{"id":77,"date":"2021-12-15T09:53:17","date_gmt":"2021-12-15T09:53:17","guid":{"rendered":"https:\/\/wp-prd.let.ethz.ch\/analysis19\/chapter\/absolute-konvergenz\/"},"modified":"2021-12-15T09:53:17","modified_gmt":"2021-12-15T09:53:17","slug":"absolute-konvergenz","status":"publish","type":"chapter","link":"https:\/\/wp-prd.let.ethz.ch\/analysis19\/chapter\/absolute-konvergenz\/","title":{"raw":"Absolute Konvergenz","rendered":"Absolute Konvergenz"},"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=\"z3583037af869\" class=\"sectionHead\"><span class=\"titlemark\">7.2 <\/span> <a id=\"x1-1920002\"><\/a>Absolute Konvergenz<\/h3> <p class=\"noindent\">In diesem Abschnitt wollen wir uns vor allem mit absolut konvergenten Reihen auseinandersetzen und einige Konvergenzkriterien beweisen. Auch m\u00f6chten wir zeigen, dass absolut konvergente Reihen im Gegensatz zu bedingt konvergenten Reihen stabilere Eigenschaften haben. <\/p> <div class=\"me metheorem\"> <p class=\"indent\"><\/p><h4 id=\"z262506282f0e\"> <a id=\"x1-192001r28\"><\/a> <span class=\"ecbx-1095\">Proposition 7.28 <\/span>(Absolute Konvergenz)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Eine absolut konvergente 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>1<\/mn><\/mrow><mrow><mi>\u221e<\/mi><\/mrow><\/msubsup><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><\/math> <span class=\"ecti-1095\">ist auch konvergent und es gilt die verallgemeinerte Dreiecksungleichung<\/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><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><msub><mrow><mi>a<\/mi><\/mrow><mrow> <mi>n<\/mi><\/mrow><\/msub><mstyle><mrow><mo fence=\"true\" form=\"prefix\"> |<\/mo><mrow \/><mo fence=\"true\" form=\"postfix\" \/><\/mrow><\/mstyle> <mo class=\"MathClass-rel\">\u2264<\/mo><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><mo class=\"MathClass-rel\">|<\/mo><msub><mrow><mi>a<\/mi><\/mrow><mrow> <mi>n<\/mi><\/mrow><\/msub><mo class=\"MathClass-rel\">|<\/mo><mo class=\"MathClass-punc\">.<\/mo><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <\/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\">Der erste Teil folgt unmittelbar aus zweimaliger Anwendung des Cauchy-Kriteriums f\u00fcr Reihen (Satz <a href=\"..\/..\/chapter\/reihen#x1-191001r26\">7.26<\/a>): Da 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>1<\/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><\/math> konvergiert, gibt es f\u00fcr <math display=\"inline\"><mi>\ud835\udf00<\/mi> <mo class=\"MathClass-rel\">&gt;<\/mo> <mn>0<\/mn><\/math> nach dem Cauchy-Kriterium ein <span class=\"maperiod\"><math display=\"inline\"><mi>N<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2115<\/mi><\/math><\/span><span class=\"period\">,<\/span> so dass f\u00fcr <math display=\"inline\"><mi>n<\/mi> <mo class=\"MathClass-rel\">\u2265<\/mo> <mi>m<\/mi> <mo class=\"MathClass-rel\">\u2265<\/mo> <mi>N<\/mi><\/math> die Absch\u00e4tzung                                                                                                                                                                           <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><munderover accent=\"false\" accentunder=\"false\"><mrow><mo> \u2211<\/mo> <\/mrow><mrow><mi>k<\/mi><mo class=\"MathClass-rel\">=<\/mo><mi>m<\/mi><\/mrow><mrow><mi>n<\/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> <mo class=\"MathClass-rel\">&lt;<\/mo> <mi>\ud835\udf00<\/mi><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">gilt. Daraus folgt <\/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><munderover accent=\"false\" accentunder=\"false\"><mrow><mo> \u2211<\/mo> <\/mrow><mrow><mi>k<\/mi><mo