<\/mi><mo class=\"MathClass-rel\">|<\/mo><\/mrow> <mrow><mi>R<\/mi><\/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\">Nach dem Wurzelkriterium konvergiert die Reihe also absolut f\u00fcr <math display=\"inline\"><mfrac><mrow><mo class=\"MathClass-rel\">|<\/mo><mi>z<\/mi><mo class=\"MathClass-rel\">|<\/mo><\/mrow> <mrow><mi>R<\/mi><\/mrow><\/mfrac> <mo class=\"MathClass-rel\">&lt;<\/mo> <mn>1<\/mn><\/math> und divergiert f\u00fcr <span class=\"maperiod\"><math display=\"inline\"><mfrac><mrow><mo class=\"MathClass-rel\">|<\/mo><mi>z<\/mi><mo class=\"MathClass-rel\">|<\/mo><\/mrow> <mrow><mi>R<\/mi><\/mrow><\/mfrac> <mo class=\"MathClass-rel\">&gt;<\/mo> <mn>1<\/mn><\/math><\/span><span class=\"period\">.<\/span> Die F\u00e4lle <math display=\"inline\"><mi>R<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mn>0<\/mn><\/math> und <math display=\"inline\"><mi>R<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mo class=\"MathClass-bin\">+<\/mo><mi>\u221e<\/mi><\/math> ergeben sich aus dem gleichen Argument (wieso?). <\/p><p class=\"indent\">Sei nun <span class=\"maperiod\"><math display=\"inline\"><mi>S<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mo class=\"MathClass-open\">(<\/mo><mn>0<\/mn><mo class=\"MathClass-punc\">,<\/mo><mi>R<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math><\/span><span class=\"period\">.<\/span> F\u00fcr den Beweis der gleichm\u00e4ssigen Konvergenz auf <math display=\"inline\"><msub><mrow><mi>B<\/mi><\/mrow><mrow><mi>S<\/mi> <\/mrow> <\/msub> <mo class=\"MathClass-open\">(<\/mo><mn>0<\/mn><mo class=\"MathClass-close\">)<\/mo><\/math> bemerken wir, dass nach obigem bereits <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><mo class=\"MathClass-rel\">|<\/mo><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><mo class=\"MathClass-rel\">|<\/mo><msup><mrow><mi>S<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msup> <mo class=\"MathClass-rel\">&lt;<\/mo> <mi>\u221e<\/mi><\/math> gilt. Daher existiert f\u00fcr jedes <math display=\"inline\"><mi>\ud835\udf00<\/mi> <mo class=\"MathClass-rel\">&gt;<\/mo> <mn>0<\/mn><\/math> ein <math display=\"inline\"><mi>N<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2115<\/mi><\/math> mit <span class=\"maperiod\"><math display=\"inline\"><msubsup><mrow><mi class=\"MathClass-op\">\u2211<\/mi><mo> <\/mo> <\/mrow><mrow><mi>n<\/mi><mo class=\"MathClass-rel\">=<\/mo><mi>N<\/mi><\/mrow><mrow><mi>\u221e<\/mi><\/mrow><\/msubsup><mo class=\"MathClass-rel\">|<\/mo><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><mo class=\"MathClass-rel\">|<\/mo><msup><mrow><mi>S<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msup> <mo class=\"MathClass-rel\">&lt;<\/mo> <mi>\ud835\udf00<\/mi><\/math><\/span><span class=\"period\">.<\/span> F\u00fcr alle <math display=\"inline\"><mi>z<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <msub><mrow><mi>B<\/mi><\/mrow><mrow><mi>S<\/mi> <\/mrow> <\/msub> <mo class=\"MathClass-open\">(<\/mo><mn>0<\/mn><mo class=\"MathClass-close\">)<\/mo><\/math> und <math display=\"inline\"><mi>n<\/mi> <mo class=\"MathClass-rel\">\u2265<\/mo> <mi>N<\/mi><\/math> gilt damit <\/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><munderover accent=\"false\" accentunder=\"false\"><mrow><mo>\u2211<\/mo> <\/mrow><mrow><mi>k<\/mi><mo class=\"MathClass-rel\">=<\/mo><mn>0<\/mn><\/mrow><mrow><mi>n<\/mi><\/mrow><\/munderover><msub><mrow><mi>a<\/mi><\/mrow><mrow> <mi>k<\/mi><\/mrow><\/msub><msup><mrow><mi>z<\/mi><\/mrow><mrow><mi>k<\/mi><\/mrow><\/msup> <mo class=\"MathClass-bin\">\u2212<\/mo><munderover accent=\"false\" accentunder=\"false\"><mrow><mo>\u2211<\/mo> <\/mrow><mrow><mi>k<\/mi><mo class=\"MathClass-rel\">=<\/mo><mn>0<\/mn><\/mrow><mrow><mi>\u221e<\/mi><\/mrow><\/munderover><msub><mrow><mi>a<\/mi><\/mrow><mrow> <mi>k<\/mi><\/mrow><\/msub><msup><mrow><mi>z<\/mi><\/mrow><mrow><mi>k<\/mi><\/mrow><\/msup><\/mrow><mo fence=\"true\" form=\"postfix\">|<\/mo><\/mrow> <mo class=\"MathClass-rel\">=<\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> |<\/mo><mrow><munderover accent=\"false\" accentunder=\"false\"><mrow><mo>\u2211<\/mo> <\/mrow><mrow><mi>k<\/mi><mo class=\"MathClass-rel\">=<\/mo><mi>n<\/mi><mo class=\"MathClass-bin\">+<\/mo><mn>1<\/mn><\/mrow><mrow><mi>\u221e<\/mi><\/mrow><\/munderover><msub><mrow><mi>a<\/mi><\/mrow><mrow> <mi>k<\/mi><\/mrow><\/msub><msup><mrow><mi>z<\/mi><\/mrow><mrow><mi>k<\/mi><\/mrow><\/msup><\/mrow><mo fence=\"true\" form=\"postfix\">|<\/mo><\/mrow> <mo class=\"MathClass-rel\">\u2264<\/mo><munderover accent=\"false\" accentunder=\"false\"><mrow><mo>\u2211<\/mo> <\/mrow><mrow><mi>k<\/mi><mo class=\"MathClass-rel\">=<\/mo><mi>N<\/mi><\/mrow><mrow><mi>\u221e<\/mi><\/mrow><\/munderover><mo class=\"MathClass-rel\">|<\/mo><msub><mrow><mi>a<\/mi><\/mrow><mrow> <mi>k<\/mi><\/mrow><\/msub><mo class=\"MathClass-rel\">|<\/mo><msup><mrow><mi>S<\/mi><\/mrow><mrow><mi>k<\/mi><\/mrow><\/msup> <mo class=\"MathClass-rel\">&lt;<\/mo> <mi>\ud835\udf00<\/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\">Dies beweist die gleichm\u00e4ssige Konvergenz der stetigen Funktionenfolge <math display=\"inline\"><msubsup><mrow><mi class=\"MathClass-op\">\u2211<\/mi><mo> <\/mo> <\/mrow><mrow><mi>k<\/mi><mo class=\"MathClass-rel\">=<\/mo><mn>0<\/mn><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msubsup><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>k<\/mi><\/mrow><\/msub><msup><mrow><mi>z<\/mi><\/mrow><mrow><mi>k<\/mi><\/mrow><\/msup><\/math> auf <math display=\"inline\"><msub><mrow><mi>B<\/mi><\/mrow><mrow><mi>S<\/mi> <\/mrow> <\/msub> <mo class=\"MathClass-open\">(<\/mo><mn>0<\/mn><mo class=\"MathClass-close\">)<\/mo><\/math> gegen <math display=\"inline\"><msubsup><mrow><mi class=\"MathClass-op\">\u2211<\/mi><mo> <\/mo> <\/mrow><mrow><mi>k<\/mi><mo class=\"MathClass-rel\">=<\/mo><mn>0<\/mn><\/mrow><mrow><mi>\u221e<\/mi><\/mrow><\/msubsup><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>k<\/mi><\/mrow><\/msub><msup><mrow><mi>z<\/mi><\/mrow><mrow><mi>k<\/mi><\/mrow><\/msup><\/math> und damit die Stetigkeit von <math display=\"inline\"><mi>z<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <msub><mrow><mi>B<\/mi><\/mrow><mrow><mi>S<\/mi><\/mrow><\/msub><mo class=\"MathClass-open\">(<\/mo><mn>0<\/mn><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-rel\">\u21a6<\/mo><msubsup><mrow><mi class=\"MathClass-op\">\u2211<\/mi><mo> <\/mo> <\/mrow><mrow><mi>k<\/mi><mo class=\"MathClass-rel\">=<\/mo><mn>0<\/mn><\/mrow><mrow><mi>\u221e<\/mi><\/mrow><\/msubsup><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>k<\/mi><\/mrow><\/msub><msup><mrow><mi>z<\/mi><\/mrow><mrow><mi>k<\/mi><\/mrow><\/msup> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2102<\/mi><\/math> nach Satz <a href=\"..\/..\/chapter\/konvergenz-von-funktionenfolgen#x1-198009r48\">7.48<\/a>. <\/p><p class=\"indent\">Insbesondere ist die Funktion <math display=\"inline\"><mi>z<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <msub><mrow><mi>B<\/mi><\/mrow><mrow><mi>R<\/mi><\/mrow><\/msub><mo class=\"MathClass-open\">(<\/mo><mn>0<\/mn><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-rel\">\u21a6<\/mo><msubsup><mrow><mi class=\"MathClass-op\">\u2211<\/mi><mo> <\/mo> <\/mrow><mrow><mi>n<\/mi><mo class=\"MathClass-rel\">=<\/mo><mn>0<\/mn><\/mrow><mrow><mi>\u221e<\/mi><\/mrow><\/msubsup><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><msup><mrow><mi>z<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msup> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2102<\/mi><\/math> stetig an jedem Punkt, da es zu <math display=\"inline\"><mi>z<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <msub><mrow><mi>B<\/mi><\/mrow><mrow><mi>R<\/mi><\/mrow><\/msub><mo class=\"MathClass-open\">(<\/mo><mn>0<\/mn><mo class=\"MathClass-close\">)<\/mo><\/math> ein <math display=\"inline\"><mi>S<\/mi> <mo class=\"MathClass-rel\">&lt;<\/mo> <mi>R<\/mi><\/math> gibt, mit <math display=\"inline\"><mi>z<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <msub><mrow><mi>B<\/mi><\/mrow><mrow><mi>S<\/mi> <\/mrow> <\/msub> <mo class=\"MathClass-open\">(<\/mo><mn>0<\/mn><mo class=\"MathClass-close\">)<\/mo><\/math> (wieso zeigt dies die Stetigkeit?). Dies beweist den Satz. <span>&nbsp;&nbsp;<\/span><\/p><div class=\"qed\">\u25a0<\/div><\/details><\/div> <div class=\"me meexample\"> <p class=\"indent\"><\/p><h4 id=\"z4b52e7ba7944\"> <a id=\"x1-200003r57\"><\/a> <span class=\"ecbx-1095\">Beispiel 7.57 <\/span>(Nicht gleichm\u00e4ssige Konvergenz)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Man k<\/span><span class=\"ecti-1095\">\u00f6<\/span><span class=\"ecti-1095\">nnte denken, dass Satz<\/span><span class=\"ecti-1095\">&nbsp;<\/span><a href=\"..\/..\/chapter\/potenzreihen#x1-200002r56\"><span class=\"ecti-1095\">7.56<\/span><\/a> <span class=\"ecti-1095\">eigentlich sagt, dass die Partialsummen<\/span><span class=\"ecti-1095\">&nbsp;<\/span><math display=\"inline\"><msubsup><mrow><mi class=\"MathClass-op\">\u2211<\/mi><mo> <\/mo> <\/mrow><mrow><mi>k<\/mi><mo class=\"MathClass-rel\">=<\/mo><mn>0<\/mn><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msubsup><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>k<\/mi><\/mrow><\/msub><msup><mrow><mi>z<\/mi><\/mrow><mrow><mi>k<\/mi><\/mrow><\/msup><\/math> <span class=\"ecti-1095\">der Potenzreihe auf ganz<\/span><span class=\"ecti-1095\">&nbsp;<\/span><math display=\"inline\"><msub><mrow><mi>B<\/mi><\/mrow><mrow><mi>R<\/mi><\/mrow><\/msub><mo class=\"MathClass-open\">(<\/mo><mn>0<\/mn><mo class=\"MathClass-close\">)<\/mo><\/math> <span class=\"ecti-1095\">gleichm<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">ssig gegen die durch die Potenzreihe definierte Funktion<\/span><span class=\"ecti-1095\">&nbsp;<\/span><math display=\"inline\"><mi>z<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <msub><mrow><mi>B<\/mi><\/mrow><mrow><mi>R<\/mi><\/mrow><\/msub><mo class=\"MathClass-open\">(<\/mo><mn>0<\/mn><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-rel\">\u21a6<\/mo><msubsup><mrow><mi class=\"MathClass-op\">\u2211<\/mi><mo> <\/mo> <\/mrow><mrow><mi>k<\/mi><mo class=\"MathClass-rel\">=<\/mo><mn>0<\/mn><\/mrow><mrow><mi>\u221e<\/mi><\/mrow><\/msubsup><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>k<\/mi><\/mrow><\/msub><msup><mrow><mi>z<\/mi><\/mrow><mrow><mi>k<\/mi><\/mrow><\/msup><\/math> <span class=\"ecti-1095\">streben, da ja<\/span><span class=\"ecti-1095\">&nbsp;<\/span><math display=\"inline\"><mi>S<\/mi> <mo class=\"MathClass-rel\">&lt;<\/mo> <mi>R<\/mi><\/math> <span class=\"ecti-1095\">beliebig ist. Dies ist aber nicht immer so (siehe auch <\/span><span class=\"ecti-1095\">\u00dc<\/span><span class=\"ecti-1095\">bung<\/span><span class=\"ecti-1095\">&nbsp;<\/span><a href=\"..\/..\/chapter\/konvergenz-von-funktionenfolgen#x1-198012r50\"><span class=\"ecti-1095\">7.50<\/span><\/a><span class=\"ecti-1095\">), wie wir hier kurz anhand<\/span> <span class=\"ecti-1095\">der geometrischen Reihe zeigen wollen.<\/span> <\/p><p class=\"indent\"><span class=\"ecti-1095\">F<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r<\/span><span class=\"ecti-1095\">&nbsp;<\/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>z<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msup><\/math> <span class=\"ecti-1095\">ist der<\/span> <span class=\"ecti-1095\">Konvergenzradius<\/span><span class=\"ecti-1095\">&nbsp;<\/span><math display=\"inline\"><mi>R<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mn>1<\/mn><\/math> <span class=\"ecti-1095\">und die mittels der Potenzreihe definierte Funktion<\/span> <span class=\"ecti-1095\">ist<\/span><span class=\"ecti-1095\">&nbsp;<\/span><math display=\"inline\"><mi>z<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <msub><mrow><mi>B<\/mi><\/mrow><mrow><mn>1<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-open\">(<\/mo><mn>0<\/mn><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-rel\">\u21a6<\/mo> <mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mn>1<\/mn><mo class=\"MathClass-bin\">\u2212<\/mo><mi>z<\/mi><\/mrow><\/mfrac><\/math><span class=\"ecti-1095\">. Falls die Konvergenz<\/span> <span class=\"ecti-1095\">auf ganz<\/span><span class=\"ecti-1095\">&nbsp;<\/span><math display=\"inline\"><msub><mrow><mi>B<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><mo class=\"MathClass-open\">(<\/mo><mn>0<\/mn><mo class=\"MathClass-close\">)<\/mo><\/math> <span class=\"ecti-1095\">gleichm<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">ssig<\/span> <span class=\"ecti-1095\">w<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">re, dann g<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">be es f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r<\/span><span class=\"ecti-1095\">&nbsp;<\/span><math display=\"inline\"><mi>\ud835\udf00<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mn>1<\/mn><\/math> <span class=\"ecti-1095\">ein<\/span><span class=\"ecti-1095\">&nbsp;<\/span><math display=\"inline\"><mi>N<\/mi><\/math> <span class=\"ecti-1095\">so dass<\/span> <span class=\"ecti-1095\">f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r alle<\/span><span class=\"ecti-1095\">&nbsp;<\/span><math display=\"inline\"><mi>n<\/mi> <mo class=\"MathClass-rel\">\u2265<\/mo> <mi>N<\/mi><\/math> <span class=\"ecti-1095\">und <\/span><math display=\"inline\"><mi>z<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <msub><mrow><mi>B<\/mi><\/mrow><mrow><mn>1<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-open\">(<\/mo><mn>0<\/mn><mo class=\"MathClass-close\">)<\/mo><\/math> <span class=\"ecti-1095\">die Absch<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">tzung<\/span> <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"> <mrow><mo fence=\"true\" form=\"prefix\"> |<\/mo><mrow><munderover accent=\"false\" accentunder=\"false\"><mrow><mo>\u2211<\/mo> <\/mrow><mrow><mi>k<\/mi><mo class=\"MathClass-rel\">=<\/mo><mn>0<\/mn><\/mrow><mrow><mi>n<\/mi><\/mrow><\/munderover><msup><mrow><mi>z<\/mi><\/mrow><mrow><mi>k<\/mi><\/mrow><\/msup> <mo class=\"MathClass-bin\">\u2212<\/mo> <mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mn>1<\/mn> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>z<\/mi><\/mrow><\/mfrac> <\/mrow><mo fence=\"true\" form=\"postfix\">|<\/mo><\/mrow> <mo class=\"MathClass-rel\">&lt;<\/mo> <mn>1<\/mn><\/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\">gelten w<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">rde. Wir setzen<\/span><span class=\"ecti-1095\">&nbsp;<\/span><math display=\"inline\"><mi>n<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mi>N<\/mi><\/math> <span class=\"ecti-1095\">und erhalten mittels der Dreiecksungleichung daraus<\/span> <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mstyle><mrow><mo fence=\"true\" form=\"prefix\"> |<\/mo><mrow \/><mo fence=\"true\" form=\"postfix\" \/><\/mrow><\/mstyle> <mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mn>1<\/mn> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>z<\/mi><\/mrow><\/mfrac><mstyle><mrow><mo fence=\"true\" form=\"prefix\"> |<\/mo><mrow \/><mo fence=\"true\" form=\"postfix\" \/><\/mrow><\/mstyle> <mo class=\"MathClass-rel\">&lt;<\/mo> <mn>1<\/mn> <mo class=\"MathClass-bin\">+<\/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>k<\/mi><mo class=\"MathClass-rel\">=<\/mo><mn>0<\/mn><\/mrow><mrow><mi>N<\/mi><\/mrow><\/munderover><msup><mrow><mi>z<\/mi><\/mrow><mrow><mi>k<\/mi><\/mrow><\/msup><mstyle><mrow><mo fence=\"true\" form=\"prefix\"> |<\/mo><mrow \/><mo fence=\"true\" form=\"postfix\" \/><\/mrow><\/mstyle> <mo class=\"MathClass-rel\">\u2264<\/mo> <mn>2<\/mn> <mo class=\"MathClass-bin\">+<\/mo> <mi>N<\/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\">f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r alle<\/span><span class=\"ecti-1095\">&nbsp;<\/span><span class=\"maperiod\"><math display=\"inline\"><mi>z<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <msub><mrow><mi>B<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><mo class=\"MathClass-open\">(<\/mo><mn>0<\/mn><mo class=\"MathClass-close\">)<\/mo><\/math><\/span><span class=\"period\">.<\/span> <span class=\"ecti-1095\">Dies ist aber ein Widerspruch, da<\/span> <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><munder class=\"msub\"><mrow><mi class=\"qopname\">lim<\/mi><mo>  <\/mo><\/mrow><mrow><mi>x<\/mi><mo class=\"MathClass-rel\">\u2197<\/mo><mn>1<\/mn><\/mrow><\/munder> <mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mn>1<\/mn> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>x<\/mi><\/mrow><\/mfrac> <mo class=\"MathClass-rel\">=<\/mo> <mo class=\"MathClass-bin\">+<\/mo><mi>\u221e<\/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> <\/div> <div class=\"me meexample\"> <p class=\"indent\"><\/p><h4 id=\"zcf911a8dd3bf\"> <a id=\"x1-200004r58\"><\/a> <span class=\"ecbx-1095\">\u00dc<\/span><span class=\"ecbx-1095\">bung 7.58 <\/span>(Konvergenzradien)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Finden Sie f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r jedes <\/span><math display=\"inline\"><mi>R<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mo class=\"MathClass-open\">[<\/mo><mn>0<\/mn><mo class=\"MathClass-punc\">,<\/mo><mi>\u221e<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">\u222a<\/mo><mrow><mo fence=\"true\" form=\"prefix\"> {<\/mo><mrow><mi>\u221e<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">}<\/mo><\/mrow><\/math> <span class=\"ecti-1095\">eine Potenzreihe mit Konvergenzradius<\/span><span class=\"ecti-1095\">&nbsp;<\/span><span class=\"maperiod\"><math display=\"inline\"><mi>R<\/mi><\/math><\/span><span class=\"period\">.<\/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\">F<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r<\/span> <math display=\"inline\"><mi>R<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mn>0<\/mn><\/math> <span class=\"ecti-1095\">m<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">ssen Sie eine Potenzreihe mit sehr schnell wachsenden Koeffizienten verwenden.