class=\"MathClass-rel\">=<\/mo><mi>m<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/munderover><msub><mrow><mi>a<\/mi><\/mrow><mrow> <mi>k<\/mi><\/mrow><\/msub><mstyle><mrow><mo fence=\"true\" form=\"prefix\"> |<\/mo><mrow \/><mo fence=\"true\" form=\"postfix\" \/><\/mrow><\/mstyle> <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>m<\/mi><\/mrow><mrow><mi>n<\/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> <mo class=\"MathClass-rel\">&lt;<\/mo> <mi>\ud835\udf00<\/mi><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">mit der Dreiecksungleichung. Da&nbsp;<math display=\"inline\"><mi>\ud835\udf00<\/mi> <mo class=\"MathClass-rel\">&gt;<\/mo> <mn>0<\/mn><\/math> beliebig war, beweist dies nach dem Cauchy-Kriterium die Konvergenz der Reihe&nbsp;<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><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><\/math><\/span><span class=\"period\">.<\/span> <\/p><p class=\"indent\">Der zweite Teil folgt nun aus der Ungleichung <\/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><munderover accent=\"false\" accentunder=\"false\"><mrow><mo> \u2211<\/mo> <\/mrow><mrow><mi>k<\/mi><mo class=\"MathClass-rel\">=<\/mo><mn>1<\/mn><\/mrow><mrow><mi>n<\/mi><\/mrow><\/munderover><msub><mrow><mi>a<\/mi><\/mrow><mrow> <mi>k<\/mi><\/mrow><\/msub><mstyle><mrow><mo fence=\"true\" form=\"prefix\"> |<\/mo><mrow \/><mo fence=\"true\" form=\"postfix\" \/><\/mrow><\/mstyle> <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><mn>1<\/mn><\/mrow><mrow><mi>n<\/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><\/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 <math display=\"inline\"><mi>n<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2115<\/mi><\/math> und dem Grenz\u00fcbergang f\u00fcr <span class=\"maperiod\"><math display=\"inline\"><mi>n<\/mi> <mo class=\"MathClass-rel\">\u2192<\/mo><mi>\u221e<\/mi><\/math><\/span><span class=\"period\">.<\/span> <span>&nbsp;&nbsp;<\/span><\/p><div class=\"qed\">\u25a0<\/div><\/details><\/div> <a id=\"x1-192002r191\"><\/a> <h4 id=\"z6207f098ef60\" class=\"subsectionHead\"><span class=\"titlemark\">7.2.1 <\/span> <a id=\"x1-1930001\"><\/a>Hinreichende Kriterien f\u00fcr absolute Konvergenz<\/h4> <p class=\"noindent\">Falls sich die Glieder einer Reihe im Absolutbetrag durch die Glieder einer konvergenten Reihe absch\u00e4tzen lassen, so ist die Reihe konvergent, wie wir in folgendem Korollar des Vergleichssatzes (Korollar <a href=\"..\/..\/chapter\/reihen#x1-188002r12\">7.12<\/a>) zeigen. <\/p> <div class=\"me metheorem\"> <p class=\"indent\"><\/p><h4 id=\"zd91bf72461cf\"> <a id=\"x1-193001r29\"><\/a> <span class=\"ecbx-1095\">Korollar 7.29 <\/span>(Majorantenkriterium von Weierstrass)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Sei <\/span><math display=\"inline\"><msub><mrow><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi> <\/mrow> <\/msub> <mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><\/math> <span class=\"ecti-1095\">eine komplexe und <\/span><math display=\"inline\"><msub><mrow><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>b<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><\/math> <span class=\"ecti-1095\">eine reelle Folge mit <\/span><math display=\"inline\"><mo class=\"MathClass-rel\">|<\/mo><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><mo class=\"MathClass-rel\">|<\/mo><mo class=\"MathClass-rel\">\u2264<\/mo> <msub><mrow><mi>b<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><\/math> <span class=\"ecti-1095\">f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r alle hinreichend grossen <\/span><span class=\"maperiod\"><math display=\"inline\"><mi>n<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2115<\/mi><\/math><\/span><span class=\"period\">.