<\/span><\/p><\/details>  <\/div> <div class=\"me meexample\"> <p class=\"indent\"><\/p><h4 id=\"z733398af5602\"> <a id=\"x1-200005r59\"><\/a> <span class=\"ecbx-1095\">\u00dc<\/span><span class=\"ecbx-1095\">bung 7.59.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Berechnen Sie den Konvergenzradius <\/span><math display=\"inline\"><mi>R<\/mi><\/math> <span class=\"ecti-1095\">der Potenzreihe<\/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>n<\/mi><mo class=\"MathClass-rel\">=<\/mo><mn>1<\/mn><\/mrow><mrow><mi>\u221e<\/mi><\/mrow><\/munderover><mfrac><mrow><msup><mrow><mo class=\"MathClass-open\">(<\/mo><msqrt><mrow><msup><mrow><mi>n<\/mi><\/mrow><mrow><mn>2<\/mn> <\/mrow> <\/msup> <mo class=\"MathClass-bin\">+<\/mo> <mi>n<\/mi><\/mrow><\/msqrt> <mo class=\"MathClass-bin\">\u2212<\/mo><msqrt><mrow><msup><mrow><mi>n<\/mi><\/mrow><mrow><mn>2<\/mn> <\/mrow> <\/msup> <mo class=\"MathClass-bin\">+<\/mo> <mn>1<\/mn><\/mrow><\/msqrt><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msup><\/mrow> <mrow><msup><mrow><mi>n<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><\/mrow><\/mfrac> <msup><mrow><mi>x<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msup><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\"><span class=\"ecti-1095\">und zeigen Sie Konvergenz der Potenzreihe bei den Punkten<\/span> <span class=\"maperiod\"><math display=\"inline\"><mo class=\"MathClass-bin\">\u2212<\/mo> <mi>R<\/mi><mo class=\"MathClass-punc\">,<\/mo> <mi>R<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u211d<\/mi><\/math><\/span><span class=\"period\">.<\/span> <\/p> <\/div> <div class=\"me melemma\"> <p class=\"indent\"><\/p><h4 id=\"za534835ee916\"> <a id=\"x1-200006r60\"><\/a> <span class=\"ecbx-1095\">Lemma 7.60 <\/span>(Konvergenzradius via Quotientenkriterium)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Sei <\/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><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><msup><mrow><mi>z<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msup><\/math> <span class=\"ecti-1095\">eine<\/span> <span class=\"ecti-1095\">Potenzreihe mit <\/span><math display=\"inline\"><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><mo class=\"MathClass-rel\">\u2260<\/mo><mn>0<\/mn><\/math> <span class=\"ecti-1095\">f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r alle <\/span><math display=\"inline\"><mi>n<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2115<\/mi><\/math><span class=\"ecti-1095\">. Der<\/span> <span class=\"ecti-1095\">Konvergenzradius <\/span><math display=\"inline\"><mi>R<\/mi><\/math> <span class=\"ecti-1095\">ist gegeben durch<\/span> <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mi>R<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><munder class=\"msub\"><mrow><mi class=\"qopname\">lim<\/mi><mo>  <\/mo><\/mrow><mrow><mi>n<\/mi><mo class=\"MathClass-rel\">\u2192<\/mo><mi>\u221e<\/mi><\/mrow><\/munder><mfrac><mrow><mo class=\"MathClass-rel\">|<\/mo><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi><mo class=\"MathClass-bin\">+<\/mo><mn>1<\/mn><\/mrow><\/msub><mo class=\"MathClass-rel\">|<\/mo><\/mrow> <mrow><mo class=\"MathClass-rel\">|<\/mo><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><mo class=\"MathClass-rel\">|<\/mo><\/mrow><\/mfrac> <\/mrow><\/mfrac> <mo class=\"MathClass-rel\">=<\/mo><munder class=\"msub\"><mrow><mi class=\"qopname\"> lim<\/mi><mo>  <\/mo><\/mrow><mrow><mi>n<\/mi><mo class=\"MathClass-rel\">\u2192<\/mo><mi>\u221e<\/mi><\/mrow><\/munder> <mfrac><mrow><mo class=\"MathClass-rel\">|<\/mo><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><mo class=\"MathClass-rel\">|<\/mo><\/mrow> <mrow><mo class=\"MathClass-rel\">|<\/mo><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi><mo class=\"MathClass-bin\">+<\/mo><mn>1<\/mn><\/mrow><\/msub><mo class=\"MathClass-rel\">|<\/mo><\/mrow><\/mfrac><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\"><span class=\"ecti-1095\">falls dieser Grenzwert existiert.<\/span> <\/p> <\/div> <div class=\"me meexample\"> <p class=\"indent\"><\/p><h4 id=\"zc0de8309f61b\"> <a id=\"x1-200007r61\"><\/a> <span class=\"ecbx-1095\">\u00dc<\/span><span class=\"ecbx-1095\">bung 7.61.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Zeigen Sie Lemma<\/span><span class=\"ecti-1095\">&nbsp;<\/span><a href=\"..\/..\/chapter\/potenzreihen#x1-200006r60\"><span class=\"ecti-1095\">7.60<\/span><\/a><span class=\"ecti-1095\">.<\/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\">Wiederholen Sie den Beweis von Satz <\/span><a href=\"..\/..\/chapter\/potenzreihen#x1-200002r56\"><span class=\"ecti-1095\">7.56<\/span><\/a><span class=\"ecti-1095\">.<\/span><\/p><\/details>  <\/div> <a id=\"x1-200008r200\"><\/a> <h4 id=\"zafe02ce450a2\" class=\"subsectionHead\"><span class=\"titlemark\">7.4.2 <\/span> <a id=\"x1-2010002\"><\/a>Addition und Multiplikation<\/h4> <div class=\"me metheorem\"> <p class=\"indent\"><\/p><h4 id=\"z41f961ddfa59\"> <a id=\"x1-201001r62\"><\/a> <span class=\"ecbx-1095\">Proposition 7.62 <\/span>(Summen und Produkte)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Seien <\/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><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><msup><mrow><mi>z<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msup><\/math> <span class=\"ecti-1095\">und<\/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><msub><mrow><mi>b<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><msup><mrow><mi>z<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msup><\/math> <span class=\"ecti-1095\">zwei Potenzreihen mit<\/span> <span class=\"ecti-1095\">Konvergenzradius <\/span><math display=\"inline\"><msub><mrow><mi>R<\/mi><\/mrow><mrow><mi>a<\/mi><\/mrow><\/msub><\/math> <span class=\"ecti-1095\">respektive <\/span><math display=\"inline\"><msub><mrow><mi>R<\/mi><\/mrow><mrow><mi>b<\/mi><\/mrow><\/msub><\/math><span class=\"ecti-1095\">. Dann<\/span> <span class=\"ecti-1095\">gilt f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r alle <\/span><math display=\"inline\"><mi>z<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2102<\/mi><\/math> <span class=\"ecti-1095\">mit <\/span><math display=\"inline\"><mo class=\"MathClass-rel\">|<\/mo><mi>z<\/mi><mo class=\"MathClass-rel\">|<\/mo> <mo class=\"MathClass-rel\">&lt;<\/mo><mi class=\"qopname\"> min<\/mi><mo>  <\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> {<\/mo><mrow><msub><mrow><mi>R<\/mi><\/mrow><mrow><mi>a<\/mi><\/mrow><\/msub><mo class=\"MathClass-punc\">,<\/mo><msub><mrow><mi>R<\/mi><\/mrow><mrow><mi>b<\/mi><\/mrow><\/msub><\/mrow><mo fence=\"true\" form=\"postfix\">}<\/mo><\/mrow><\/math> <\/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>n<\/mi><mo class=\"MathClass-rel\">=<\/mo><mn>0<\/mn><\/mrow><mrow><mi>\u221e<\/mi><\/mrow><\/munderover><msub><mrow><mi>a<\/mi><\/mrow><mrow> <mi>n<\/mi><\/mrow><\/msub><msup><mrow><mi>z<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msup> <mo class=\"MathClass-bin\">+<\/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><msub><mrow><mi>b<\/mi><\/mrow><mrow> <mi>n<\/mi><\/mrow><\/msub><msup><mrow><mi>z<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msup><\/mtd> <mtd class=\"align-even\"> <mo class=\"MathClass-rel\">=<\/mo><munderover accent=\"false\" accentunder=\"false\"><mrow><mo> \u2211<\/mo> <\/mrow><mrow><mi>n<\/mi><mo class=\"MathClass-rel\">=<\/mo><mn>0<\/mn><\/mrow><mrow><mi>\u221e<\/mi><\/mrow><\/munderover><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>a<\/mi><\/mrow><mrow> <mi>n<\/mi><\/mrow><\/msub> <mo class=\"MathClass-bin\">+<\/mo> <msub><mrow><mi>b<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><msup><mrow><mi>z<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msup><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\"><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>n<\/mi><mo class=\"MathClass-rel\">=<\/mo><mn>0<\/mn><\/mrow><mrow><mi>\u221e<\/mi><\/mrow><\/munderover><msub><mrow><mi>a<\/mi><\/mrow><mrow> <mi>n<\/mi><\/mrow><\/msub><msup><mrow><mi>z<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msup><mstyle><mrow><mo fence=\"true\" form=\"prefix\"> )<\/mo><mrow \/><mo fence=\"true\" form=\"postfix\" \/><\/mrow><\/mstyle><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>n<\/mi><mo class=\"MathClass-rel\">=<\/mo><mn>0<\/mn><\/mrow><mrow><mi>\u221e<\/mi><\/mrow><\/munderover><msub><mrow><mi>b<\/mi><\/mrow><mrow> <mi>n<\/mi><\/mrow><\/msub><msup><mrow><mi>z<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msup><mstyle><mrow><mo fence=\"true\" form=\"prefix\"> )<\/mo><mrow \/><mo fence=\"true\" form=\"postfix\" \/><\/mrow><\/mstyle><\/mtd> <mtd class=\"align-even\"> <mo class=\"MathClass-rel\">=<\/mo><munderover accent=\"false\" accentunder=\"false\"><mrow><mo> \u2211<\/mo> <\/mrow><mrow><mi>n<\/mi><mo class=\"MathClass-rel\">=<\/mo><mn>0<\/mn><\/mrow><mrow><mi>\u221e<\/mi><\/mrow><\/munderover><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>k<\/mi><mo class=\"MathClass-rel\">=<\/mo><mn>0<\/mn><\/mrow><mrow><mi>n<\/mi><\/mrow><\/munderover><msub><mrow><mi>a<\/mi><\/mrow><mrow> <mi>n<\/mi><mo class=\"MathClass-bin\">\u2212<\/mo><mi>k<\/mi><\/mrow><\/msub><msub><mrow><mi>b<\/mi><\/mrow><mrow><mi>k<\/mi><\/mrow><\/msub><mstyle><mrow><mo fence=\"true\" form=\"prefix\"> )<\/mo><mrow \/><mo fence=\"true\" form=\"postfix\" \/><\/mrow><\/mstyle><msup><mrow><mi>z<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msup><mo class=\"MathClass-punc\">.<\/mo><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\" \/> <mtd class=\"align-label\"> <mspace width=\"2em\" \/><\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\"><span class=\"ecti-1095\">Insbesondere ist der Konvergenzradius der Potenzreihen auf der rechten Seite mindestens<\/span> <span class=\"maperiod\"><math display=\"inline\"><mi class=\"qopname\">min<\/mi><mo>  <\/mo><mrow><mo fence=\"true\" form=\"prefix\"> {<\/mo><mrow><msub><mrow><mi>R<\/mi><\/mrow><mrow><mi>a<\/mi> <\/mrow> <\/msub> <mo class=\"MathClass-punc\">,<\/mo> <msub><mrow><mi>R<\/mi><\/mrow><mrow><mi>b<\/mi> <\/mrow> <\/msub><\/mrow><mo fence=\"true\" form=\"postfix\">}<\/mo><\/mrow><\/math><\/span><span class=\"period\">.<\/span> <\/p> <\/div> <p class=\"indent\"> <\/p> <div class=\"proof\"> <p class=\"indent\"><span class=\"head\"><\/span><\/p><details open><summary><b>Beweis.<\/b><\/summary><p class=\"indent\" style=\"margin-top: 10\">Die erste Eigenschaft folgt aus Linearit\u00e4t des Grenzwerts. Die zweite verwendet noch Korollar <a href=\"..\/..\/chapter\/absolute-konvergenz#x1-195003r37\">7.37<\/a>. <span>&nbsp;&nbsp;<\/span><\/p><div class=\"qed\">\u25a0<\/div><\/details><\/div> <div class=\"me meexample\"> <p class=\"indent\"><\/p><h4 id=\"z46ec4c2b4ed2\"> <a id=\"x1-201002r63\"><\/a> <span class=\"ecbx-1095\">Beispiel 7.63.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Falls <\/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><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><msup><mrow><mi>z<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msup><\/math> <span class=\"ecti-1095\">mindestens<\/span> <span class=\"ecti-1095\">Konvergenzradius <\/span><math display=\"inline\"><mn>1<\/mn><\/math> <span class=\"ecti-1095\">hat, so gilt<\/span> <\/p><math display=\"block\"><mtable class=\"align\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"> <mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mn>1<\/mn> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>z<\/mi><\/mrow><\/mfrac><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><msub><mrow><mi>a<\/mi><\/mrow><mrow> <mi>n<\/mi><\/mrow><\/msub><msup><mrow><mi>z<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msup> <mo class=\"MathClass-rel\">=<\/mo><munderover accent=\"false\" accentunder=\"false\"><mrow><mo> \u2211<\/mo> <\/mrow><mrow><mi>n<\/mi><mo class=\"MathClass-rel\">=<\/mo><mn>0<\/mn><\/mrow><mrow><mi>\u221e<\/mi><\/mrow><\/munderover><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>a<\/mi><\/mrow><mrow> <mn>0<\/mn><\/mrow><\/msub> <mo class=\"MathClass-bin\">+<\/mo> <mo>\u2026<\/mo> <mo class=\"MathClass-bin\">+<\/mo> <msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><msup><mrow><mi>z<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msup><mo class=\"MathClass-punc\">.<\/mo><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"><mstyle class=\"label\" id=\"x1-201003r6\" \/><mstyle class=\"maketag\"><mtext>(7.6)<\/mtext><\/mstyle><mspace class=\"nbsp\" width=\"0.33em\" \/> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\"><span class=\"ecti-1095\">f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r alle <\/span><math display=\"inline\"><mi>z<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2102<\/mi><\/math> <span class=\"ecti-1095\">mit<\/span> <math display=\"inline\"><mo class=\"MathClass-rel\">|<\/mo><mi>z<\/mi><mo class=\"MathClass-rel\">|<\/mo> <mo class=\"MathClass-rel\">&lt;<\/mo> <mn>1<\/mn><\/math><span class=\"ecti-1095\">. In der Tat hat<\/span> <span class=\"ecti-1095\">die Potenzreihe <\/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>z<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msup><\/math> <span class=\"ecti-1095\">Konvergenzradius <\/span><math display=\"inline\"><mn>1<\/mn><\/math> <span class=\"ecti-1095\">und f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r <\/span><math display=\"inline\"><mi>z<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2102<\/mi><\/math> <span class=\"ecti-1095\">mit <\/span><math display=\"inline\"><mo class=\"MathClass-rel\">|<\/mo><mi>z<\/mi><mo class=\"MathClass-rel\">|<\/mo> <mo class=\"MathClass-rel\">&lt;<\/mo> <mn>1<\/mn><\/math> <span class=\"ecti-1095\">gilt<\/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>z<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msup> <mo class=\"MathClass-rel\">=<\/mo> <mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mn>1<\/mn><mo class=\"MathClass-bin\">\u2212<\/mo><mi>z<\/mi><\/mrow><\/mfrac><\/math><span class=\"ecti-1095\">, womit<\/span> (<a href=\"..\/..\/chapter\/potenzreihen#x1-201003r6\">7.6<\/a>)<span class=\"ecti-1095\">aus Proposition <\/span><a href=\"..\/..\/chapter\/potenzreihen#x1-201001r62\"><span class=\"ecti-1095\">7.62<\/span><\/a> <span class=\"ecti-1095\">folgt.<\/span> <\/p> <\/div> <div class=\"me meexample\"> <p class=\"indent\"><\/p><h4 id=\"zdf83828b0955\"> <a id=\"x1-201004r64\"><\/a> <span class=\"ecbx-1095\">\u00dc<\/span><span class=\"ecbx-1095\">bung 7.64.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Berechnen Sie <\/span><span class=\"maperiod\"><math display=\"inline\"><msubsup><mrow><mi class=\"MathClass-op\">\u2211<\/mi><mo> <\/mo> <\/mrow><mrow><mi>n<\/mi><mo class=\"MathClass-rel\">=<\/mo><mn>1<\/mn><\/mrow><mrow><mi>\u221e<\/mi><\/mrow><\/msubsup><mi>n<\/mi><msup><mrow><mn>2<\/mn><\/mrow><mrow><mo class=\"MathClass-bin\">\u2212<\/mo><mi>n<\/mi><\/mrow><\/msup><\/math><\/span><span class=\"period\">.<\/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\">Der                                          Wert                                          von<\/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>q<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msup><\/math> <span class=\"ecti-1095\">f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r                                                                                                         alle<\/span> <math display=\"inline\"><mi>q<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2102<\/mi><\/math> <span class=\"ecti-1095\">mit<\/span> <math display=\"inline\"><mo class=\"MathClass-rel\">|<\/mo><mi>q<\/mi><mo class=\"MathClass-rel\">|<\/mo> <mo class=\"MathClass-rel\">&lt;<\/mo> <mn>1<\/mn><\/math> <span class=\"ecti-1095\">ist bereits bekannt.<\/span><\/p><\/details>  <\/div> <a id=\"x1-201005r201\"><\/a> <h4 id=\"z31639062cec2\" class=\"subsectionHead\"><span class=\"titlemark\">7.4.3 <\/span> <a id=\"x1-2020003\"><\/a>Stetigkeit bei Randpunkten<\/h4> <p class=\"noindent\">Wir wollen nun eine Potenzreihe <math display=\"inline\"><msubsup><mrow><mi class=\"MathClass-op\"> \u2211<\/mi><mo> <\/mo> <\/mrow><mrow><mi>n<\/mi><mo class=\"MathClass-rel\">=<\/mo><mn>0<\/mn><\/mrow><mrow><mi>\u221e<\/mi><\/mrow><\/msubsup><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><msup><mrow><mi>x<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msup><\/math> betrachten, wobei wir <math display=\"inline\"><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2102<\/mi><\/math> f\u00fcr <math display=\"inline\"><mi>n<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2115<\/mi><\/math> erlauben, aber nur reelle Zahlen <math display=\"inline\"><mi>x<\/mi><\/math> einsetzen wollen. Nach Satz <a href=\"..\/..\/chapter\/potenzreihen#x1-200002r56\">7.56<\/a> existiert ein <span class=\"maperiod\"><math display=\"inline\"><mi>R<\/mi> <mo class=\"MathClass-rel\">\u2265<\/mo> <mn>0<\/mn><\/math><\/span><span class=\"period\">,<\/span> so dass die Funktion                                                                                                                                                                           <\/p><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> <mo class=\"MathClass-open\">(<\/mo><mo class=\"MathClass-bin\">\u2212<\/mo><mi>R<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>R<\/mi><mo class=\"MathClass-close\">)<\/mo><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><msub><mrow><mi>a<\/mi><\/mrow><mrow> <mi>n<\/mi><\/mrow><\/msub><msup><mrow><mi>x<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msup> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2102<\/mi><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">wohldefiniert und stetig ist, aber <math display=\"inline\"><msubsup><mrow><mi class=\"MathClass-op\"> \u2211<\/mi><mo> <\/mo> <\/mrow><mrow><mi>n<\/mi><mo class=\"MathClass-rel\">=<\/mo><mn>0<\/mn><\/mrow><mrow><mi>\u221e<\/mi><\/mrow><\/msubsup><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><msup><mrow><mi>x<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msup><\/math> f\u00fcr alle <math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u211d<\/mi><\/math> mit <math display=\"inline\"><mo class=\"MathClass-rel\">|<\/mo><mi>x<\/mi><mo class=\"MathClass-rel\">|<\/mo> <mo class=\"MathClass-rel\">&gt;<\/mo> <mi>R<\/mi><\/math> divergiert. Wir nehmen nun weiter an, dass <math display=\"inline\"><mi>R<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mo class=\"MathClass-open\">(<\/mo><mn>0<\/mn><mo class=\"MathClass-punc\">,<\/mo><mi>\u221e<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math> und <math display=\"inline\"><msubsup><mrow><mi class=\"MathClass-op\"> \u2211<\/mi><mo> <\/mo> <\/mrow><mrow><mi>n<\/mi><mo class=\"MathClass-rel\">=<\/mo><mn>0<\/mn><\/mrow><mrow><mi>\u221e<\/mi><\/mrow><\/msubsup><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><msup><mrow><mi>R<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msup><\/math> ebenfalls konvergiert. Satz <a href=\"..\/..\/chapter\/potenzreihen#x1-200002r56\">7.56<\/a> sagt in diesem Fall \u00fcberhaupt nichts \u00fcber die erweiterte Funktion <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mover accent=\"true\"><mrow><mi>f<\/mi><\/mrow><mo accent=\"true\">\u00af<\/mo><\/mover> <mo class=\"MathClass-punc\">:<\/mo> <mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mo class=\"MathClass-open\">(<\/mo><mo class=\"MathClass-bin\">\u2212<\/mo><mi>R<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>R<\/mi><mo class=\"MathClass-close\">]<\/mo><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><msub><mrow><mi>a<\/mi><\/mrow><mrow> <mi>n<\/mi><\/mrow><\/msub><msup><mrow><mi>x<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msup> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2102<\/mi><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">aus. <\/p> <div class=\"me metheorem\"> <p class=\"indent\"><\/p><h4 id=\"z26cf5e909c27\"> <a id=\"x1-202001r65\"><\/a> <span class=\"ecbx-1095\">Satz 7.65 <\/span>(Abelscher Grenzwertsatz)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Unter obigen Annahmen ist auch <\/span><math display=\"inline\"><mover accent=\"true\"><mrow><mi>f<\/mi><\/mrow><mo accent=\"true\">\u00af<\/mo><\/mover><\/math> <span class=\"ecti-1095\">stetig. Das heisst,<\/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>n<\/mi><mo class=\"MathClass-rel\">=<\/mo><mn>0<\/mn><\/mrow><mrow><mi>\u221e<\/mi><\/mrow><\/munderover><msub><mrow><mi>a<\/mi><\/mrow><mrow> <mi>n<\/mi><\/mrow><\/msub><msup><mrow><mi>R<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msup> <mo class=\"MathClass-rel\">=<\/mo> <mover accent=\"true\"><mrow><mi>f<\/mi><\/mrow><mo accent=\"true\">\u00af<\/mo><\/mover> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>R<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-rel\">=<\/mo><munder class=\"msub\"><mrow><mi class=\"qopname\"> lim<\/mi><mo>  <\/mo><\/mrow><mrow> <mi>x<\/mi><mo class=\"MathClass-rel\">\u2197<\/mo><mi>R<\/mi><\/mrow><\/munder><mover accent=\"true\"><mrow><mi>f<\/mi><\/mrow><mo accent=\"true\">\u00af<\/mo><\/mover> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-rel\">=<\/mo><munder class=\"msub\"><mrow><mi class=\"qopname\"> lim<\/mi><mo>  <\/mo><\/mrow><mrow><mi>x<\/mi><mo class=\"MathClass-rel\">\u2197<\/mo><mi>R<\/mi><\/mrow><\/munder><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><msub><mrow><mi>a<\/mi><\/mrow><mrow> <mi>n<\/mi><\/mrow><\/msub><msup><mrow><mi>x<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msup><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\"><span class=\"ecti-1095\">Eine analoge Aussage gilt, falls <\/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><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><msup><mrow><mo class=\"MathClass-open\">(<\/mo><mo class=\"MathClass-bin\">\u2212<\/mo><mi>R<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msup><\/math> <span class=\"ecti-1095\">konvergiert.<\/span> <\/p> <\/div> <div class=\"me meexample\"> <p class=\"indent\"><\/p><h4 id=\"z8d6daf8598a0\"> <a id=\"x1-202002r66\"><\/a> <span class=\"ecbx-1095\">Beispiel 7.66 <\/span>(Zwei alternierende Potenzreihen)<span class=\"ecbx-1095\">.<\/span> <\/h4> <dl class=\"enumerate\"><dt class=\"enumerate\"> <span class=\"ecti-1095\">(i)<\/span><\/dt><dd class=\"enumerate\"><span class=\"ecti-1095\">F<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r <\/span><math display=\"inline\"><msub><mrow><mi>a<\/mi><\/mrow><mrow><mn>0<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-rel\">=<\/mo> <mn>0<\/mn><\/math> <span class=\"ecti-1095\">und <\/span><math display=\"inline\"><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi> <\/mrow> <\/msub> <mo class=\"MathClass-rel\">=<\/mo> <mfrac><mrow><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><mo class=\"MathClass-bin\">+<\/mo><mn>1<\/mn><\/mrow><\/msup><\/mrow> <mrow><mi>n<\/mi><\/mrow><\/mfrac> <\/math> <span class=\"ecti-1095\">f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r alle <\/span><math display=\"inline\"><mi>n<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2115<\/mi><\/math> <span class=\"ecti-1095\">ist der Konvergenzradius der Potenzreihe <\/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><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><msup><mrow><mi>z<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msup><\/math> <span class=\"ecti-1095\">durch <\/span><math display=\"inline\"><mi>R<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mn>1<\/mn><\/math> <span class=\"ecti-1095\">gegeben und <\/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>1<\/mn><\/mrow><mrow><mi>\u221e<\/mi><\/mrow><\/msubsup><mfrac><mrow><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><mo class=\"MathClass-bin\">+<\/mo><mn>1<\/mn><\/mrow><\/msup><\/mrow> <mrow><mi>n<\/mi><\/mrow><\/mfrac> <\/math> <span class=\"ecti-1095\">konvergiert, womit der Abelsche Grenzwertsatz (Satz <\/span><a href=\"..\/..\/chapter\/potenzreihen#x1-202001r65\"><span class=\"ecti-1095\">7.65<\/span><\/a><span class=\"ecti-1095\">) angewendet werden kann.<\/span> <span class=\"ecti-1095\">Sobald wir die Funktion<\/span><span class=\"ecti-1095\">&nbsp;<\/span><math display=\"inline\"><mi>f<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-rel\">=<\/mo><msubsup><mrow><mi class=\"MathClass-op\"> \u2211<\/mi><mo> <\/mo> <\/mrow><mrow><mi>n<\/mi><mo class=\"MathClass-rel\">=<\/mo><mn>1<\/mn><\/mrow><mrow><mi>\u221e<\/mi><\/mrow><\/msubsup><mfrac><mrow><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><mo class=\"MathClass-bin\">+<\/mo><mn>1<\/mn><\/mrow><\/msup><\/mrow> <mrow><mi>n<\/mi><\/mrow><\/mfrac> <msup><mrow><mi>x<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msup><\/math> <span class=\"ecti-1095\">f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r<\/span><span class=\"ecti-1095\">&nbsp;<\/span><math display=\"inline\"><mo class=\"MathClass-rel\">|<\/mo><mi>x<\/mi><mo class=\"MathClass-rel\">|<\/mo> <mo class=\"MathClass-rel\">&lt;<\/mo> <mn>1<\/mn><\/math> <span class=\"ecti-1095\">kennen, k<\/span><span class=\"ecti-1095\">\u00f6<\/span><span class=\"ecti-1095\">nnen wir damit auch <\/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>1<\/mn><\/mrow><mrow><mi>\u221e<\/mi><\/mrow><\/msubsup><mfrac><mrow><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><mo class=\"MathClass-bin\">+<\/mo><mn>1<\/mn><\/mrow><\/msup><\/mrow> <mrow><mi>n<\/mi><\/mrow><\/mfrac> <\/math> <span class=\"ecti-1095\">berechnen.<\/span> <\/dd><dt class=\"enumerate\"> <span class=\"ecti-1095\">(ii)<\/span><\/dt><dd class=\"enumerate\"><span class=\"ecti-1095\">F<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r <\/span><math display=\"inline\"><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi> <\/mrow> <\/msub> <mo class=\"MathClass-rel\">=<\/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><\/math> <span class=\"ecti-1095\">f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r alle <\/span><math display=\"inline\"><mi>n<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <msub><mrow><mi>\u2115<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><\/math> <span class=\"ecti-1095\">ist <\/span><math display=\"inline\"><mi>R<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mn>1<\/mn><\/math> <span class=\"ecti-1095\">und der Abelsche Grenzwertsatz (Satz <\/span><a href=\"..\/..\/chapter\/potenzreihen#x1-202001r65\"><span class=\"ecti-1095\">7.65<\/span><\/a><span class=\"ecti-1095\">) kann nicht angewendet werden, da <\/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><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><\/math> <span class=\"ecti-1095\">divergiert.<\/span><\/dd><\/dl> <\/div> <div class=\"me meexample\"> <p class=\"indent\"><\/p><h4 id=\"z104e6a94857e\"> <span class=\"ecti-1095\">Bemerkung.<\/span><\/h4> <p class=\"indent\">Hierzu eine   historische   Anmerkung:   Euler   (1707-1783)   und   seine   Zeitgenossen hatten noch einen  anderen  Zugang  zu  Reihen  und  wiesen  auf  Grund  der  Gleichung <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><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>x<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msup> <mo class=\"MathClass-rel\">=<\/mo> <mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mn>1<\/mn><mo class=\"MathClass-bin\">+<\/mo><mi>x<\/mi><\/mrow><\/mfrac><\/math> f\u00fcr <math display=\"inline\"><mo class=\"MathClass-rel\">|<\/mo><mi>x<\/mi><mo class=\"MathClass-rel\">|<\/mo> <mo class=\"MathClass-rel\">&lt;<\/mo> <mn>1<\/mn><\/math> der                                                                                                              Reihe <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><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> <mo class=\"MathClass-rel\">=<\/mo> <mn>1<\/mn> <mo class=\"MathClass-bin\">\u2212<\/mo> <mn>1<\/mn> <mo class=\"MathClass-bin\">+<\/mo> <mn>1<\/mn> <mo class=\"MathClass-bin\">\u2212<\/mo> <mn>1<\/mn> <mo class=\"MathClass-bin\">+<\/mo> <mo class=\"MathClass-rel\">\u22ef<\/mo><mspace class=\"thinspace\" width=\"0.17em\" \/><\/math> den                                                                                                               Wert <math display=\"inline\"><mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mn>2<\/mn><\/mrow><\/mfrac><\/math> zu, was aber unserem  modernerem  Konvergenzbegriff  (und  insbesondere  Proposition&nbsp;<a href=\"..\/..\/chapter\/reihen#x1-187002r2\">7.2<\/a>) widerspricht. <\/p> <\/div> <p class=\"indent\"> <\/p> <div class=\"proof\"> <p class=\"indent\"><span class=\"head\"><\/span><\/p><details open><summary><b>Beweis des Abelschen Grenzwertsatzes.<\/b><\/summary><p class=\"indent\" style=\"margin-top: 10\"> Wir nehmen ohne Beschr\u00e4nkung der Allgemeinheit an, dass der Konvergenzradius <math display=\"inline\"><mn>1<\/mn><\/math> ist (sonst ersetzt man <math display=\"inline\"><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><\/math> mit <math display=\"inline\"><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi> <\/mrow> <\/msub> <msup><mrow><mi>R<\/mi><\/mrow><mrow><mi>n<\/mi> <\/mrow> <\/msup> <\/math> f\u00fcr alle <math display=\"inline\"><mi>n<\/mi><\/math>). Nach Beispiel&nbsp;<a href=\"..\/..\/chapter\/potenzreihen#x1-201002r63\">7.63<\/a> gilt <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"> <mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mn>1<\/mn> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>x<\/mi><\/mrow><\/mfrac><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><msub><mrow><mi>a<\/mi><\/mrow><mrow> <mi>n<\/mi><\/mrow><\/msub><msup><mrow><mi>x<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msup> <mo class=\"MathClass-rel\">=<\/mo><munderover accent=\"false\" accentunder=\"false\"><mrow><mo> \u2211<\/mo> <\/mrow><mrow><mi>n<\/mi><mo class=\"MathClass-rel\">=<\/mo><mn>0<\/mn><\/mrow><mrow><mi>\u221e<\/mi><\/mrow><\/munderover><mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><msub><mrow><mi>a<\/mi><\/mrow><mrow> <mn>0<\/mn><\/mrow><\/msub> <mo class=\"MathClass-bin\">+<\/mo> <mo>\u2026<\/mo> <mo class=\"MathClass-bin\">+<\/mo> <msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <msup><mrow><mi>x<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msup><mo class=\"MathClass-punc\">.<\/mo><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">f\u00fcr alle <span class=\"maperiod\"><math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mo class=\"MathClass-open\">(<\/mo><mo class=\"MathClass-bin\">\u2212<\/mo><mn>1<\/mn><mo class=\"MathClass-punc\">,<\/mo><mn>1<\/mn><mo class=\"MathClass-close\">)<\/mo><\/math><\/span><span class=\"period\">.<\/span> Wir definieren <span class=\"maperiod\"><math display=\"inline\"><msub><mrow><mi>A<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub> <mo class=\"MathClass-rel\">=<\/mo> <msub><mrow><mi>a<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub> <mo class=\"MathClass-bin\">+<\/mo> <mi class=\"MathClass-op\">\u2026<\/mi><mo> <\/mo> <mo class=\"MathClass-bin\">+<\/mo> <msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><\/math><\/span><span class=\"period\">,<\/span> <math display=\"inline\"><mi>A<\/mi> <mo class=\"MathClass-rel\">=<\/mo><munder class=\"msub\"><mrow><mi class=\"qopname\"> lim<\/mi><mo>  <\/mo> <\/mrow><mrow><mi>n<\/mi><mo class=\"MathClass-rel\">\u2192<\/mo><mi>\u221e<\/mi><\/mrow><\/munder><msub><mrow><mi>A<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub> <mo class=\"MathClass-rel\">=<\/mo><msubsup><mrow><mi class=\"qopname\"> \u2211<\/mi><mo>  <\/mo> <\/mrow><mrow><mi>n<\/mi><mo class=\"MathClass-rel\">=<\/mo><mn>0<\/mn><\/mrow><mrow><mi>\u221e<\/mi><\/mrow><\/msubsup><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><\/math> (was nach Annahme existiert) und erhalten mit <math display=\"inline\"><msub><mrow><mi>b<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub> <mo class=\"MathClass-rel\">=<\/mo> <msub><mrow><mi>A<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>A<\/mi><\/math> f\u00fcr <math display=\"inline\"><mi>n<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2115<\/mi><\/math>                                                                                                                                                                           die Gleichung <\/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> <mo class=\"MathClass-rel\">=<\/mo><munderover accent=\"false\" accentunder=\"false\"><mrow><mo> \u2211<\/mo> <\/mrow><mrow><mi>n<\/mi><mo class=\"MathClass-rel\">=<\/mo><mn>0<\/mn><\/mrow><mrow><mi>\u221e<\/mi><\/mrow><\/munderover><msub><mrow><mi>a<\/mi><\/mrow><mrow> <mi>n<\/mi><\/mrow><\/msub><msup><mrow><mi>x<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msup><\/mtd> <mtd class=\"align-even\"> <mo class=\"MathClass-rel\">=<\/mo> <mo class=\"MathClass-open\">(<\/mo><mn>1<\/mn> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>x<\/mi><mo class=\"MathClass-close\">)<\/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><msub><mrow><mi>A<\/mi><\/mrow><mrow> <mi>n<\/mi><\/mrow><\/msub><msup><mrow><mi>x<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msup><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\" \/> <mtd class=\"align-label\"> <mspace width=\"2em\" \/><\/mtd><\/mtr><mtr><mtd class=\"align-odd\" columnalign=\"right\" \/> <mtd class=\"align-even\"> <mo class=\"MathClass-rel\">=<\/mo> <mo class=\"MathClass-open\">(<\/mo><mn>1<\/mn> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>x<\/mi><mo class=\"MathClass-close\">)<\/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><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>b<\/mi><\/mrow><mrow> <mi>n<\/mi><\/mrow><\/msub> <mo class=\"MathClass-bin\">+<\/mo> <mi>A<\/mi><mo class=\"MathClass-close\">)<\/mo><msup><mrow><mi>x<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msup><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\" \/> <mtd class=\"align-label\"> <mspace width=\"2em\" \/><\/mtd><\/mtr><mtr><mtd class=\"align-odd\" columnalign=\"right\" \/> <mtd class=\"align-even\"> <mo class=\"MathClass-rel\">=<\/mo> <mo class=\"MathClass-open\">(<\/mo><mn>1<\/mn> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>x<\/mi><mo class=\"MathClass-close\">)<\/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><msub><mrow><mi>b<\/mi><\/mrow><mrow> <mi>n<\/mi><\/mrow><\/msub><msup><mrow><mi>x<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msup> <mo class=\"MathClass-bin\">+<\/mo> <mi>A<\/mi><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\" \/> <mtd class=\"align-label\"> <mspace width=\"2em\" \/><\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">f\u00fcr alle <span class=\"maperiod\"><math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mo class=\"MathClass-open\">(<\/mo><mo class=\"MathClass-bin\">\u2212<\/mo><mn>1<\/mn><mo class=\"MathClass-punc\">,<\/mo><mn>1<\/mn><mo class=\"MathClass-close\">)<\/mo><\/math><\/span><span class=\"period\">.<\/span> Obige Formelmanipulationen m\u00f6gen vielleicht vom Himmel gefallen sein; ab jetzt wird das Argument jedoch wenig \u00dcberraschungen bieten. Sei <span class=\"maperiod\"><math display=\"inline\"><mi>\ud835\udf00<\/mi> <mo class=\"MathClass-rel\">&gt;<\/mo> <mn>0<\/mn><\/math><\/span><span class=\"period\">.