<\/span> <span class=\"ecti-1095\">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>1<\/mn><\/mrow><mrow><mi>\u221e<\/mi><\/mrow><\/msubsup><msub><mrow><mi>b<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><\/math> <span class=\"ecti-1095\">konvergiert, dann ist <\/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><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><\/math> <span class=\"ecti-1095\">absolut konvergent und daher auch konvergent.<\/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\">Da endlich viele Startglieder vernachl\u00e4ssigt werden k\u00f6nnen, d\u00fcrfen wir annehmen, dass die Ungleichung <math display=\"inline\"><mo class=\"MathClass-rel\">|<\/mo><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><mo class=\"MathClass-rel\">|<\/mo><mo class=\"MathClass-rel\">\u2264<\/mo> <msub><mrow><mi>b<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><\/math> f\u00fcr alle <math display=\"inline\"><mi>n<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2115<\/mi><\/math> gilt. Nach dem Vergleichssatz (Korollar <a href=\"..\/..\/chapter\/reihen#x1-188002r12\">7.12<\/a>) ist <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><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><\/math> dann absolut konvergent und die Konvergenz folgt aus Proposition&nbsp;<a href=\"..\/..\/chapter\/absolute-konvergenz#x1-192001r28\">7.28<\/a>. <span>&nbsp;&nbsp;<\/span><\/p><div class=\"qed\">\u25a0<\/div><\/details><\/div> <p class=\"indent\">Wir m\u00f6chten nun zwei Korollare des Majorantenkriteriums diskutieren. <\/p> <div class=\"me metheorem\"> <p class=\"indent\"><\/p><h4 id=\"za08d8396415e\"> <a id=\"x1-193002r30\"><\/a> <span class=\"ecbx-1095\">Korollar 7.30 <\/span>(Cauchy-Wurzelkriterium)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Sei <\/span><math display=\"inline\"><msub><mrow><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi> <\/mrow> <\/msub> <mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><\/math> <span class=\"ecti-1095\">eine Folge komplexer Zahlen und<\/span> <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mi>\u03b1<\/mi> <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> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u211d<\/mi> <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><mo class=\"MathClass-punc\">.<\/mo><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\"><span class=\"ecti-1095\">Dann gilt<\/span> <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mi>\u03b1<\/mi> <mo class=\"MathClass-rel\">&lt;<\/mo> <mn>1<\/mn><\/mtd> <mtd class=\"align-even\"><mspace class=\"thickpace\" width=\"0.28em\" \/><mo class=\"MathClass-rel\">\u21d2<\/mo><mspace class=\"thickpace\" width=\"0.28em\" \/><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><msub><mrow><mi>a<\/mi><\/mrow><mrow> <mi>n<\/mi><\/mrow><\/msub><mstyle class=\"text\"><mtext>&nbsp;ist&nbsp;absolut&nbsp;konvergent<\/mtext><\/mstyle><mo class=\"MathClass-punc\">,<\/mo><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\" \/> <mtd class=\"align-label\"> <mspace width=\"2em\" \/><\/mtd><\/mtr><mtr><mtd class=\"align-odd\" columnalign=\"right\"><mi>\u03b1<\/mi> <mo class=\"MathClass-rel\">&gt;<\/mo> <mn>1<\/mn><\/mtd> <mtd class=\"align-even\"><mspace class=\"thickpace\" width=\"0.28em\" \/><mo class=\"MathClass-rel\">\u21d2<\/mo><mspace class=\"thickpace\" width=\"0.28em\" \/><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><msub><mrow><mi>a<\/mi><\/mrow><mrow> <mi>n<\/mi><\/mrow><\/msub><mstyle class=\"text\"><mtext>&nbsp;ist&nbsp;divergent&nbsp;und&nbsp;<\/mtext><\/mstyle><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><\/mrow><\/msub><mstyle class=\"text\"><mtext>&nbsp;ist&nbsp;keine&nbsp;Nullfolge<\/mtext><\/mstyle><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> <\/div> <p class=\"indent\">Sehr schwammig ausgedr\u00fcckt l\u00e4sst sich eine Folge <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> <\/mrow> <\/msub> <\/math> f\u00fcr <math display=\"inline\"><mi>\u03b1<\/mi> <mo class=\"MathClass-rel\">&lt;<\/mo> <mn>1<\/mn><\/math> wie in Korollar <a href=\"..