<\/span> Dann existiert ein <math display=\"inline\"><mi>N<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2115<\/mi><\/math> mit <math display=\"inline\"><mo class=\"MathClass-rel\">|<\/mo><msub><mrow><mi>b<\/mi><\/mrow><mrow><mi>n<\/mi> <\/mrow> <\/msub> <mo class=\"MathClass-rel\">|<\/mo> <mo class=\"MathClass-rel\">&lt;<\/mo> <mi>\ud835\udf00<\/mi><\/math> f\u00fcr alle <span class=\"maperiod\"><math display=\"inline\"><mi>n<\/mi> <mo class=\"MathClass-rel\">\u2265<\/mo> <mi>N<\/mi><\/math><\/span><span class=\"period\">.<\/span> Daraus folgt f\u00fcr <span class=\"maperiod\"><math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mo class=\"MathClass-open\">[<\/mo><mn>0<\/mn><mo class=\"MathClass-punc\">,<\/mo><mn>1<\/mn><mo class=\"MathClass-close\">)<\/mo><\/math><\/span><span class=\"period\">,<\/span> dass <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><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> <mi>A<\/mi><mo class=\"MathClass-rel\">|<\/mo><\/mtd> <mtd class=\"align-even\"> <mo class=\"MathClass-rel\">=<\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> |<\/mo><mrow><mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mn>1<\/mn> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><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><msub><mrow><mi>b<\/mi><\/mrow><mrow> <mi>n<\/mi><\/mrow><\/msub><msup><mrow><mi>x<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msup><\/mrow><mo fence=\"true\" form=\"postfix\">|<\/mo><\/mrow><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\" \/> <mtd class=\"align-label\"> <mspace width=\"2em\" \/><\/mtd><\/mtr><mtr><mtd class=\"align-odd\" columnalign=\"right\" \/> <mtd class=\"align-even\"> <mo class=\"MathClass-rel\">\u2264<\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> |<\/mo><mrow><mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mn>1<\/mn> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><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>N<\/mi><\/mrow><\/munderover><msub><mrow><mi>b<\/mi><\/mrow><mrow> <mi>n<\/mi><\/mrow><\/msub><msup><mrow><mi>x<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msup><\/mrow><mo fence=\"true\" form=\"postfix\">|<\/mo><\/mrow> <mo class=\"MathClass-bin\">+<\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mn>1<\/mn> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><mi>\ud835\udf00<\/mi><munderover accent=\"false\" accentunder=\"false\"><mrow><mo>\u2211<\/mo> <\/mrow><mrow><mi>n<\/mi><mo class=\"MathClass-rel\">=<\/mo><mi>N<\/mi><mo class=\"MathClass-bin\">+<\/mo><mn>1<\/mn><\/mrow><mrow><mi>\u221e<\/mi><\/mrow><\/munderover><msup><mrow><mi>x<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msup><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\" \/> <mtd class=\"align-label\"> <mspace width=\"2em\" \/><\/mtd><\/mtr><mtr><mtd class=\"align-odd\" columnalign=\"right\" \/> <mtd class=\"align-even\"> <mo class=\"MathClass-rel\">\u2264<\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> |<\/mo><mrow><mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mn>1<\/mn> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><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>N<\/mi><\/mrow><\/munderover><msub><mrow><mi>b<\/mi><\/mrow><mrow> <mi>n<\/mi><\/mrow><\/msub><msup><mrow><mi>x<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msup><\/mrow><mo fence=\"true\" form=\"postfix\">|<\/mo><\/mrow> <mo class=\"MathClass-bin\">+<\/mo> <mi>\ud835\udf00<\/mi><mo class=\"MathClass-punc\">.<\/mo><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\" \/> <mtd class=\"align-label\"> <mspace width=\"2em\" \/><\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">Da aber das Polynom <math display=\"inline\"> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mn>1<\/mn> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><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>N<\/mi><\/mrow><\/msubsup><msub><mrow><mi>b<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><msup><mrow><mi>x<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msup><\/math> auf <math display=\"inline\"><mi>\u211d<\/mi><\/math> stetig ist und bei <math display=\"inline\"><mn>1<\/mn><\/math> verschwindet, gibt es weiters ein <span class=\"maperiod\"><math display=\"inline\"><mi>\u03b4<\/mi> <mo class=\"MathClass-rel\">&gt;<\/mo> <mn>0<\/mn><\/math><\/span><span class=\"period\">,<\/span> so dass <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mn>1<\/mn> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>\u03b4<\/mi><mo class=\"MathClass-punc\">,<\/mo><mn>1<\/mn><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><mspace class=\"thickpace\" width=\"0.28em\" \/><mo class=\"MathClass-rel\">\u21d2<\/mo><mspace class=\"thickpace\" width=\"0.28em\" \/> <mrow><mo fence=\"true\" form=\"prefix\"> |<\/mo><mrow><mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mn>1<\/mn> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><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>N<\/mi><\/mrow><\/munderover><msub><mrow><mi>b<\/mi><\/mrow><mrow> <mi>n<\/mi><\/mrow><\/msub><msup><mrow><mi>x<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msup><\/mrow><mo fence=\"true\" form=\"postfix\">|<\/mo><\/mrow> <mo class=\"MathClass-rel\">&lt;<\/mo> <mi>\ud835\udf00<\/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\">Daher gilt <math display=\"inline\"><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> <mi>A<\/mi><mo class=\"MathClass-rel\">|<\/mo> <mo class=\"MathClass-rel\">&lt;<\/mo> <mn>2<\/mn><mi>\ud835\udf00<\/mi><\/math> f\u00fcr alle <math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mo class=\"MathClass-open\">(<\/mo><mn>1<\/mn> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>\u03b4<\/mi><mo class=\"MathClass-punc\">,<\/mo><mn>1<\/mn><mo class=\"MathClass-close\">)<\/mo><\/math> und der Satz folgt. <span>&nbsp;&nbsp;<\/span><\/p><div class=\"qed\">\u25a0<\/div><\/details><\/div> <div class=\"me meexample\"> <p class=\"indent\"><\/p><h4 id=\"z034eaee0f668\"> <a id=\"x1-202005r67\"><\/a> <span class=\"ecbx-1095\">\u00dc<\/span><span class=\"ecbx-1095\">bung 7.67.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Wo wurde im obigen Beweis verwendet, dass<\/span><span class=\"ecti-1095\">&nbsp;<\/span><math display=\"inline\"><mi>x<\/mi><\/math> <span class=\"ecti-1095\">reell ist?<\/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\">F<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r <\/span><math display=\"inline\"><mi>z<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <msub><mrow><mi>B<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><mo class=\"MathClass-open\">(<\/mo><mn>0<\/mn><mo class=\"MathClass-close\">)<\/mo><\/math> <span class=\"ecti-1095\">ist mitunter <\/span><math display=\"inline\"><mn>1<\/mn> <mo class=\"MathClass-bin\">\u2212<\/mo><mo class=\"MathClass-rel\">|<\/mo><mi>z<\/mi><mo class=\"MathClass-rel\">|<\/mo><\/math> <span class=\"ecti-1095\">kleiner als <\/span><span class=\"maperiod\"><math display=\"inline\"><mo class=\"MathClass-rel\">|<\/mo><mn>1<\/mn> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>z<\/mi><mo class=\"MathClass-rel\">|<\/mo><\/math><\/span><span class=\"period\">,<\/span> <span class=\"ecti-1095\">doch gilt <\/span><math display=\"inline\"><mn>1<\/mn> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>x<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mo class=\"MathClass-rel\">|<\/mo><mn>1<\/mn> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>x<\/mi><mo class=\"MathClass-rel\">|<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <mn>1<\/mn> <mo class=\"MathClass-bin\">\u2212<\/mo><mo class=\"MathClass-rel\">|<\/mo><mi>x<\/mi><mo class=\"MathClass-rel\">|<\/mo><\/math> <span class=\"ecti-1095\">f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r <\/span><span class=\"maperiod\"><math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mo class=\"MathClass-open\">(<\/mo><mn>0<\/mn><mo class=\"MathClass-punc\">,<\/mo> <mn>1<\/mn><mo class=\"MathClass-close\">)<\/mo><\/math><\/span><span class=\"period\">.<\/span><\/p><\/details>  <\/div> <a id=\"x1-202006r199\"><\/a> \n","rendered":"\n<style scoped=\"scoped\">.cmr-5{font-size:50%;}\n.cmr-7{font-size:70%;}\n.cmmi-5{font-size:50%;font-style: italic;}\n.cmmi-7{font-size:70%;font-style: italic;}\n.cmmi-10{font-style: italic;}\n.cmsy-5{font-size:50%;}\n.cmsy-7{font-size:70%;}\n.cmbx-10{ font-weight: bold;}\n.cmbsy-10{font-weight: bold;}\n.cmbsy-10{font-weight: bold;}\n.cmbsy-10{font-weight: bold;}\n.cmbsy-7{font-size:70%;font-weight: bold;}\n.cmbsy-7{font-weight: bold;}\n.cmbsy-7{font-weight: bold;}\n.cmbsy-5{font-size:50%;font-weight: bold;}\n.cmbsy-5{font-weight: bold;}\n.cmbsy-5{font-weight: 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font-weight:bold; clear:left; float:left;}\ndd {width:100%; padding-left:1em; padding-top: 0px; clear:right;}\ndd + dd {float:right; clear:both;}\ndd + dt {clear:both;}\ndt + dt {width: 100%; float: none; padding: 0 70% 0 0;}\ndt + dt + dd {margin-top: -2em;}\ndt + dt + dd + dt {margin-top: 2em;}\n<\/style>\n<style scoped=\"scoped\">\n\/* CSS Analysis-Skript D-Math ETHZ *\/\n\n\/* Uniform Font, also for headers *\/\nh3 {\n\tfont-family: \"Times New Roman\", serif;\n\tmargin-bottom: 35px;\n}\nh4 {\n\tfont-family: \"Times New Roman\", serif;\n}\nh5 {\n\tfont-family: \"Times New Roman\", serif;\n}\n\n\/* Bold font, e.g. for definitions *\/\n.ecbx-1095 {font-weight: 550 ;}\n\n\n\/* Uniform spacing, indent: larger, noindent, enumerate, itemize *\/\np.indent {\n\tmargin: 25px 0px 0px 0px;\n\ttext-indent: 0px; \n}\np.noindent {\n\tmargin: 15px 0px 0px 0px;\n\ttext-indent: 0px; \n}\ndl.enumerate {\n\tmargin: 0px 0px 0px 0px;\n}\ndl.enumerate dt, dl.enumerate dd {\n\tmargin-top: 15px;\n\tmargin-bottom: 0px;\n}\ndiv.custom-itemize {\n\tmargin: 0px 0px 0px 0px;\n}\ndiv.custom-itemize div.item-head {\n\tmargin-top: 15px;\n\tmargin-bottom: 0px;\n\ttext-align: center;\n}\ndiv.custom-itemize div.item-head:first-of-type {\n\tmargin-top: 0px;\n} \ndiv.custom-itemize div.item-content {\n\tmargin-top: 15px;\n\tmargin-bottom: 0px;\n}\n.MJXc-display {\n\tmargin: 15px 0px 0px 0px;\n}\n\n\n\n\/* green metheorem\/melemma CSS class for more\/medium important latex-theorem-environments *\/\n\/* metheorem box+header *\/\ndiv.metheorem {\n    margin-bottom: 40px;\n    margin-top: 40px;\n\tpadding: 0px 15px 15px 15px;\n    border: 1px solid #333;\n    border-color: #4eb79e;\n    background: #c7e4da;\n}\ndiv.metheorem h4 {\n    background: #4eb79e;\n    color: white;\n\tmargin-top: 12px;\n\tmargin-left: -15px;\n\tmargin-right: -15px;\n\tpadding: 0px 15px 0px 15px;\n}\n\/* melemma box+header *\/\ndiv.melemma {\n    margin-bottom: 40px;\n    margin-top: 40px;\n\tpadding: 0px 15px 15px 15px;\n    border: 1px solid #333;\n    border-color: #4eb79e;\n    background: #F2F2F2;\n}\ndiv.melemma h4 {\n    background: #4eb79e;\n    color: white;\n\tmargin-top: 12px;\n\tmargin-left: -15px;\n\tmargin-right: -15px;\n\tpadding: 0px 15px 0px 15px;\n}\n\/* meexample box+header *\/\ndiv.meexample {\n    margin-bottom: 30px;\n    margin-top: 30px;\n\tpadding: 0px 15px 15px 15px;\n\tborder-color: gainsboro;\n\tborder-style: solid;\n\tborder-width: thin;\n}\ndiv.meexample h4 {\n\tfont-size: inherit;\n\tfont-weight: bold;\n    padding: 15px 0px 0px 0px;\n\tmargin-top: 0px;\n\tmargin-bottom: 5px;\n}\ndiv.meexample h4+p.noindent, div.meexample h4+p.indent {\n\tmargin-top: 5px;\n\ttext-indent: 0px;\n}\n\/* padding and margins for stuff inside these boxes, CSS-selector &gt; doesn't work in WP *\/\ndiv.me details {\n\tmargin: 10px 0px 0px 0px;\n}\ndiv.me dd {\n    width: calc(100% - 30px);\n}\t\n\n\n\/* fixing background of pictures *\/\nimg {\n\tbackground: white;\n}\n\n\/* div-container for centered geoapplet *\/\ndiv.geoapplet {\n\tmargin-left: auto;\n\tmargin-right: auto;\n\tmargin-top: 15px;\n\tmax-width: 100%;\n}\ndiv.geoapplet iframe {\n\tborder-style: none;\n\tmax-height: 110vw;\n}\n\n\/* div-container for centered squeezed tables *\/\ndiv.websqueeze {\n\tmargin-left: auto;\n\tmargin-right: auto;\n}\n\n\/* two containers for squeezing text sizes *\/\ndiv.mesmalltext, div.mesmalltext * {\n\tfont-size: 15px;\n}\nspan.metinytext, span.metinytext * {\n\tfont-size: 12px;\n}\n\n\n\/* removing grid lines in equations *\/\n#content table.equation tr td, #content table.equation tr th {\n    border: none;\n}\n#content table.equation {\n    border: none;\n}\n\n\/* hover\/click-solution for short inline explanations and footnotes *\/\n.hover-text {    \/* hidden part *\/\n    display: none;\n}\n.marginpar {     \/* style for footnote as marginpar *\/\n\ttext-decoration: none;\n\tborder: solid;\n\tborder-width: 1pt;\n\tpadding: 3pt;\t\n\twidth: 30%;\n\tbackground: white;\n}\n.hover-trigger { \/* style for hover\/click-trigger text\/symbol *\/\n\tbackground: none;\n\tborder: none;\n\tpadding: 0;\n\toutline: inherit;\t\n\ttext-transform: none;\n\tfont: inherit;\n\tposition: inherit;\n\tvertical-align: baseline;\n    color: #FF7F00;\n\tcursor: help;\n}\n.hover-trigger:hover +.hover-text{\n    display: inline;\n}\n.hover-trigger:active +.hover-text{\n    display: inline;\n}\n\n\/* simplifying style of details\/summary, removing triangle *\/\ndetails summary {\n  background: none;\n  list-style: none;\n  outline: none;\n  cursor: pointer;\n}\ndetails summary::-webkit-details-marker { \n  display: inline;\n  display: none;\n}\n\n\/* MC-True\/False as inline details\/summary *\/\ndetails.mcquest, div.me details.mcquest {\n\tdisplay: inline;\n\tmargin-top: 0px;\n}\nsummary.mcquest {\n\tdisplay: inline;\n\tcolor: #FF7F00;\n\tcursor: help;\n}\n\n\/* proof style: simple black box with gray background \n                little black square at the end on the right *\/\ndiv.proof {\n\tborder-color: black;\n\tborder-style: solid;\n\tborder-width: thin;\n\tbackground-color: #F2F2F2;\n\tpadding: 15px;\n\tmargin-top: 1em; \n}\ndiv.proof p:first-of-type {\n\tmargin: 0px;\n}\ndiv.qed {\n\tmargin-top: -25px;\n\tmargin-bottom: -7px;\n\ttext-align: right;\n}\ntable.equation+div.qed {\n\tmargin-top: -65px;\n}\n\n\/* The following is making also math-formulas inside the headers of Lemmas, etc., white. *\/\ndiv.melemma h4 span {\n    color: white;\n}\ndiv.metheorem h4 span {\n    color: white;\n}\n\n\/* The following are used to avoid fullstop, period, colon, semicolon, and endquote (broader) to move by itself to the next line after a formula.\n   The math-environment before needs to be wrapped in span.maperiod and the fullstop etc. in a span.period --- together they achieve what we want.  *\/\nspan.maperiod {\n       margin-right: 5px;\n}\nspan.period {\n       display: inline-block;\n       width: 0px;\n       margin-left: -5px;\n       margin-right: 4.9px;\n\t   text-indent: 0px;\n}\nspan.maendquote {\n       margin-right: 8px;\n}\nspan.endquote {\n       display: inline-block;\n       width: 0px;\n       margin-left: -8px;\n       margin-right: 7.9px;\n}\n\n\n\/* The following is removing an extra space left of the equation side in aligned equations *\/\nspan.mjx-mtd {\n    padding-left: 0em !important;\n}\n\n\/* The following fixes the weird problem that math appears smaller if it was rendered while the details tag was closed. *\/\ndetails span.mjx-chtml, details span.MathJax_CHTML {\n font-size: 100% !important;\n}\n\n\/* trying to fix line breaks in verbatim, new lines are missing *\/\npre.verbatim {\n\twhite-space: pre-wrap;\n\tfont-size: small;\n}\n<\/style><h3 id=\"zc8703823ed71\" class=\"sectionHead\"><span class=\"titlemark\">7.4 <\/span> <a id=\"x1-1990004\"><\/a>Potenzreihen<\/h3> <div class=\"me metheorem\"> <div class=\"wp-nocaption \"><\/div><h4 id=\"z15eba0fbb4a0\"> <a id=\"x1-199001r54\"><\/a> <span class=\"ecbx-1095\">Definition 7.54 <\/span>(Potenzreihe)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\">F\u00fcr jedes <math display=\"inline\"><mi>n<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <msub><mrow><mi>\u2115<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><\/math> sei <span class=\"maperiod\"><math display=\"inline\"><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi> <\/mrow> <\/msub> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2102<\/mi><\/math><\/span><span class=\"period\">.<\/span> Dann ist der formale Ausdruck <\/p><math display=\"block\"><mtable class=\"align\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><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><msub><mrow><mi>a<\/mi><\/mrow><mrow> <mi>n<\/mi><\/mrow><\/msub><msup><mrow><mi>z<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msup><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"><mstyle class=\"label\" id=\"x1-199002r5\" \/><mstyle class=\"maketag\"><mtext>(7.5)<\/mtext><\/mstyle><mspace class=\"nbsp\" width=\"0.33em\" \/> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">eine <span class=\"ecbx-1095\">Potenzreihe <\/span>in der Variable <span class=\"maperiod\"><math display=\"inline\"><mi>z<\/mi><\/math><\/span><span class=\"period\">.<\/span> <\/p> <\/div> <p class=\"indent\">Es dr\u00e4ngt sich bei obiger Definition ein Vergleich zur Definition eines Polynoms in Definition <a href=\"..\/..\/chapter\/polynome#x1-81005r13\">3.13<\/a> auf. Im Gegensatz zur Diskussion von Polynomen ist aber eine Potenzreihe vorerst nur ein formaler Ausdruck. Es ist nicht klar, bei welchen komplexen Zahlen man eine Potenzreihe auswerten darf. Insbesondere wissen wir (noch) nicht, ob wir diesem formalen Ausdruck \u00fcberhaupt eine Funktion auf <math display=\"inline\"><mi>\u2102<\/mi><\/math> oder einer bestimmten Teilmenge von <math display=\"inline\"><mi>\u2102<\/mi><\/math> zuordnen k\u00f6nnen. Diese Frage h\u00e4ngt stark von den Koeffizienten <math display=\"inline\"><msub><mrow><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi> <\/mrow> <\/msub> <mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mi>n<\/mi><mo class=\"MathClass-rel\">\u2208<\/mo><msub><mrow><mi>\u2115<\/mi><\/mrow><mrow><mn>0<\/mn> <\/mrow> <\/msub> <\/mrow> <\/msub> <\/math> ab und wird in Satz <a href=\"..\/..\/chapter\/potenzreihen#x1-200002r56\">7.56<\/a> beantwortet. <\/p><p class=\"indent\">Wie schon bei Polynomen in Definition <a href=\"..\/..\/chapter\/polynome#x1-81001r11\">3.11<\/a> ist auch hier die Definition (<a href=\"..\/..\/chapter\/summen-und-produkte#x1-77003r2\">3.2<\/a>) \u00e4usserst sinnvoll, damit die Potenzreihe (<a href=\"..\/..\/chapter\/potenzreihen#x1-199002r5\">7.5<\/a>)bei <math display=\"inline\"><mi>z<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mn>0<\/mn><\/math> auf jeden Fall konvergiert und den Wert <math display=\"inline\"><msub><mrow><mi>a<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><\/math> hat. <a id=\"x1-199003r198\"><\/a> <\/p> <h4 id=\"z1dd1e58054f4\" class=\"subsectionHead\"><span class=\"titlemark\">7.4.1 <\/span> <a id=\"x1-2000001\"><\/a>Konvergenzradius<\/h4> <div class=\"me metheorem\"> <div class=\"wp-nocaption \"><\/div><h4 id=\"zf0ed7d793817\"> <a id=\"x1-200001r55\"><\/a> <span class=\"ecbx-1095\">Definition 7.55.<\/span> <\/h4> <p class=\"indent\">Sei <math display=\"inline\"><msubsup><mrow><mi class=\"MathClass-op\"> \u2211<\/mi><mo> <\/mo> <\/mrow><mrow><mi>n<\/mi><mo class=\"MathClass-rel\">=<\/mo><mn>0<\/mn><\/mrow><mrow><mi>\u221e<\/mi><\/mrow><\/msubsup><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><msup><mrow><mi>z<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msup><\/math> eine Potenzreihe mit komplexen Koeffizienten <span class=\"maperiod\"><math display=\"inline\"><msub><mrow><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mi>n<\/mi><mo class=\"MathClass-rel\">\u2208<\/mo><msub><mrow><mi>\u2115<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><\/mrow><\/msub><\/math><\/span><span class=\"period\">.<\/span> Wir definieren den <span class=\"ecbx-1095\">Konvergenzradius <\/span>durch <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mi>R<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><munder class=\"msub\"><mrow><mi class=\"qopname\">limsup<\/mi><mo>  <\/mo><\/mrow><mrow><mi>n<\/mi><mo class=\"MathClass-rel\">\u2192<\/mo><mi>\u221e<\/mi><\/mrow><\/munder><mroot><mrow><mo class=\"MathClass-rel\">|<\/mo><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><mo class=\"MathClass-rel\">|<\/mo><\/mrow><mrow><mi>n<\/mi><\/mrow><\/mroot><\/mrow><\/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\">wobei wir <math display=\"inline\"> <mfrac> <mrow> <mn>1<\/mn><\/mrow> <mrow><mo class=\"MathClass-bin\">+<\/mo><mi>\u221e<\/mi><\/mrow><\/mfrac> <mo class=\"MathClass-rel\">=<\/mo> <mn>0<\/mn><\/math> setzen und hier (aber auch nur hier) die Vereinbarung <math display=\"inline\"><mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mn>0<\/mn><\/mrow><\/mfrac> <mo class=\"MathClass-rel\">=<\/mo> <mo class=\"MathClass-bin\">+<\/mo><mi>\u221e<\/mi><\/math> treffen. <\/p> <\/div> <div class=\"me metheorem\"> <div class=\"wp-nocaption \"><\/div><h4 id=\"z481151657781\"> <a id=\"x1-200002r56\"><\/a> <span class=\"ecbx-1095\">Satz 7.56 <\/span>(\u00dcber den Konvergenzradius)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Sei <\/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><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><msup><mrow><mi>z<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msup><\/math> <span class=\"ecti-1095\">eine Potenzreihe<\/span> <span class=\"ecti-1095\">und <\/span><math display=\"inline\"><mi>R<\/mi><\/math> <span class=\"ecti-1095\">ihr Konvergenzradius.<\/span> <span class=\"ecti-1095\">Dann konvergiert 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><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><msup><mrow><mi>z<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msup><\/math> <span class=\"ecti-1095\">f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r alle <\/span><math display=\"inline\"><mi>z<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2102<\/mi><\/math> <span class=\"ecti-1095\">mit<\/span> <math display=\"inline\"><mo class=\"MathClass-rel\">|<\/mo><mi>z<\/mi><mo class=\"MathClass-rel\">|<\/mo> <mo class=\"MathClass-rel\">&lt;<\/mo> <mi>R<\/mi><\/math> <span class=\"ecti-1095\">absolut und divergiert<\/span> <span class=\"ecti-1095\">f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r alle <\/span><math display=\"inline\"><mi>z<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2102<\/mi><\/math> <span class=\"ecti-1095\">mit<\/span> <math display=\"inline\"><mo class=\"MathClass-rel\">|<\/mo><mi>z<\/mi><mo class=\"MathClass-rel\">|<\/mo> <mo class=\"MathClass-rel\">&gt;<\/mo> <mi>R<\/mi><\/math><span class=\"ecti-1095\">. Weiters konvergiert<\/span> <span class=\"ecti-1095\">die Funktionenfolge <\/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>N<\/mi><\/mrow><\/msubsup><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><msup><mrow><mi>z<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msup><\/math> <span class=\"ecti-1095\">gleichm<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">ssig gegen <\/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><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><msup><mrow><mi>z<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msup><\/math> <span class=\"ecti-1095\">auf<\/span> <span class=\"ecti-1095\">jeder Kreisscheibe der Form <\/span><math display=\"inline\"><msub><mrow><mi>B<\/mi><\/mrow><mrow><mi>S<\/mi><\/mrow><\/msub><mo class=\"MathClass-open\">(<\/mo><mn>0<\/mn><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> {<\/mo><mrow><mi>z<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2102<\/mi><mo class=\"MathClass-rel\">\u2223<\/mo><mo class=\"MathClass-rel\">|<\/mo><mi>z<\/mi><mo class=\"MathClass-rel\">|<\/mo> <mo class=\"MathClass-rel\">&lt;<\/mo> <mi>S<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">}<\/mo><\/mrow><\/math> <span class=\"ecti-1095\">f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r jedes <\/span><span class=\"maperiod\"><math display=\"inline\"><mi>S<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mo class=\"MathClass-open\">(<\/mo><mn>0<\/mn><mo class=\"MathClass-punc\">,<\/mo><mi>R<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math><\/span><span class=\"period\">.<\/span> <span class=\"ecti-1095\">Insbesondere definiert die Potenzreihe die stetige Abbildung<\/span> <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mi>z<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <msub><mrow><mi>B<\/mi><\/mrow><mrow><mi>R<\/mi><\/mrow><\/msub><mo class=\"MathClass-open\">(<\/mo><mn>0<\/mn><mo class=\"MathClass-close\">)<\/mo><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><msub><mrow><mi>a<\/mi><\/mrow><mrow> <mi>n<\/mi><\/mrow><\/msub><msup><mrow><mi>z<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msup> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2102<\/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> <\/div> <div class=\"center\"> <div class=\"wp-nocaption \"><\/div><div class=\"wp-nocaption \"><\/div><div class=\"mefigcentered\" id=\"wpsize=467&amp;url=Pictures\/reihen\/konvradius.pdf\"><img decoding=\"async\" id=\"z81fb2cb33b47\" alt=\"PIC\" src=\"https:\/\/people.math.ethz.ch\/~einsiedl\/Pictures\/reihen\/konvradius.svg\" width=\"467\" \/><\/div>  <\/div> <div class=\"wp-nocaption \"><\/div> <div class=\"proof\"> <p class=\"indent\"><span class=\"head\"><\/span><\/p><details open=\"open\"><summary><b>Beweis.<\/b><\/summary><p class=\"indent\" style=\"margin-top: 10\">Wir verwenden das Wurzelkriterium aus Korollar <a href=\"..\/..\/chapter\/absolute-konvergenz#x1-193002r30\">7.30<\/a> f\u00fcr ein beliebiges <math display=\"inline\"><mi>z<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2102<\/mi><\/math> und die Reihe <math display=\"inline\"><msubsup><mrow><mi class=\"MathClass-op\"> \u2211<\/mi><mo> <\/mo> <\/mrow><mrow><mi>n<\/mi><mo class=\"MathClass-rel\">=<\/mo><mn>0<\/mn><\/mrow><mrow><mi>\u221e<\/mi><\/mrow><\/msubsup><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><msup><mrow><mi>z<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msup><\/math> und berechnen deswegen <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><munder class=\"msub\"><mrow><mi class=\"qopname\"> limsup<\/mi><mo>  <\/mo><\/mrow><mrow><mi>n<\/mi><mo class=\"MathClass-rel\">\u2192<\/mo><mi>\u221e<\/mi><\/mrow><\/munder><mroot><mrow><mo class=\"MathClass-rel\">|<\/mo><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><msup><mrow><mi>z<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msup><mo class=\"MathClass-rel\">|<\/mo><\/mrow><mrow><mi>n<\/mi><\/mrow><\/mroot> <mo class=\"MathClass-rel\">=<\/mo><munder class=\"msub\"><mrow><mi class=\"qopname\"> limsup<\/mi><mo>  <\/mo><\/mrow><mrow> <mi>n<\/mi><mo class=\"MathClass-rel\">\u2192<\/mo><mi>\u221e<\/mi><\/mrow><\/munder><mroot><mrow><mo class=\"MathClass-rel\">|<\/mo><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><mo class=\"MathClass-rel\">|<\/mo><\/mrow><mrow><mi>n<\/mi><\/mrow><\/mroot> <mrow><mo fence=\"true\" form=\"prefix\"> |<\/mo><mrow><mi>z<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">|<\/mo><\/mrow> <mo class=\"MathClass-rel\">=<\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> |<\/mo><mrow><mi>z<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">|<\/mo><\/mrow><munder class=\"msub\"><mrow><mi class=\"qopname\">limsup<\/mi><mo>  <\/mo><\/mrow><mrow><mi>n<\/mi><mo class=\"MathClass-rel\">\u2192<\/mo><mi>\u221e<\/mi><\/mrow><\/munder><mroot><mrow><mo class=\"MathClass-rel\">|<\/mo><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><mo class=\"MathClass-rel\">|<\/mo><\/mrow><mrow><mi>n<\/mi><\/mrow><\/mroot> <mo class=\"MathClass-rel\">=<\/mo> <mfrac><mrow><mo class=\"MathClass-rel\">|<\/mo><mi>z<\/mi><mo class=\"MathClass-rel\">|<\/mo><\/mrow> <mrow><mi>R<\/mi><\/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\">Nach dem Wurzelkriterium konvergiert die Reihe also absolut f\u00fcr <math display=\"inline\"><mfrac><mrow><mo class=\"MathClass-rel\">|<\/mo><mi>z<\/mi><mo class=\"MathClass-rel\">|<\/mo><\/mrow> <mrow><mi>R<\/mi><\/mrow><\/mfrac> <mo class=\"MathClass-rel\">&lt;<\/mo> <mn>1<\/mn><\/math> und divergiert f\u00fcr <span class=\"maperiod\"><math display=\"inline\"><mfrac><mrow><mo class=\"MathClass-rel\">|<\/mo><mi>z<\/mi><mo class=\"MathClass-rel\">|<\/mo><\/mrow> <mrow><mi>R<\/mi><\/mrow><\/mfrac> <mo class=\"MathClass-rel\">&gt;<\/mo> <mn>1<\/mn><\/math><\/span><span class=\"period\">.<\/span> Die F\u00e4lle <math display=\"inline\"><mi>R<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mn>0<\/mn><\/math> und <math display=\"inline\"><mi>R<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mo class=\"MathClass-bin\">+<\/mo><mi>\u221e<\/mi><\/math> ergeben sich aus dem gleichen Argument (wieso?). <\/p><p class=\"indent\">Sei nun <span class=\"maperiod\"><math display=\"inline\"><mi>S<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mo class=\"MathClass-open\">(<\/mo><mn>0<\/mn><mo class=\"MathClass-punc\">,<\/mo><mi>R<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math><\/span><span class=\"period\">.<\/span> F\u00fcr den Beweis der gleichm\u00e4ssigen Konvergenz auf <math display=\"inline\"><msub><mrow><mi>B<\/mi><\/mrow><mrow><mi>S<\/mi> <\/mrow> <\/msub> <mo class=\"MathClass-open\">(<\/mo><mn>0<\/mn><mo class=\"MathClass-close\">)<\/mo><\/math> bemerken wir, dass nach obigem bereits <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><mo class=\"MathClass-rel\">|<\/mo><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><mo class=\"MathClass-rel\">|<\/mo><msup><mrow><mi>S<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msup> <mo class=\"MathClass-rel\">&lt;<\/mo> <mi>\u221e<\/mi><\/math> gilt. Daher existiert f\u00fcr jedes <math display=\"inline\"><mi>\ud835\udf00<\/mi> <mo class=\"MathClass-rel\">&gt;<\/mo> <mn>0<\/mn><\/math> ein <math display=\"inline\"><mi>N<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2115<\/mi><\/math> mit <span class=\"maperiod\"><math display=\"inline\"><msubsup><mrow><mi class=\"MathClass-op\">\u2211<\/mi><mo> <\/mo> <\/mrow><mrow><mi>n<\/mi><mo class=\"MathClass-rel\">=<\/mo><mi>N<\/mi><\/mrow><mrow><mi>\u221e<\/mi><\/mrow><\/msubsup><mo class=\"MathClass-rel\">|<\/mo><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><mo class=\"MathClass-rel\">|<\/mo><msup><mrow><mi>S<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msup> <mo class=\"MathClass-rel\">&lt;<\/mo> <mi>\ud835\udf00<\/mi><\/math><\/span><span class=\"period\">.<\/span> F\u00fcr alle <math display=\"inline\"><mi>z<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <msub><mrow><mi>B<\/mi><\/mrow><mrow><mi>S<\/mi> <\/mrow> <\/msub> <mo class=\"MathClass-open\">(<\/mo><mn>0<\/mn><mo class=\"MathClass-close\">)<\/mo><\/math> und <math display=\"inline\"><mi>n<\/mi> <mo class=\"MathClass-rel\">\u2265<\/mo> <mi>N<\/mi><\/math> gilt damit <\/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><munderover accent=\"false\" accentunder=\"false\"><mrow><mo>\u2211<\/mo> <\/mrow><mrow><mi>k<\/mi><mo class=\"MathClass-rel\">=<\/mo><mn>0<\/mn><\/mrow><mrow><mi>n<\/mi><\/mrow><\/munderover><msub><mrow><mi>a<\/mi><\/mrow><mrow> <mi>k<\/mi><\/mrow><\/msub><msup><mrow><mi>z<\/mi><\/mrow><mrow><mi>k<\/mi><\/mrow><\/msup> <mo class=\"MathClass-bin\">\u2212<\/mo><munderover accent=\"false\" accentunder=\"false\"><mrow><mo>\u2211<\/mo> <\/mrow><mrow><mi>k<\/mi><mo class=\"MathClass-rel\">=<\/mo><mn>0<\/mn><\/mrow><mrow><mi>\u221e<\/mi><\/mrow><\/munderover><msub><mrow><mi>a<\/mi><\/mrow><mrow> <mi>k<\/mi><\/mrow><\/msub><msup><mrow><mi>z<\/mi><\/mrow><mrow><mi>k<\/mi><\/mrow><\/msup><\/mrow><mo fence=\"true\" form=\"postfix\">|<\/mo><\/mrow> <mo class=\"MathClass-rel\">=<\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> |<\/mo><mrow><munderover accent=\"false\" accentunder=\"false\"><mrow><mo>\u2211<\/mo> <\/mrow><mrow><mi>k<\/mi><mo class=\"MathClass-rel\">=<\/mo><mi>n<\/mi><mo class=\"MathClass-bin\">+<\/mo><mn>1<\/mn><\/mrow><mrow><mi>\u221e<\/mi><\/mrow><\/munderover><msub><mrow><mi>a<\/mi><\/mrow><mrow> <mi>k<\/mi><\/mrow><\/msub><msup><mrow><mi>z<\/mi><\/mrow><mrow><mi>k<\/mi><\/mrow><\/msup><\/mrow><mo fence=\"true\" form=\"postfix\">|<\/mo><\/mrow> <mo class=\"MathClass-rel\">\u2264<\/mo><munderover accent=\"false\" accentunder=\"false\"><mrow><mo>\u2211<\/mo> <\/mrow><mrow><mi>k<\/mi><mo class=\"MathClass-rel\">=<\/mo><mi>N<\/mi><\/mrow><mrow><mi>\u221e<\/mi><\/mrow><\/munderover><mo class=\"MathClass-rel\">|<\/mo><msub><mrow><mi>a<\/mi><\/mrow><mrow> <mi>k<\/mi><\/mrow><\/msub><mo class=\"MathClass-rel\">|<\/mo><msup><mrow><mi>S<\/mi><\/mrow><mrow><mi>k<\/mi><\/mrow><\/msup> <mo class=\"MathClass-rel\">&lt;<\/mo> <mi>\ud835\udf00<\/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\">Dies beweist die gleichm\u00e4ssige Konvergenz der stetigen Funktionenfolge <math display=\"inline\"><msubsup><mrow><mi class=\"MathClass-op\">\u2211<\/mi><mo> <\/mo> <\/mrow><mrow><mi>k<\/mi><mo class=\"MathClass-rel\">=<\/mo><mn>0<\/mn><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msubsup><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>k<\/mi><\/mrow><\/msub><msup><mrow><mi>z<\/mi><\/mrow><mrow><mi>k<\/mi><\/mrow><\/msup><\/math> auf <math display=\"inline\"><msub><mrow><mi>B<\/mi><\/mrow><mrow><mi>S<\/mi> <\/mrow> <\/msub> <mo class=\"MathClass-open\">(<\/mo><mn>0<\/mn><mo class=\"MathClass-close\">)<\/mo><\/math> gegen <math display=\"inline\"><msubsup><mrow><mi class=\"MathClass-op\">\u2211<\/mi><mo> <\/mo> <\/mrow><mrow><mi>k<\/mi><mo class=\"MathClass-rel\">=<\/mo><mn>0<\/mn><\/mrow><mrow><mi>\u221e<\/mi><\/mrow><\/msubsup><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>k<\/mi><\/mrow><\/msub><msup><mrow><mi>z<\/mi><\/mrow><mrow><mi>k<\/mi><\/mrow><\/msup><\/math> und damit die Stetigkeit von <math display=\"inline\"><mi>z<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <msub><mrow><mi>B<\/mi><\/mrow><mrow><mi>S<\/mi><\/mrow><\/msub><mo class=\"MathClass-open\">(<\/mo><mn>0<\/mn><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-rel\">\u21a6<\/mo><msubsup><mrow><mi class=\"MathClass-op\">\u2211<\/mi><mo> <\/mo> <\/mrow><mrow><mi>k<\/mi><mo class=\"MathClass-rel\">=<\/mo><mn>0<\/mn><\/mrow><mrow><mi>\u221e<\/mi><\/mrow><\/msubsup><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>k<\/mi><\/mrow><\/msub><msup><mrow><mi>z<\/mi><\/mrow><mrow><mi>k<\/mi><\/mrow><\/msup> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2102<\/mi><\/math> nach Satz <a href=\"..\/..\/chapter\/konvergenz-von-funktionenfolgen#x1-198009r48\">7.48<\/a>. <\/p><p class=\"indent\">Insbesondere ist die Funktion <math display=\"inline\"><mi>z<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <msub><mrow><mi>B<\/mi><\/mrow><mrow><mi>R<\/mi><\/mrow><\/msub><mo class=\"MathClass-open\">(<\/mo><mn>0<\/mn><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-rel\">\u21a6<\/mo><msubsup><mrow><mi class=\"MathClass-op\">\u2211<\/mi><mo> <\/mo> <\/mrow><mrow><mi>n<\/mi><mo class=\"MathClass-rel\">=<\/mo><mn>0<\/mn><\/mrow><mrow><mi>\u221e<\/mi><\/mrow><\/msubsup><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><msup><mrow><mi>z<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msup> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2102<\/mi><\/math> stetig an jedem Punkt, da es zu <math display=\"inline\"><mi>z<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <msub><mrow><mi>B<\/mi><\/mrow><mrow><mi>R<\/mi><\/mrow><\/msub><mo class=\"MathClass-open\">(<\/mo><mn>0<\/mn><mo class=\"MathClass-close\">)<\/mo><\/math> ein <math display=\"inline\"><mi>S<\/mi> <mo class=\"MathClass-rel\">&lt;<\/mo> <mi>R<\/mi><\/math> gibt, mit <math display=\"inline\"><mi>z<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <msub><mrow><mi>B<\/mi><\/mrow><mrow><mi>S<\/mi> <\/mrow> <\/msub> <mo class=\"MathClass-open\">(<\/mo><mn>0<\/mn><mo class=\"MathClass-close\">)<\/mo><\/math> (wieso zeigt dies die Stetigkeit?). Dies beweist den Satz. <span>&nbsp;&nbsp;<\/span><\/p><div class=\"qed\">\u25a0<\/div><\/details><\/div> <div class=\"me meexample\"> <div class=\"wp-nocaption \"><\/div><h4 id=\"z4b52e7ba7944\"> <a id=\"x1-200003r57\"><\/a> <span class=\"ecbx-1095\">Beispiel 7.57 <\/span>(Nicht gleichm\u00e4ssige Konvergenz)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Man k<\/span><span class=\"ecti-1095\">\u00f6<\/span><span class=\"ecti-1095\">nnte denken, dass Satz<\/span><span class=\"ecti-1095\">&nbsp;<\/span><a href=\"..