\/..\/chapter\/absolute-konvergenz#x1-193002r30\">7.30<\/a> bis auf endlich viele Glieder und einen kleinen Fehler <math display=\"inline\"><mi>\ud835\udf00<\/mi> <mo class=\"MathClass-rel\">&gt;<\/mo> <mn>0<\/mn><\/math> von oben durch die geometrische Folge <math display=\"inline\"><msup><mrow><mo class=\"MathClass-open\">(<\/mo><mi>\u03b1<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mi>\ud835\udf00<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msup><\/math> absch\u00e4tzen. Somit kann man Korollar <a href=\"..\/..\/chapter\/absolute-konvergenz#x1-193001r29\">7.29<\/a> anwenden. Nun aber genauer. <\/p><p class=\"indent\"> <\/p> <div class=\"proof\"> <p class=\"indent\"><span class=\"head\"><\/span><\/p><details open><summary><b>Beweis.<\/b><\/summary><p class=\"indent\" style=\"margin-top: 10\">Angenommen <span class=\"maperiod\"><math display=\"inline\"><mi>\u03b1<\/mi> <mo class=\"MathClass-rel\">&lt;<\/mo> <mn>1<\/mn><\/math><\/span><span class=\"period\">.<\/span> Dann gibt es ein <math display=\"inline\"><mi>n<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2115<\/mi><\/math> mit <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><munder class=\"msub\"><mrow><mi class=\"qopname\"> sup<\/mi><mo>  <\/mo><\/mrow><mrow><mi>k<\/mi><mo class=\"MathClass-rel\">\u2265<\/mo><mi>n<\/mi><\/mrow><\/munder><mroot><mrow><mo class=\"MathClass-rel\">|<\/mo><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>k<\/mi><\/mrow><\/msub><mo class=\"MathClass-rel\">|<\/mo><\/mrow><mrow><mi>k<\/mi><\/mrow><\/mroot> <mo class=\"MathClass-rel\">&lt;<\/mo> <mi>q<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mfrac><mrow><mn>1<\/mn> <mo class=\"MathClass-bin\">+<\/mo> <mi>\u03b1<\/mi><\/mrow> <mrow><mn>2<\/mn><\/mrow><\/mfrac> <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\">und somit <math display=\"inline\"><mo class=\"MathClass-rel\">|<\/mo><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>k<\/mi><\/mrow><\/msub><mo class=\"MathClass-rel\">|<\/mo> <mo class=\"MathClass-rel\">&lt;<\/mo> <msup><mrow><mi>q<\/mi><\/mrow><mrow><mi>k<\/mi><\/mrow><\/msup><\/math> f\u00fcr alle <span class=\"maperiod\"><math display=\"inline\"><mi>k<\/mi> <mo class=\"MathClass-rel\">\u2265<\/mo> <mi>n<\/mi><\/math><\/span><span class=\"period\">.<\/span> Die Reihe <math display=\"inline\"><msubsup><mrow><mi class=\"MathClass-op\"> \u2211<\/mi><mo> <\/mo> <\/mrow><mrow><mi>k<\/mi><mo class=\"MathClass-rel\">=<\/mo><mn>1<\/mn><\/mrow><mrow><mi>\u221e<\/mi><\/mrow><\/msubsup><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>k<\/mi><\/mrow><\/msub><\/math> konvergiert somit absolut wegen dem Majorantenkriterium (Korollar <a href=\"..\/..\/chapter\/absolute-konvergenz#x1-193001r29\">7.29<\/a>) und der geometrischen Reihe in Beispiel <a href=\"..\/..\/chapter\/reihen#x1-187003r3\">7.3<\/a>. <\/p><p class=\"indent\">Falls <math display=\"inline\"><mi>\u03b1<\/mi> <mo class=\"MathClass-rel\">&gt;<\/mo> <mn>1<\/mn><\/math> gilt, gibt es nach Satz <a href=\"..\/..\/chapter\/reelle-folgen#x1-160001r15\">6.15<\/a> eine Teilfolge <math display=\"inline\"><msub><mrow><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>a<\/mi><\/mrow><mrow><msub><mrow><mi>n<\/mi><\/mrow><mrow><mi>k<\/mi><\/mrow><\/msub><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mi>k<\/mi><\/mrow><\/msub><\/math> mit <math display=\"inline\"><mroot><mrow><mo class=\"MathClass-rel\">|<\/mo><msub><mrow><mi>a<\/mi><\/mrow><mrow><msub><mrow><mi>n<\/mi><\/mrow><mrow><mi>k<\/mi> <\/mrow> <\/msub> <\/mrow> <\/msub> <mo class=\"MathClass-rel\">|<\/mo><\/mrow> <mrow> <msub><mrow><mi>n<\/mi><\/mrow><mrow><mi>k<\/mi> <\/mrow> <\/msub> <\/mrow> <\/mroot> <mo class=\"MathClass-rel\">&gt;<\/mo> <mn>1<\/mn><\/math> f\u00fcr alle <span class=\"maperiod\"><math display=\"inline\"><mi>k<\/mi><\/math><\/span><span class=\"period\">.<\/span> Daraus folgt aber <span class=\"maperiod\"><math display=\"inline\"><mo class=\"MathClass-rel\">|<\/mo><msub><mrow><mi>a<\/mi><\/mrow><mrow><msub><mrow><mi>n<\/mi><\/mrow><mrow><mi>k<\/mi> <\/mrow> <\/msub> <\/mrow> <\/msub> <mo class=\"MathClass-rel\">|<\/mo> <mo class=\"MathClass-rel\">&gt;<\/mo> <mn>1<\/mn><\/math><\/span><span class=\"period\">.<\/span> Insbesondere ist <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> <\/mrow> <\/msub> <\/math> keine Nullfolge 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>1<\/mn><\/mrow><mrow><mi>\u221e<\/mi><\/mrow><\/msubsup><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><\/math> divergiert nach Proposition <a href=\"..\/..\/chapter\/reihen#x1-187002r2\">7.2<\/a>. <span>&nbsp;&nbsp;<\/span><\/p><div class=\"qed\">\u25a0<\/div><\/details><\/div> <div class=\"me meexample\"> <p class=\"indent\"><\/p><h4 id=\"z18d1e762804c\"> <a id=\"x1-193003r31\"><\/a> <span class=\"ecbx-1095\">Beispiel 7.31 <\/span>(Der Fall <math display=\"inline\"><mi>\u03b1<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mn>1<\/mn><\/math> in Korollar <a href=\"..\/..\/chapter\/absolute-konvergenz#x1-193002r30\">7.30<\/a>)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Sei <\/span><math display=\"inline\"><msub><mrow><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi> <\/mrow> <\/msub> <mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><\/math> <span class=\"ecti-1095\">eine Folge komplexer Zahlen<\/span> <span class=\"ecti-1095\">und <\/span><math display=\"inline\"><mi>\u03b1<\/mi> <mo class=\"MathClass-rel\">=<\/mo><msub><mrow><mi class=\"qopname\"> limsup<\/mi><mo>  <\/mo><\/mrow><mrow><mi>n<\/mi><mo class=\"MathClass-rel\">\u2192<\/mo><mi>\u221e<\/mi><\/mrow><\/msub><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><\/math> <span class=\"ecti-1095\">wie im Wurzelkriterium<\/span> <span class=\"ecti-1095\">(Korollar <\/span><a href=\"..\/..\/chapter\/absolute-konvergenz#x1-193002r30\"><span class=\"ecti-1095\">7.30<\/span><\/a><span class=\"ecti-1095\">). Falls <\/span><span class=\"maperiod\"><math display=\"inline\"><mi>\u03b1<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mn>1<\/mn><\/math><\/span><span class=\"period\">,<\/span> <span class=\"ecti-1095\">dann kann anhand des Wurzelkriteriums keine Entscheidung <\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">ber Konvergenz oder Divergenz der<\/span> <span class=\"ecti-1095\">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>1<\/mn><\/mrow><mrow><mi>\u221e<\/mi><\/mrow><\/msubsup><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><\/math> <span class=\"ecti-1095\">getroffen werden.<\/span> <\/p> <div class=\"custom-itemize\"><div class=\"item-head\"> <span class=\"tcrm-1095\">\u2022<\/span><\/div><div class=\"item-content\"><span class=\"ecti-1095\">Ist <\/span><math display=\"inline\"><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi> <\/mrow> <\/msub> <mo class=\"MathClass-rel\">=<\/mo> <mfrac> <mrow> <mn>1<\/mn><\/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><span class=\"maperiod\"><math display=\"inline\"><mi>n<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2115<\/mi><\/math><\/span><span class=\"period\">,<\/span> <span class=\"ecti-1095\">dann gilt <\/span><math display=\"inline\"><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><mn>1<\/mn><\/mrow> <mrow><mroot><mrow><mi>n<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/mroot><\/mrow><\/mfrac> <mo class=\"MathClass-rel\">\u2192<\/mo> <mn>1<\/mn><\/math> <span class=\"ecti-1095\">f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r <\/span><math display=\"inline\"><mi>n<\/mi> <mo class=\"MathClass-rel\">\u2192<\/mo> <mi>\u221e<\/mi><\/math> <span class=\"ecti-1095\">wegen Beispiel <\/span><a href=\"..