\/..\/chapter\/potenzreihen#x1-200002r56\"><span class=\"ecti-1095\">7.56<\/span><\/a> <span class=\"ecti-1095\">eigentlich sagt, dass die Partialsummen<\/span><span class=\"ecti-1095\">&nbsp;<\/span><math display=\"inline\"><msubsup><mrow><mi class=\"MathClass-op\">\u2211<\/mi><mo> <\/mo> <\/mrow><mrow><mi>k<\/mi><mo class=\"MathClass-rel\">=<\/mo><mn>0<\/mn><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msubsup><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>k<\/mi><\/mrow><\/msub><msup><mrow><mi>z<\/mi><\/mrow><mrow><mi>k<\/mi><\/mrow><\/msup><\/math> <span class=\"ecti-1095\">der Potenzreihe auf ganz<\/span><span class=\"ecti-1095\">&nbsp;<\/span><math display=\"inline\"><msub><mrow><mi>B<\/mi><\/mrow><mrow><mi>R<\/mi><\/mrow><\/msub><mo class=\"MathClass-open\">(<\/mo><mn>0<\/mn><mo class=\"MathClass-close\">)<\/mo><\/math> <span class=\"ecti-1095\">gleichm<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">ssig gegen die durch die Potenzreihe definierte Funktion<\/span><span class=\"ecti-1095\">&nbsp;<\/span><math display=\"inline\"><mi>z<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <msub><mrow><mi>B<\/mi><\/mrow><mrow><mi>R<\/mi><\/mrow><\/msub><mo class=\"MathClass-open\">(<\/mo><mn>0<\/mn><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-rel\">\u21a6<\/mo><msubsup><mrow><mi class=\"MathClass-op\">\u2211<\/mi><mo> <\/mo> <\/mrow><mrow><mi>k<\/mi><mo class=\"MathClass-rel\">=<\/mo><mn>0<\/mn><\/mrow><mrow><mi>\u221e<\/mi><\/mrow><\/msubsup><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>k<\/mi><\/mrow><\/msub><msup><mrow><mi>z<\/mi><\/mrow><mrow><mi>k<\/mi><\/mrow><\/msup><\/math> <span class=\"ecti-1095\">streben, da ja<\/span><span class=\"ecti-1095\">&nbsp;<\/span><math display=\"inline\"><mi>S<\/mi> <mo class=\"MathClass-rel\">&lt;<\/mo> <mi>R<\/mi><\/math> <span class=\"ecti-1095\">beliebig ist. Dies ist aber nicht immer so (siehe auch <\/span><span class=\"ecti-1095\">\u00dc<\/span><span class=\"ecti-1095\">bung<\/span><span class=\"ecti-1095\">&nbsp;<\/span><a href=\"..\/..\/chapter\/konvergenz-von-funktionenfolgen#x1-198012r50\"><span class=\"ecti-1095\">7.50<\/span><\/a><span class=\"ecti-1095\">), wie wir hier kurz anhand<\/span> <span class=\"ecti-1095\">der geometrischen Reihe zeigen wollen.<\/span> <\/p><p class=\"indent\"><span class=\"ecti-1095\">F<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r<\/span><span class=\"ecti-1095\">&nbsp;<\/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>z<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msup><\/math> <span class=\"ecti-1095\">ist der<\/span> <span class=\"ecti-1095\">Konvergenzradius<\/span><span class=\"ecti-1095\">&nbsp;<\/span><math display=\"inline\"><mi>R<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mn>1<\/mn><\/math> <span class=\"ecti-1095\">und die mittels der Potenzreihe definierte Funktion<\/span> <span class=\"ecti-1095\">ist<\/span><span class=\"ecti-1095\">&nbsp;<\/span><math display=\"inline\"><mi>z<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <msub><mrow><mi>B<\/mi><\/mrow><mrow><mn>1<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-open\">(<\/mo><mn>0<\/mn><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-rel\">\u21a6<\/mo> <mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mn>1<\/mn><mo class=\"MathClass-bin\">\u2212<\/mo><mi>z<\/mi><\/mrow><\/mfrac><\/math><span class=\"ecti-1095\">. Falls die Konvergenz<\/span> <span class=\"ecti-1095\">auf ganz<\/span><span class=\"ecti-1095\">&nbsp;<\/span><math display=\"inline\"><msub><mrow><mi>B<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><mo class=\"MathClass-open\">(<\/mo><mn>0<\/mn><mo class=\"MathClass-close\">)<\/mo><\/math> <span class=\"ecti-1095\">gleichm<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">ssig<\/span> <span class=\"ecti-1095\">w<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">re, dann g<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">be es f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r<\/span><span class=\"ecti-1095\">&nbsp;<\/span><math display=\"inline\"><mi>\ud835\udf00<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mn>1<\/mn><\/math> <span class=\"ecti-1095\">ein<\/span><span class=\"ecti-1095\">&nbsp;<\/span><math display=\"inline\"><mi>N<\/mi><\/math> <span class=\"ecti-1095\">so dass<\/span> <span class=\"ecti-1095\">f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r alle<\/span><span class=\"ecti-1095\">&nbsp;<\/span><math display=\"inline\"><mi>n<\/mi> <mo class=\"MathClass-rel\">\u2265<\/mo> <mi>N<\/mi><\/math> <span class=\"ecti-1095\">und <\/span><math display=\"inline\"><mi>z<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <msub><mrow><mi>B<\/mi><\/mrow><mrow><mn>1<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-open\">(<\/mo><mn>0<\/mn><mo class=\"MathClass-close\">)<\/mo><\/math> <span class=\"ecti-1095\">die Absch<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">tzung<\/span> <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"> <mrow><mo fence=\"true\" form=\"prefix\"> |<\/mo><mrow><munderover accent=\"false\" accentunder=\"false\"><mrow><mo>\u2211<\/mo> <\/mrow><mrow><mi>k<\/mi><mo class=\"MathClass-rel\">=<\/mo><mn>0<\/mn><\/mrow><mrow><mi>n<\/mi><\/mrow><\/munderover><msup><mrow><mi>z<\/mi><\/mrow><mrow><mi>k<\/mi><\/mrow><\/msup> <mo class=\"MathClass-bin\">\u2212<\/mo> <mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mn>1<\/mn> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>z<\/mi><\/mrow><\/mfrac> <\/mrow><mo fence=\"true\" form=\"postfix\">|<\/mo><\/mrow> <mo class=\"MathClass-rel\">&lt;<\/mo> <mn>1<\/mn><\/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\">gelten w<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">rde. Wir setzen<\/span><span class=\"ecti-1095\">&nbsp;<\/span><math display=\"inline\"><mi>n<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mi>N<\/mi><\/math> <span class=\"ecti-1095\">und erhalten mittels der Dreiecksungleichung daraus<\/span> <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mstyle><mrow><mo fence=\"true\" form=\"prefix\"> |<\/mo><mrow \/><mo fence=\"true\" form=\"postfix\" \/><\/mrow><\/mstyle> <mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mn>1<\/mn> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>z<\/mi><\/mrow><\/mfrac><mstyle><mrow><mo fence=\"true\" form=\"prefix\"> |<\/mo><mrow \/><mo fence=\"true\" form=\"postfix\" \/><\/mrow><\/mstyle> <mo class=\"MathClass-rel\">&lt;<\/mo> <mn>1<\/mn> <mo class=\"MathClass-bin\">+<\/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>k<\/mi><mo class=\"MathClass-rel\">=<\/mo><mn>0<\/mn><\/mrow><mrow><mi>N<\/mi><\/mrow><\/munderover><msup><mrow><mi>z<\/mi><\/mrow><mrow><mi>k<\/mi><\/mrow><\/msup><mstyle><mrow><mo fence=\"true\" form=\"prefix\"> |<\/mo><mrow \/><mo fence=\"true\" form=\"postfix\" \/><\/mrow><\/mstyle> <mo class=\"MathClass-rel\">\u2264<\/mo> <mn>2<\/mn> <mo class=\"MathClass-bin\">+<\/mo> <mi>N<\/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\">f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r alle<\/span><span class=\"ecti-1095\">&nbsp;<\/span><span class=\"maperiod\"><math display=\"inline\"><mi>z<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <msub><mrow><mi>B<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><mo class=\"MathClass-open\">(<\/mo><mn>0<\/mn><mo class=\"MathClass-close\">)<\/mo><\/math><\/span><span class=\"period\">.<\/span> <span class=\"ecti-1095\">Dies ist aber ein Widerspruch, da<\/span> <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><munder class=\"msub\"><mrow><mi class=\"qopname\">lim<\/mi><mo>  <\/mo><\/mrow><mrow><mi>x<\/mi><mo class=\"MathClass-rel\">\u2197<\/mo><mn>1<\/mn><\/mrow><\/munder> <mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mn>1<\/mn> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>x<\/mi><\/mrow><\/mfrac> <mo class=\"MathClass-rel\">=<\/mo> <mo class=\"MathClass-bin\">+<\/mo><mi>\u221e<\/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> <\/div> <div class=\"me meexample\"> <div class=\"wp-nocaption \"><\/div><h4 id=\"zcf911a8dd3bf\"> <a id=\"x1-200004r58\"><\/a> <span class=\"ecbx-1095\">\u00dc<\/span><span class=\"ecbx-1095\">bung 7.58 <\/span>(Konvergenzradien)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Finden Sie f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r jedes <\/span><math display=\"inline\"><mi>R<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mo class=\"MathClass-open\">[<\/mo><mn>0<\/mn><mo class=\"MathClass-punc\">,<\/mo><mi>\u221e<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">\u222a<\/mo><mrow><mo fence=\"true\" form=\"prefix\"> {<\/mo><mrow><mi>\u221e<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">}<\/mo><\/mrow><\/math> <span class=\"ecti-1095\">eine Potenzreihe mit Konvergenzradius<\/span><span class=\"ecti-1095\">&nbsp;<\/span><span class=\"maperiod\"><math display=\"inline\"><mi>R<\/mi><\/math><\/span><span class=\"period\">.<\/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\">F<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r<\/span> <math display=\"inline\"><mi>R<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mn>0<\/mn><\/math> <span class=\"ecti-1095\">m<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">ssen Sie eine Potenzreihe mit sehr schnell wachsenden Koeffizienten verwenden.<\/span><\/p><\/details>  <\/div> <div class=\"me meexample\"> <div class=\"wp-nocaption \"><\/div><h4 id=\"z733398af5602\"> <a id=\"x1-200005r59\"><\/a> <span class=\"ecbx-1095\">\u00dc<\/span><span class=\"ecbx-1095\">bung 7.59.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Berechnen Sie den Konvergenzradius <\/span><math display=\"inline\"><mi>R<\/mi><\/math> <span class=\"ecti-1095\">der Potenzreihe<\/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>n<\/mi><mo class=\"MathClass-rel\">=<\/mo><mn>1<\/mn><\/mrow><mrow><mi>\u221e<\/mi><\/mrow><\/munderover><mfrac><mrow><msup><mrow><mo class=\"MathClass-open\">(<\/mo><msqrt><mrow><msup><mrow><mi>n<\/mi><\/mrow><mrow><mn>2<\/mn> <\/mrow> <\/msup> <mo class=\"MathClass-bin\">+<\/mo> <mi>n<\/mi><\/mrow><\/msqrt> <mo class=\"MathClass-bin\">\u2212<\/mo><msqrt><mrow><msup><mrow><mi>n<\/mi><\/mrow><mrow><mn>2<\/mn> <\/mrow> <\/msup> <mo class=\"MathClass-bin\">+<\/mo> <mn>1<\/mn><\/mrow><\/msqrt><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msup><\/mrow> <mrow><msup><mrow><mi>n<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><\/mrow><\/mfrac> <msup><mrow><mi>x<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msup><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\"><span class=\"ecti-1095\">und zeigen Sie Konvergenz der Potenzreihe bei den Punkten<\/span> <span class=\"maperiod\"><math display=\"inline\"><mo class=\"MathClass-bin\">\u2212<\/mo> <mi>R<\/mi><mo class=\"MathClass-punc\">,<\/mo> <mi>R<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u211d<\/mi><\/math><\/span><span class=\"period\">.<\/span> <\/p> <\/div> <div class=\"me melemma\"> <div class=\"wp-nocaption \"><\/div><h4 id=\"za534835ee916\"> <a id=\"x1-200006r60\"><\/a> <span class=\"ecbx-1095\">Lemma 7.60 <\/span>(Konvergenzradius via Quotientenkriterium)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Sei <\/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><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><msup><mrow><mi>z<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msup><\/math> <span class=\"ecti-1095\">eine<\/span> <span class=\"ecti-1095\">Potenzreihe mit <\/span><math display=\"inline\"><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><mo class=\"MathClass-rel\">\u2260<\/mo><mn>0<\/mn><\/math> <span class=\"ecti-1095\">f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r alle <\/span><math display=\"inline\"><mi>n<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2115<\/mi><\/math><span class=\"ecti-1095\">. Der<\/span> <span class=\"ecti-1095\">Konvergenzradius <\/span><math display=\"inline\"><mi>R<\/mi><\/math> <span class=\"ecti-1095\">ist gegeben durch<\/span> <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mi>R<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><munder class=\"msub\"><mrow><mi class=\"qopname\">lim<\/mi><mo>  <\/mo><\/mrow><mrow><mi>n<\/mi><mo class=\"MathClass-rel\">\u2192<\/mo><mi>\u221e<\/mi><\/mrow><\/munder><mfrac><mrow><mo class=\"MathClass-rel\">|<\/mo><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi><mo class=\"MathClass-bin\">+<\/mo><mn>1<\/mn><\/mrow><\/msub><mo class=\"MathClass-rel\">|<\/mo><\/mrow> <mrow><mo class=\"MathClass-rel\">|<\/mo><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><mo class=\"MathClass-rel\">|<\/mo><\/mrow><\/mfrac> <\/mrow><\/mfrac> <mo class=\"MathClass-rel\">=<\/mo><munder class=\"msub\"><mrow><mi class=\"qopname\"> lim<\/mi><mo>  <\/mo><\/mrow><mrow><mi>n<\/mi><mo class=\"MathClass-rel\">\u2192<\/mo><mi>\u221e<\/mi><\/mrow><\/munder> <mfrac><mrow><mo class=\"MathClass-rel\">|<\/mo><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><mo class=\"MathClass-rel\">|<\/mo><\/mrow> <mrow><mo class=\"MathClass-rel\">|<\/mo><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi><mo class=\"MathClass-bin\">+<\/mo><mn>1<\/mn><\/mrow><\/msub><mo class=\"MathClass-rel\">|<\/mo><\/mrow><\/mfrac><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\"><span class=\"ecti-1095\">falls dieser Grenzwert existiert.<\/span> <\/p> <\/div> <div class=\"me meexample\"> <div class=\"wp-nocaption \"><\/div><h4 id=\"zc0de8309f61b\"> <a id=\"x1-200007r61\"><\/a> <span class=\"ecbx-1095\">\u00dc<\/span><span class=\"ecbx-1095\">bung 7.61.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Zeigen Sie Lemma<\/span><span class=\"ecti-1095\">&nbsp;<\/span><a href=\"..\/..\/chapter\/potenzreihen#x1-200006r60\"><span class=\"ecti-1095\">7.60<\/span><\/a><span class=\"ecti-1095\">.<\/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\">Wiederholen Sie den Beweis von Satz <\/span><a href=\"..\/..\/chapter\/potenzreihen#x1-200002r56\"><span class=\"ecti-1095\">7.56<\/span><\/a><span class=\"ecti-1095\">.<\/span><\/p><\/details>  <\/div> <a id=\"x1-200008r200\"><\/a> <h4 id=\"zafe02ce450a2\" class=\"subsectionHead\"><span class=\"titlemark\">7.4.2 <\/span> <a id=\"x1-2010002\"><\/a>Addition und Multiplikation<\/h4> <div class=\"me metheorem\"> <div class=\"wp-nocaption \"><\/div><h4 id=\"z41f961ddfa59\"> <a id=\"x1-201001r62\"><\/a> <span class=\"ecbx-1095\">Proposition 7.62 <\/span>(Summen und Produkte)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Seien <\/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><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><msup><mrow><mi>z<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msup><\/math> <span class=\"ecti-1095\">und<\/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><msub><mrow><mi>b<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><msup><mrow><mi>z<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msup><\/math> <span class=\"ecti-1095\">zwei Potenzreihen mit<\/span> <span class=\"ecti-1095\">Konvergenzradius <\/span><math display=\"inline\"><msub><mrow><mi>R<\/mi><\/mrow><mrow><mi>a<\/mi><\/mrow><\/msub><\/math> <span class=\"ecti-1095\">respektive <\/span><math display=\"inline\"><msub><mrow><mi>R<\/mi><\/mrow><mrow><mi>b<\/mi><\/mrow><\/msub><\/math><span class=\"ecti-1095\">. Dann<\/span> <span class=\"ecti-1095\">gilt f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r alle <\/span><math display=\"inline\"><mi>z<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2102<\/mi><\/math> <span class=\"ecti-1095\">mit <\/span><math display=\"inline\"><mo class=\"MathClass-rel\">|<\/mo><mi>z<\/mi><mo class=\"MathClass-rel\">|<\/mo> <mo class=\"MathClass-rel\">&lt;<\/mo><mi class=\"qopname\"> min<\/mi><mo>  <\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> {<\/mo><mrow><msub><mrow><mi>R<\/mi><\/mrow><mrow><mi>a<\/mi><\/mrow><\/msub><mo class=\"MathClass-punc\">,<\/mo><msub><mrow><mi>R<\/mi><\/mrow><mrow><mi>b<\/mi><\/mrow><\/msub><\/mrow><mo fence=\"true\" form=\"postfix\">}<\/mo><\/mrow><\/math> <\/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>n<\/mi><mo class=\"MathClass-rel\">=<\/mo><mn>0<\/mn><\/mrow><mrow><mi>\u221e<\/mi><\/mrow><\/munderover><msub><mrow><mi>a<\/mi><\/mrow><mrow> <mi>n<\/mi><\/mrow><\/msub><msup><mrow><mi>z<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msup> <mo class=\"MathClass-bin\">+<\/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><msub><mrow><mi>b<\/mi><\/mrow><mrow> <mi>n<\/mi><\/mrow><\/msub><msup><mrow><mi>z<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msup><\/mtd> <mtd class=\"align-even\"> <mo class=\"MathClass-rel\">=<\/mo><munderover accent=\"false\" accentunder=\"false\"><mrow><mo> \u2211<\/mo> <\/mrow><mrow><mi>n<\/mi><mo class=\"MathClass-rel\">=<\/mo><mn>0<\/mn><\/mrow><mrow><mi>\u221e<\/mi><\/mrow><\/munderover><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>a<\/mi><\/mrow><mrow> <mi>n<\/mi><\/mrow><\/msub> <mo class=\"MathClass-bin\">+<\/mo> <msub><mrow><mi>b<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><msup><mrow><mi>z<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msup><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\"><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>n<\/mi><mo class=\"MathClass-rel\">=<\/mo><mn>0<\/mn><\/mrow><mrow><mi>\u221e<\/mi><\/mrow><\/munderover><msub><mrow><mi>a<\/mi><\/mrow><mrow> <mi>n<\/mi><\/mrow><\/msub><msup><mrow><mi>z<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msup><mstyle><mrow><mo fence=\"true\" form=\"prefix\"> )<\/mo><mrow \/><mo fence=\"true\" form=\"postfix\" \/><\/mrow><\/mstyle><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>n<\/mi><mo class=\"MathClass-rel\">=<\/mo><mn>0<\/mn><\/mrow><mrow><mi>\u221e<\/mi><\/mrow><\/munderover><msub><mrow><mi>b<\/mi><\/mrow><mrow> <mi>n<\/mi><\/mrow><\/msub><msup><mrow><mi>z<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msup><mstyle><mrow><mo fence=\"true\" form=\"prefix\"> )<\/mo><mrow \/><mo fence=\"true\" form=\"postfix\" \/><\/mrow><\/mstyle><\/mtd> <mtd class=\"align-even\"> <mo class=\"MathClass-rel\">=<\/mo><munderover accent=\"false\" accentunder=\"false\"><mrow><mo> \u2211<\/mo> <\/mrow><mrow><mi>n<\/mi><mo class=\"MathClass-rel\">=<\/mo><mn>0<\/mn><\/mrow><mrow><mi>\u221e<\/mi><\/mrow><\/munderover><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>k<\/mi><mo class=\"MathClass-rel\">=<\/mo><mn>0<\/mn><\/mrow><mrow><mi>n<\/mi><\/mrow><\/munderover><msub><mrow><mi>a<\/mi><\/mrow><mrow> <mi>n<\/mi><mo class=\"MathClass-bin\">\u2212<\/mo><mi>k<\/mi><\/mrow><\/msub><msub><mrow><mi>b<\/mi><\/mrow><mrow><mi>k<\/mi><\/mrow><\/msub><mstyle><mrow><mo fence=\"true\" form=\"prefix\"> )<\/mo><mrow \/><mo fence=\"true\" form=\"postfix\" \/><\/mrow><\/mstyle><msup><mrow><mi>z<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msup><mo class=\"MathClass-punc\">.<\/mo><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\" \/> <mtd class=\"align-label\"> <mspace width=\"2em\" \/><\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\"><span class=\"ecti-1095\">Insbesondere ist der Konvergenzradius der Potenzreihen auf der rechten Seite mindestens<\/span> <span class=\"maperiod\"><math display=\"inline\"><mi class=\"qopname\">min<\/mi><mo>  <\/mo><mrow><mo fence=\"true\" form=\"prefix\"> {<\/mo><mrow><msub><mrow><mi>R<\/mi><\/mrow><mrow><mi>a<\/mi> <\/mrow> <\/msub> <mo class=\"MathClass-punc\">,<\/mo> <msub><mrow><mi>R<\/mi><\/mrow><mrow><mi>b<\/mi> <\/mrow> <\/msub><\/mrow><mo fence=\"true\" form=\"postfix\">}<\/mo><\/mrow><\/math><\/span><span class=\"period\">.<\/span> <\/p> <\/div> <div class=\"wp-nocaption \"><\/div> <div class=\"proof\"> <p class=\"indent\"><span class=\"head\"><\/span><\/p><details open=\"open\"><summary><b>Beweis.<\/b><\/summary><p class=\"indent\" style=\"margin-top: 10\">Die erste Eigenschaft folgt aus Linearit\u00e4t des Grenzwerts. Die zweite verwendet noch Korollar <a href=\"..\/..\/chapter\/absolute-konvergenz#x1-195003r37\">7.37<\/a>. <span>&nbsp;&nbsp;<\/span><\/p><div class=\"qed\">\u25a0<\/div><\/details><\/div> <div class=\"me meexample\"> <div class=\"wp-nocaption \"><\/div><h4 id=\"z46ec4c2b4ed2\"> <a id=\"x1-201002r63\"><\/a> <span class=\"ecbx-1095\">Beispiel 7.63.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Falls <\/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><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><msup><mrow><mi>z<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msup><\/math> <span class=\"ecti-1095\">mindestens<\/span> <span class=\"ecti-1095\">Konvergenzradius <\/span><math display=\"inline\"><mn>1<\/mn><\/math> <span class=\"ecti-1095\">hat, so gilt<\/span> <\/p><math display=\"block\"><mtable class=\"align\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"> <mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mn>1<\/mn> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>z<\/mi><\/mrow><\/mfrac><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><msub><mrow><mi>a<\/mi><\/mrow><mrow> <mi>n<\/mi><\/mrow><\/msub><msup><mrow><mi>z<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msup> <mo class=\"MathClass-rel\">=<\/mo><munderover accent=\"false\" accentunder=\"false\"><mrow><mo> \u2211<\/mo> <\/mrow><mrow><mi>n<\/mi><mo class=\"MathClass-rel\">=<\/mo><mn>0<\/mn><\/mrow><mrow><mi>\u221e<\/mi><\/mrow><\/munderover><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>a<\/mi><\/mrow><mrow> <mn>0<\/mn><\/mrow><\/msub> <mo class=\"MathClass-bin\">+<\/mo> <mo>\u2026<\/mo> <mo class=\"MathClass-bin\">+<\/mo> <msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><msup><mrow><mi>z<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msup><mo class=\"MathClass-punc\">.<\/mo><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"><mstyle class=\"label\" id=\"x1-201003r6\" \/><mstyle class=\"maketag\"><mtext>(7.6)<\/mtext><\/mstyle><mspace class=\"nbsp\" width=\"0.33em\" \/> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\"><span class=\"ecti-1095\">f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r alle <\/span><math display=\"inline\"><mi>z<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2102<\/mi><\/math> <span class=\"ecti-1095\">mit<\/span> <math display=\"inline\"><mo class=\"MathClass-rel\">|<\/mo><mi>z<\/mi><mo class=\"MathClass-rel\">|<\/mo> <mo class=\"MathClass-rel\">&lt;<\/mo> <mn>1<\/mn><\/math><span class=\"ecti-1095\">. In der Tat hat<\/span> <span class=\"ecti-1095\">die Potenzreihe <\/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>z<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msup><\/math> <span class=\"ecti-1095\">Konvergenzradius <\/span><math display=\"inline\"><mn>1<\/mn><\/math> <span class=\"ecti-1095\">und f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r <\/span><math display=\"inline\"><mi>z<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2102<\/mi><\/math> <span class=\"ecti-1095\">mit <\/span><math display=\"inline\"><mo class=\"MathClass-rel\">|<\/mo><mi>z<\/mi><mo class=\"MathClass-rel\">|<\/mo> <mo class=\"MathClass-rel\">&lt;<\/mo> <mn>1<\/mn><\/math> <span class=\"ecti-1095\">gilt<\/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>z<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msup> <mo class=\"MathClass-rel\">=<\/mo> <mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mn>1<\/mn><mo class=\"MathClass-bin\">\u2212<\/mo><mi>z<\/mi><\/mrow><\/mfrac><\/math><span class=\"ecti-1095\">, womit<\/span> (<a href=\"..\/..\/chapter\/potenzreihen#x1-201003r6\">7.6<\/a>)<span class=\"ecti-1095\">aus Proposition <\/span><a href=\"..\/..\/chapter\/potenzreihen#x1-201001r62\"><span class=\"ecti-1095\">7.62<\/span><\/a> <span class=\"ecti-1095\">folgt.<\/span> <\/p> <\/div> <div class=\"me meexample\"> <div class=\"wp-nocaption \"><\/div><h4 id=\"zdf83828b0955\"> <a id=\"x1-201004r64\"><\/a> <span class=\"ecbx-1095\">\u00dc<\/span><span class=\"ecbx-1095\">bung 7.64.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Berechnen Sie <\/span><span class=\"maperiod\"><math display=\"inline\"><msubsup><mrow><mi class=\"MathClass-op\">\u2211<\/mi><mo> <\/mo> <\/mrow><mrow><mi>n<\/mi><mo class=\"MathClass-rel\">=<\/mo><mn>1<\/mn><\/mrow><mrow><mi>\u221e<\/mi><\/mrow><\/msubsup><mi>n<\/mi><msup><mrow><mn>2<\/mn><\/mrow><mrow><mo class=\"MathClass-bin\">\u2212<\/mo><mi>n<\/mi><\/mrow><\/msup><\/math><\/span><span class=\"period\">.<\/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\">Der                                          Wert                                          von<\/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>q<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msup><\/math> <span class=\"ecti-1095\">f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r                                                                                                         alle<\/span> <math display=\"inline\"><mi>q<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2102<\/mi><\/math> <span class=\"ecti-1095\">mit<\/span> <math display=\"inline\"><mo class=\"MathClass-rel\">|<\/mo><mi>q<\/mi><mo class=\"MathClass-rel\">|<\/mo> <mo class=\"MathClass-rel\">&lt;<\/mo> <mn>1<\/mn><\/math> <span class=\"ecti-1095\">ist bereits bekannt.<\/span><\/p><\/details>  <\/div> <a id=\"x1-201005r201\"><\/a> <h4 id=\"z31639062cec2\" class=\"subsectionHead\"><span class=\"titlemark\">7.4.3 <\/span> <a id=\"x1-2020003\"><\/a>Stetigkeit bei Randpunkten<\/h4> <p class=\"noindent\">Wir wollen nun eine Potenzreihe <math display=\"inline\"><msubsup><mrow><mi class=\"MathClass-op\"> \u2211<\/mi><mo> <\/mo> <\/mrow><mrow><mi>n<\/mi><mo class=\"MathClass-rel\">=<\/mo><mn>0<\/mn><\/mrow><mrow><mi>\u221e<\/mi><\/mrow><\/msubsup><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><msup><mrow><mi>x<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msup><\/math> betrachten, wobei wir <math display=\"inline\"><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2102<\/mi><\/math> f\u00fcr <math display=\"inline\"><mi>n<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2115<\/mi><\/math> erlauben, aber nur reelle Zahlen <math display=\"inline\"><mi>x<\/mi><\/math> einsetzen wollen. Nach Satz <a href=\"..\/..\/chapter\/potenzreihen#x1-200002r56\">7.56<\/a> existiert ein <span class=\"maperiod\"><math display=\"inline\"><mi>R<\/mi> <mo class=\"MathClass-rel\">\u2265<\/mo> <mn>0<\/mn><\/math><\/span><span class=\"period\">,<\/span> so dass die Funktion                                                                                                                                                                           <\/p><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> <mo class=\"MathClass-open\">(<\/mo><mo class=\"MathClass-bin\">\u2212<\/mo><mi>R<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>R<\/mi><mo class=\"MathClass-close\">)<\/mo><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><msub><mrow><mi>a<\/mi><\/mrow><mrow> <mi>n<\/mi><\/mrow><\/msub><msup><mrow><mi>x<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msup> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2102<\/mi><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">wohldefiniert und stetig ist, aber <math display=\"inline\"><msubsup><mrow><mi class=\"MathClass-op\"> \u2211<\/mi><mo> <\/mo> <\/mrow><mrow><mi>n<\/mi><mo class=\"MathClass-rel\">=<\/mo><mn>0<\/mn><\/mrow><mrow><mi>\u221e<\/mi><\/mrow><\/msubsup><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><msup><mrow><mi>x<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msup><\/math> f\u00fcr alle <math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u211d<\/mi><\/math> mit <math display=\"inline\"><mo class=\"MathClass-rel\">|<\/mo><mi>x<\/mi><mo class=\"MathClass-rel\">|<\/mo> <mo class=\"MathClass-rel\">&gt;<\/mo> <mi>R<\/mi><\/math> divergiert. Wir nehmen nun weiter an, dass <math display=\"inline\"><mi>R<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mo class=\"MathClass-open\">(<\/mo><mn>0<\/mn><mo class=\"MathClass-punc\">,<\/mo><mi>\u221e<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math> und <math display=\"inline\"><msubsup><mrow><mi class=\"MathClass-op\"> \u2211<\/mi><mo> <\/mo> <\/mrow><mrow><mi>n<\/mi><mo class=\"MathClass-rel\">=<\/mo><mn>0<\/mn><\/mrow><mrow><mi>\u221e<\/mi><\/mrow><\/msubsup><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><msup><mrow><mi>R<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msup><\/math> ebenfalls konvergiert. Satz <a href=\"..\/..\/chapter\/potenzreihen#x1-200002r56\">7.56<\/a> sagt in diesem Fall \u00fcberhaupt nichts \u00fcber die erweiterte Funktion <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mover accent=\"true\"><mrow><mi>f<\/mi><\/mrow><mo accent=\"true\">\u00af<\/mo><\/mover> <mo class=\"MathClass-punc\">:<\/mo> <mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mo class=\"MathClass-open\">(<\/mo><mo class=\"MathClass-bin\">\u2212<\/mo><mi>R<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>R<\/mi><mo class=\"MathClass-close\">]<\/mo><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><msub><mrow><mi>a<\/mi><\/mrow><mrow> <mi>n<\/mi><\/mrow><\/msub><msup><mrow><mi>x<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msup> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2102<\/mi><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">aus. <\/p> <div class=\"me metheorem\"> <div class=\"wp-nocaption \"><\/div><h4 id=\"z26cf5e909c27\"> <a id=\"x1-202001r65\"><\/a> <span class=\"ecbx-1095\">Satz 7.65 <\/span>(Abelscher Grenzwertsatz)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Unter obigen Annahmen ist auch <\/span><math display=\"inline\"><mover accent=\"true\"><mrow><mi>f<\/mi><\/mrow><mo accent=\"true\">\u00af<\/mo><\/mover><\/math> <span class=\"ecti-1095\">stetig. Das heisst,<\/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>n<\/mi><mo class=\"MathClass-rel\">=<\/mo><mn>0<\/mn><\/mrow><mrow><mi>\u221e<\/mi><\/mrow><\/munderover><msub><mrow><mi>a<\/mi><\/mrow><mrow> <mi>n<\/mi><\/mrow><\/msub><msup><mrow><mi>R<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msup> <mo class=\"MathClass-rel\">=<\/mo> <mover accent=\"true\"><mrow><mi>f<\/mi><\/mrow><mo accent=\"true\">\u00af<\/mo><\/mover> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>R<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-rel\">=<\/mo><munder class=\"msub\"><mrow><mi class=\"qopname\"> lim<\/mi><mo>  <\/mo><\/mrow><mrow> <mi>x<\/mi><mo class=\"MathClass-rel\">\u2197<\/mo><mi>R<\/mi><\/mrow><\/munder><mover accent=\"true\"><mrow><mi>f<\/mi><\/mrow><mo accent=\"true\">\u00af<\/mo><\/mover> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-rel\">=<\/mo><munder class=\"msub\"><mrow><mi class=\"qopname\"> lim<\/mi><mo>  <\/mo><\/mrow><mrow><mi>x<\/mi><mo class=\"MathClass-rel\">\u2197<\/mo><mi>R<\/mi><\/mrow><\/munder><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><msub><mrow><mi>a<\/mi><\/mrow><mrow> <mi>n<\/mi><\/mrow><\/msub><msup><mrow><mi>x<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msup><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\"><span class=\"ecti-1095\">Eine analoge Aussage gilt, falls <\/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><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><msup><mrow><mo class=\"MathClass-open\">(<\/mo><mo class=\"MathClass-bin\">\u2212<\/mo><mi>R<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msup><\/math> <span class=\"ecti-1095\">konvergiert.<\/span> <\/p> <\/div> <div class=\"me meexample\"> <div class=\"wp-nocaption \"><\/div><h4 id=\"z8d6daf8598a0\"> <a id=\"x1-202002r66\"><\/a> <span class=\"ecbx-1095\">Beispiel 7.66 <\/span>(Zwei alternierende Potenzreihen)<span class=\"ecbx-1095\">.<\/span> <\/h4> <dl class=\"enumerate\"><dt class=\"enumerate\"> <span class=\"ecti-1095\">(i)<\/span><\/dt><dd class=\"enumerate\"><span class=\"ecti-1095\">F<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r <\/span><math display=\"inline\"><msub><mrow><mi>a<\/mi><\/mrow><mrow><mn>0<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-rel\">=<\/mo> <mn>0<\/mn><\/math> <span class=\"ecti-1095\">und <\/span><math display=\"inline\"><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi> <\/mrow> <\/msub> <mo class=\"MathClass-rel\">=<\/mo> <mfrac><mrow><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><mo class=\"MathClass-bin\">+<\/mo><mn>1<\/mn><\/mrow><\/msup><\/mrow> <mrow><mi>n<\/mi><\/mrow><\/mfrac> <\/math> <span class=\"ecti-1095\">f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r alle <\/span><math display=\"inline\"><mi>n<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2115<\/mi><\/math> <span class=\"ecti-1095\">ist der Konvergenzradius der Potenzreihe <\/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><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><msup><mrow><mi>z<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msup><\/math> <span class=\"ecti-1095\">durch <\/span><math display=\"inline\"><mi>R<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mn>1<\/mn><\/math> <span class=\"ecti-1095\">gegeben und <\/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>1<\/mn><\/mrow><mrow><mi>\u221e<\/mi><\/mrow><\/msubsup><mfrac><mrow><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><mo class=\"MathClass-bin\">+<\/mo><mn>1<\/mn><\/mrow><\/msup><\/mrow> <mrow><mi>n<\/mi><\/mrow><\/mfrac> <\/math> <span class=\"ecti-1095\">konvergiert, womit der Abelsche Grenzwertsatz (Satz <\/span><a href=\"..\/..\/chapter\/potenzreihen#x1-202001r65\"><span class=\"ecti-1095\">7.65<\/span><\/a><span class=\"ecti-1095\">) angewendet werden kann.<\/span> <span class=\"ecti-1095\">Sobald wir die Funktion<\/span><span class=\"ecti-1095\">&nbsp;<\/span><math display=\"inline\"><mi>f<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-rel\">=<\/mo><msubsup><mrow><mi class=\"MathClass-op\"> \u2211<\/mi><mo> <\/mo> <\/mrow><mrow><mi>n<\/mi><mo class=\"MathClass-rel\">=<\/mo><mn>1<\/mn><\/mrow><mrow><mi>\u221e<\/mi><\/mrow><\/msubsup><mfrac><mrow><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><mo class=\"MathClass-bin\">+<\/mo><mn>1<\/mn><\/mrow><\/msup><\/mrow> <mrow><mi>n<\/mi><\/mrow><\/mfrac> <msup><mrow><mi>x<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msup><\/math> <span class=\"ecti-1095\">f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r<\/span><span class=\"ecti-1095\">&nbsp;<\/span><math display=\"inline\"><mo class=\"MathClass-rel\">|<\/mo><mi>x<\/mi><mo class=\"MathClass-rel\">|<\/mo> <mo class=\"MathClass-rel\">&lt;<\/mo> <mn>1<\/mn><\/math> <span class=\"ecti-1095\">kennen, k<\/span><span class=\"ecti-1095\">\u00f6<\/span><span class=\"ecti-1095\">nnen wir damit auch <\/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>1<\/mn><\/mrow><mrow><mi>\u221e<\/mi><\/mrow><\/msubsup><mfrac><mrow><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><mo class=\"MathClass-bin\">+<\/mo><mn>1<\/mn><\/mrow><\/msup><\/mrow> <mrow><mi>n<\/mi><\/mrow><\/mfrac> <\/math> <span class=\"ecti-1095\">berechnen.