\/..\/chapter\/reelle-folgen#x1-157006r4\"><span class=\"ecti-1095\">6.4<\/span><\/a> <span class=\"ecti-1095\">und wegen Beispiel <\/span><a href=\"..\/..\/chapter\/reihen#x1-187004r4\"><span class=\"ecti-1095\">7.4<\/span><\/a> <span class=\"ecti-1095\">divergiert 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>1<\/mn><\/mrow><mrow><mi>\u221e<\/mi><\/mrow><\/msubsup><mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mi>n<\/mi><\/mrow><\/mfrac><\/math> <span class=\"ecti-1095\">gegen Unendlich.<\/span> <\/div><div class=\"item-head\"> <span class=\"tcrm-1095\">\u2022<\/span><\/div><div class=\"item-content\"><span class=\"ecti-1095\">Ist <\/span><math display=\"inline\"><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi> <\/mrow> <\/msub> <mo class=\"MathClass-rel\">=<\/mo> <mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><msup><mrow><mi>n<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><\/mrow><\/mfrac><\/math> <span class=\"ecti-1095\">f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r alle <\/span><span class=\"maperiod\"><math display=\"inline\"><mi>n<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2115<\/mi><\/math><\/span><span class=\"period\">,<\/span> <span class=\"ecti-1095\">dann gilt <\/span><math display=\"inline\"><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><mn>1<\/mn><\/mrow> <mrow><msup><mrow><mroot><mrow><mi>n<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/mroot><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><\/mrow><\/mfrac> <mo class=\"MathClass-rel\">\u2192<\/mo> <mn>1<\/mn><\/math> <span class=\"ecti-1095\">f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r <\/span><math display=\"inline\"><mi>n<\/mi> <mo class=\"MathClass-rel\">\u2192<\/mo> <mi>\u221e<\/mi><\/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>1<\/mn><\/mrow><mrow><mi>\u221e<\/mi><\/mrow><\/msubsup> <mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><msup><mrow><mi>n<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><\/mrow><\/mfrac><\/math> <span class=\"ecti-1095\">konvergiert nach Beispiel <\/span><a href=\"..\/..\/chapter\/reihen#x1-188004r14\"><span class=\"ecti-1095\">7.14<\/span><\/a><span class=\"ecti-1095\">.<\/span><\/div><\/div> <\/div> <div class=\"me metheorem\"> <p class=\"indent\"><\/p><h4 id=\"zd87963d5bdaf\"> <a id=\"x1-193004r32\"><\/a> <span class=\"ecbx-1095\">Korollar 7.32 <\/span>(D\u2019Alemberts Quotientenkriterium)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Sei <\/span><math display=\"inline\"><msub><mrow><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi> <\/mrow> <\/msub> <mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><\/math> <span class=\"ecti-1095\">eine Folge<\/span> <span class=\"ecti-1095\">komplexer Zahlen 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><span class=\"maperiod\"><math display=\"inline\"><mi>n<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2115<\/mi><\/math><\/span><span class=\"period\">,<\/span> <span class=\"ecti-1095\">so dass<\/span> <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mi>\u03b1<\/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><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> <\/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\">existiert. Dann gilt<\/span> <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mi>\u03b1<\/mi> <mo class=\"MathClass-rel\">&lt;<\/mo> <mn>1<\/mn><\/mtd> <mtd class=\"align-even\"><mspace class=\"thickpace\" width=\"0.28em\" \/><mo class=\"MathClass-rel\">\u21d2<\/mo><mspace class=\"thickpace\" width=\"0.28em\" \/><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><msub><mrow><mi>a<\/mi><\/mrow><mrow> <mi>n<\/mi><\/mrow><\/msub><mstyle class=\"text\"><mtext>&nbsp;ist&nbsp;absolut&nbsp;konvergent.