<\/span> <\/dd><dt class=\"enumerate\"> <span class=\"ecti-1095\">(ii)<\/span><\/dt><dd class=\"enumerate\"><span class=\"ecti-1095\">F<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r <\/span><math display=\"inline\"><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi> <\/mrow> <\/msub> <mo class=\"MathClass-rel\">=<\/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><\/math> <span class=\"ecti-1095\">f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r alle <\/span><math display=\"inline\"><mi>n<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <msub><mrow><mi>\u2115<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><\/math> <span class=\"ecti-1095\">ist <\/span><math display=\"inline\"><mi>R<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mn>1<\/mn><\/math> <span class=\"ecti-1095\">und der Abelsche Grenzwertsatz (Satz <\/span><a href=\"..\/..\/chapter\/potenzreihen#x1-202001r65\"><span class=\"ecti-1095\">7.65<\/span><\/a><span class=\"ecti-1095\">) kann nicht angewendet werden, da <\/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><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><\/math> <span class=\"ecti-1095\">divergiert.<\/span><\/dd><\/dl> <\/div> <div class=\"me meexample\"> <div class=\"wp-nocaption \"><\/div><h4 id=\"z104e6a94857e\"> <span class=\"ecti-1095\">Bemerkung.<\/span><\/h4> <p class=\"indent\">Hierzu eine   historische   Anmerkung:   Euler   (1707-1783)   und   seine   Zeitgenossen hatten noch einen  anderen  Zugang  zu  Reihen  und  wiesen  auf  Grund  der  Gleichung <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><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>x<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msup> <mo class=\"MathClass-rel\">=<\/mo> <mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mn>1<\/mn><mo class=\"MathClass-bin\">+<\/mo><mi>x<\/mi><\/mrow><\/mfrac><\/math> f\u00fcr <math display=\"inline\"><mo class=\"MathClass-rel\">|<\/mo><mi>x<\/mi><mo class=\"MathClass-rel\">|<\/mo> <mo class=\"MathClass-rel\">&lt;<\/mo> <mn>1<\/mn><\/math> der                                                                                                              Reihe <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><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> <mo class=\"MathClass-rel\">=<\/mo> <mn>1<\/mn> <mo class=\"MathClass-bin\">\u2212<\/mo> <mn>1<\/mn> <mo class=\"MathClass-bin\">+<\/mo> <mn>1<\/mn> <mo class=\"MathClass-bin\">\u2212<\/mo> <mn>1<\/mn> <mo class=\"MathClass-bin\">+<\/mo> <mo class=\"MathClass-rel\">\u22ef<\/mo><mspace class=\"thinspace\" width=\"0.17em\" \/><\/math> den                                                                                                               Wert <math display=\"inline\"><mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mn>2<\/mn><\/mrow><\/mfrac><\/math> zu, was aber unserem  modernerem  Konvergenzbegriff  (und  insbesondere  Proposition&nbsp;<a href=\"..\/..\/chapter\/reihen#x1-187002r2\">7.2<\/a>) widerspricht. <\/p> <\/div> <div class=\"wp-nocaption \"><\/div> <div class=\"proof\"> <p class=\"indent\"><span class=\"head\"><\/span><\/p><details open=\"open\"><summary><b>Beweis des Abelschen Grenzwertsatzes.<\/b><\/summary><p class=\"indent\" style=\"margin-top: 10\"> Wir nehmen ohne Beschr\u00e4nkung der Allgemeinheit an, dass der Konvergenzradius <math display=\"inline\"><mn>1<\/mn><\/math> ist (sonst ersetzt man <math display=\"inline\"><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><\/math> mit <math display=\"inline\"><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi> <\/mrow> <\/msub> <msup><mrow><mi>R<\/mi><\/mrow><mrow><mi>n<\/mi> <\/mrow> <\/msup> <\/math> f\u00fcr alle <math display=\"inline\"><mi>n<\/mi><\/math>). Nach Beispiel&nbsp;<a href=\"..\/..\/chapter\/potenzreihen#x1-201002r63\">7.63<\/a> gilt <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"> <mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mn>1<\/mn> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>x<\/mi><\/mrow><\/mfrac><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><msub><mrow><mi>a<\/mi><\/mrow><mrow> <mi>n<\/mi><\/mrow><\/msub><msup><mrow><mi>x<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msup> <mo class=\"MathClass-rel\">=<\/mo><munderover accent=\"false\" accentunder=\"false\"><mrow><mo> \u2211<\/mo> <\/mrow><mrow><mi>n<\/mi><mo class=\"MathClass-rel\">=<\/mo><mn>0<\/mn><\/mrow><mrow><mi>\u221e<\/mi><\/mrow><\/munderover><mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><msub><mrow><mi>a<\/mi><\/mrow><mrow> <mn>0<\/mn><\/mrow><\/msub> <mo class=\"MathClass-bin\">+<\/mo> <mo>\u2026<\/mo> <mo class=\"MathClass-bin\">+<\/mo> <msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <msup><mrow><mi>x<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msup><mo class=\"MathClass-punc\">.<\/mo><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">f\u00fcr alle <span class=\"maperiod\"><math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mo class=\"MathClass-open\">(<\/mo><mo class=\"MathClass-bin\">\u2212<\/mo><mn>1<\/mn><mo class=\"MathClass-punc\">,<\/mo><mn>1<\/mn><mo class=\"MathClass-close\">)<\/mo><\/math><\/span><span class=\"period\">.<\/span> Wir definieren <span class=\"maperiod\"><math display=\"inline\"><msub><mrow><mi>A<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub> <mo class=\"MathClass-rel\">=<\/mo> <msub><mrow><mi>a<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub> <mo class=\"MathClass-bin\">+<\/mo> <mi class=\"MathClass-op\">\u2026<\/mi><mo> <\/mo> <mo class=\"MathClass-bin\">+<\/mo> <msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><\/math><\/span><span class=\"period\">,<\/span> <math display=\"inline\"><mi>A<\/mi> <mo class=\"MathClass-rel\">=<\/mo><munder class=\"msub\"><mrow><mi class=\"qopname\"> lim<\/mi><mo>  <\/mo> <\/mrow><mrow><mi>n<\/mi><mo class=\"MathClass-rel\">\u2192<\/mo><mi>\u221e<\/mi><\/mrow><\/munder><msub><mrow><mi>A<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub> <mo class=\"MathClass-rel\">=<\/mo><msubsup><mrow><mi class=\"qopname\"> \u2211<\/mi><mo>  <\/mo> <\/mrow><mrow><mi>n<\/mi><mo class=\"MathClass-rel\">=<\/mo><mn>0<\/mn><\/mrow><mrow><mi>\u221e<\/mi><\/mrow><\/msubsup><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><\/math> (was nach Annahme existiert) und erhalten mit <math display=\"inline\"><msub><mrow><mi>b<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub> <mo class=\"MathClass-rel\">=<\/mo> <msub><mrow><mi>A<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>A<\/mi><\/math> f\u00fcr <math display=\"inline\"><mi>n<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2115<\/mi><\/math>                                                                                                                                                                           die Gleichung <\/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> <mo class=\"MathClass-rel\">=<\/mo><munderover accent=\"false\" accentunder=\"false\"><mrow><mo> \u2211<\/mo> <\/mrow><mrow><mi>n<\/mi><mo class=\"MathClass-rel\">=<\/mo><mn>0<\/mn><\/mrow><mrow><mi>\u221e<\/mi><\/mrow><\/munderover><msub><mrow><mi>a<\/mi><\/mrow><mrow> <mi>n<\/mi><\/mrow><\/msub><msup><mrow><mi>x<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msup><\/mtd> <mtd class=\"align-even\"> <mo class=\"MathClass-rel\">=<\/mo> <mo class=\"MathClass-open\">(<\/mo><mn>1<\/mn> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>x<\/mi><mo class=\"MathClass-close\">)<\/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><msub><mrow><mi>A<\/mi><\/mrow><mrow> <mi>n<\/mi><\/mrow><\/msub><msup><mrow><mi>x<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msup><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\" \/> <mtd class=\"align-label\"> <mspace width=\"2em\" \/><\/mtd><\/mtr><mtr><mtd class=\"align-odd\" columnalign=\"right\" \/> <mtd class=\"align-even\"> <mo class=\"MathClass-rel\">=<\/mo> <mo class=\"MathClass-open\">(<\/mo><mn>1<\/mn> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>x<\/mi><mo class=\"MathClass-close\">)<\/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><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>b<\/mi><\/mrow><mrow> <mi>n<\/mi><\/mrow><\/msub> <mo class=\"MathClass-bin\">+<\/mo> <mi>A<\/mi><mo class=\"MathClass-close\">)<\/mo><msup><mrow><mi>x<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msup><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\" \/> <mtd class=\"align-label\"> <mspace width=\"2em\" \/><\/mtd><\/mtr><mtr><mtd class=\"align-odd\" columnalign=\"right\" \/> <mtd class=\"align-even\"> <mo class=\"MathClass-rel\">=<\/mo> <mo class=\"MathClass-open\">(<\/mo><mn>1<\/mn> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>x<\/mi><mo class=\"MathClass-close\">)<\/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><msub><mrow><mi>b<\/mi><\/mrow><mrow> <mi>n<\/mi><\/mrow><\/msub><msup><mrow><mi>x<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msup> <mo class=\"MathClass-bin\">+<\/mo> <mi>A<\/mi><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\" \/> <mtd class=\"align-label\"> <mspace width=\"2em\" \/><\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">f\u00fcr alle <span class=\"maperiod\"><math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mo class=\"MathClass-open\">(<\/mo><mo class=\"MathClass-bin\">\u2212<\/mo><mn>1<\/mn><mo class=\"MathClass-punc\">,<\/mo><mn>1<\/mn><mo class=\"MathClass-close\">)<\/mo><\/math><\/span><span class=\"period\">.<\/span> Obige Formelmanipulationen m\u00f6gen vielleicht vom Himmel gefallen sein; ab jetzt wird das Argument jedoch wenig \u00dcberraschungen bieten. Sei <span class=\"maperiod\"><math display=\"inline\"><mi>\ud835\udf00<\/mi> <mo class=\"MathClass-rel\">&gt;<\/mo> <mn>0<\/mn><\/math><\/span><span class=\"period\">.<\/span> Dann existiert ein <math display=\"inline\"><mi>N<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2115<\/mi><\/math> mit <math display=\"inline\"><mo class=\"MathClass-rel\">|<\/mo><msub><mrow><mi>b<\/mi><\/mrow><mrow><mi>n<\/mi> <\/mrow> <\/msub> <mo class=\"MathClass-rel\">|<\/mo> <mo class=\"MathClass-rel\">&lt;<\/mo> <mi>\ud835\udf00<\/mi><\/math> f\u00fcr alle <span class=\"maperiod\"><math display=\"inline\"><mi>n<\/mi> <mo class=\"MathClass-rel\">\u2265<\/mo> <mi>N<\/mi><\/math><\/span><span class=\"period\">.<\/span> Daraus folgt f\u00fcr <span class=\"maperiod\"><math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mo class=\"MathClass-open\">[<\/mo><mn>0<\/mn><mo class=\"MathClass-punc\">,<\/mo><mn>1<\/mn><mo class=\"MathClass-close\">)<\/mo><\/math><\/span><span class=\"period\">,<\/span> dass <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><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> <mi>A<\/mi><mo class=\"MathClass-rel\">|<\/mo><\/mtd> <mtd class=\"align-even\"> <mo class=\"MathClass-rel\">=<\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> |<\/mo><mrow><mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mn>1<\/mn> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><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><msub><mrow><mi>b<\/mi><\/mrow><mrow> <mi>n<\/mi><\/mrow><\/msub><msup><mrow><mi>x<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msup><\/mrow><mo fence=\"true\" form=\"postfix\">|<\/mo><\/mrow><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\" \/> <mtd class=\"align-label\"> <mspace width=\"2em\" \/><\/mtd><\/mtr><mtr><mtd class=\"align-odd\" columnalign=\"right\" \/> <mtd class=\"align-even\"> <mo class=\"MathClass-rel\">\u2264<\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> |<\/mo><mrow><mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mn>1<\/mn> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><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>N<\/mi><\/mrow><\/munderover><msub><mrow><mi>b<\/mi><\/mrow><mrow> <mi>n<\/mi><\/mrow><\/msub><msup><mrow><mi>x<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msup><\/mrow><mo fence=\"true\" form=\"postfix\">|<\/mo><\/mrow> <mo class=\"MathClass-bin\">+<\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mn>1<\/mn> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><mi>\ud835\udf00<\/mi><munderover accent=\"false\" accentunder=\"false\"><mrow><mo>\u2211<\/mo> <\/mrow><mrow><mi>n<\/mi><mo class=\"MathClass-rel\">=<\/mo><mi>N<\/mi><mo class=\"MathClass-bin\">+<\/mo><mn>1<\/mn><\/mrow><mrow><mi>\u221e<\/mi><\/mrow><\/munderover><msup><mrow><mi>x<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msup><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\" \/> <mtd class=\"align-label\"> <mspace width=\"2em\" \/><\/mtd><\/mtr><mtr><mtd class=\"align-odd\" columnalign=\"right\" \/> <mtd class=\"align-even\"> <mo class=\"MathClass-rel\">\u2264<\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> |<\/mo><mrow><mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mn>1<\/mn> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><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>N<\/mi><\/mrow><\/munderover><msub><mrow><mi>b<\/mi><\/mrow><mrow> <mi>n<\/mi><\/mrow><\/msub><msup><mrow><mi>x<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msup><\/mrow><mo fence=\"true\" form=\"postfix\">|<\/mo><\/mrow> <mo class=\"MathClass-bin\">+<\/mo> <mi>\ud835\udf00<\/mi><mo class=\"MathClass-punc\">.<\/mo><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\" \/> <mtd class=\"align-label\"> <mspace width=\"2em\" \/><\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">Da aber das Polynom <math display=\"inline\"> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mn>1<\/mn> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><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>N<\/mi><\/mrow><\/msubsup><msub><mrow><mi>b<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><msup><mrow><mi>x<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msup><\/math> auf <math display=\"inline\"><mi>\u211d<\/mi><\/math> stetig ist und bei <math display=\"inline\"><mn>1<\/mn><\/math> verschwindet, gibt es weiters ein <span class=\"maperiod\"><math display=\"inline\"><mi>\u03b4<\/mi> <mo class=\"MathClass-rel\">&gt;<\/mo> <mn>0<\/mn><\/math><\/span><span class=\"period\">,<\/span> so dass <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mn>1<\/mn> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>\u03b4<\/mi><mo class=\"MathClass-punc\">,<\/mo><mn>1<\/mn><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><mspace class=\"thickpace\" width=\"0.28em\" \/><mo class=\"MathClass-rel\">\u21d2<\/mo><mspace class=\"thickpace\" width=\"0.28em\" \/> <mrow><mo fence=\"true\" form=\"prefix\"> |<\/mo><mrow><mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mn>1<\/mn> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><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>N<\/mi><\/mrow><\/munderover><msub><mrow><mi>b<\/mi><\/mrow><mrow> <mi>n<\/mi><\/mrow><\/msub><msup><mrow><mi>x<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msup><\/mrow><mo fence=\"true\" form=\"postfix\">|<\/mo><\/mrow> <mo class=\"MathClass-rel\">&lt;<\/mo> <mi>\ud835\udf00<\/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\">Daher gilt <math display=\"inline\"><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> <mi>A<\/mi><mo class=\"MathClass-rel\">|<\/mo> <mo class=\"MathClass-rel\">&lt;<\/mo> <mn>2<\/mn><mi>\ud835\udf00<\/mi><\/math> f\u00fcr alle <math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mo class=\"MathClass-open\">(<\/mo><mn>1<\/mn> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>\u03b4<\/mi><mo class=\"MathClass-punc\">,<\/mo><mn>1<\/mn><mo class=\"MathClass-close\">)<\/mo><\/math> und der Satz folgt. <span>&nbsp;&nbsp;<\/span><\/p><div class=\"qed\">\u25a0<\/div><\/details><\/div> <div class=\"me meexample\"> <div class=\"wp-nocaption \"><\/div><h4 id=\"z034eaee0f668\"> <a id=\"x1-202005r67\"><\/a> <span class=\"ecbx-1095\">\u00dc<\/span><span class=\"ecbx-1095\">bung 7.67.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Wo wurde im obigen Beweis verwendet, dass<\/span><span class=\"ecti-1095\">&nbsp;<\/span><math display=\"inline\"><mi>x<\/mi><\/math> <span class=\"ecti-1095\">reell ist?<\/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\">F<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r <\/span><math display=\"inline\"><mi>z<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <msub><mrow><mi>B<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><mo class=\"MathClass-open\">(<\/mo><mn>0<\/mn><mo class=\"MathClass-close\">)<\/mo><\/math> <span class=\"ecti-1095\">ist mitunter <\/span><math display=\"inline\"><mn>1<\/mn> <mo class=\"MathClass-bin\">\u2212<\/mo><mo class=\"MathClass-rel\">|<\/mo><mi>z<\/mi><mo class=\"MathClass-rel\">|<\/mo><\/math> <span class=\"ecti-1095\">kleiner als <\/span><span class=\"maperiod\"><math display=\"inline\"><mo class=\"MathClass-rel\">|<\/mo><mn>1<\/mn> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>z<\/mi><mo class=\"MathClass-rel\">|<\/mo><\/math><\/span><span class=\"period\">,<\/span> <span class=\"ecti-1095\">doch gilt <\/span><math display=\"inline\"><mn>1<\/mn> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>x<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mo class=\"MathClass-rel\">|<\/mo><mn>1<\/mn> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>x<\/mi><mo class=\"MathClass-rel\">|<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <mn>1<\/mn> <mo class=\"MathClass-bin\">\u2212<\/mo><mo class=\"MathClass-rel\">|<\/mo><mi>x<\/mi><mo class=\"MathClass-rel\">|<\/mo><\/math> <span class=\"ecti-1095\">f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r <\/span><span class=\"maperiod\"><math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mo class=\"MathClass-open\">(<\/mo><mn>0<\/mn><mo class=\"MathClass-punc\">,<\/mo> <mn>1<\/mn><mo class=\"MathClass-close\">)<\/mo><\/math><\/span><span class=\"period\">.<\/span><\/p><\/details>  <\/div> <a id=\"x1-202006r199\"><\/a> 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