<\/mtext><\/mstyle><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\" \/> <mtd class=\"align-label\"> <mspace width=\"2em\" \/><\/mtd><\/mtr><mtr><mtd class=\"align-odd\" columnalign=\"right\"><mi>\u03b1<\/mi> <mo class=\"MathClass-rel\">&gt;<\/mo> <mn>1<\/mn><\/mtd> <mtd class=\"align-even\"><mspace class=\"thickpace\" width=\"0.28em\" \/><mo class=\"MathClass-rel\">\u21d2<\/mo><mspace class=\"thickpace\" width=\"0.28em\" \/><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><msub><mrow><mi>a<\/mi><\/mrow><mrow> <mi>n<\/mi><\/mrow><\/msub><mstyle class=\"text\"><mtext>&nbsp;ist&nbsp;divergent&nbsp;und&nbsp;<\/mtext><\/mstyle><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><\/mrow><\/msub><mstyle class=\"text\"><mtext>&nbsp;konvergiert&nbsp;nicht&nbsp;gegen&nbsp;Null.<\/mtext><\/mstyle><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\" \/> <mtd class=\"align-label\"> <mspace width=\"2em\" \/><\/mtd><\/mtr><\/mtable><\/math> <\/div> <div class=\"me melemma\"> <p class=\"indent\"><\/p><h4 id=\"z6338726d86e3\"> <a id=\"x1-193005r33\"><\/a> <span class=\"ecbx-1095\">Wichtige <\/span><span class=\"ecbx-1095\">\u00dc<\/span><span class=\"ecbx-1095\">bung 7.33.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Beweisen Sie Korollar <\/span><a href=\"..\/..\/chapter\/absolute-konvergenz#x1-193004r32\"><span class=\"ecti-1095\">7.32<\/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\">Gehen Sie wie im Beweis von Korollar <\/span><a href=\"..\/..\/chapter\/absolute-konvergenz#x1-193002r30\"><span class=\"ecti-1095\">7.30<\/span><\/a> <span class=\"ecti-1095\">vor. F<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r <\/span><math display=\"inline\"><mi>\u03b1<\/mi> <mo class=\"MathClass-rel\">&lt;<\/mo> <mn>1<\/mn><\/math> <span class=\"ecti-1095\">finden Sie ein <\/span><span class=\"maperiod\"><math display=\"inline\"><mi>q<\/mi> <mo class=\"MathClass-rel\">&lt;<\/mo> <mn>1<\/mn><\/math><\/span><span class=\"period\">,<\/span> <span class=\"ecti-1095\">so dass <\/span><math display=\"inline\"><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> <mo class=\"MathClass-rel\">&lt;<\/mo> <mi>q<\/mi><\/math> <span class=\"ecti-1095\">f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r alle bis auf endlich viele <\/span><span class=\"maperiod\"><math display=\"inline\"><mi>n<\/mi><\/math><\/span><span class=\"period\">.<\/span><\/p><\/details>  <\/div> <div class=\"me meexample\"> <p class=\"indent\"><\/p><h4 id=\"z0fe7afa80b15\"> <a id=\"x1-193006r34\"><\/a> <span class=\"ecbx-1095\">\u00dc<\/span><span class=\"ecbx-1095\">bung 7.34.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Wieso kann man im Quotientenkriterium (Korollar <\/span><a href=\"..\/..\/chapter\/absolute-konvergenz#x1-193004r32\"><span class=\"ecti-1095\">7.32<\/span><\/a><span class=\"ecti-1095\">) nicht auch den Limes superior<\/span> <span class=\"ecti-1095\">anstelle des Limes verwenden?<\/span> <\/p><p class=\"indent\"><\/p><details><summary style=\"color:#FF7F00\"><span class=\"ecti-1095\">L<\/span><span class=\"ecti-1095\">\u00f6<\/span><span class=\"ecti-1095\">sung.<\/span><\/summary><p class=\"indent\" style=\"margin-top: 0\"> <span class=\"ecti-1095\">In der Tat folgt aus <\/span><span class=\"maperiod\"><math display=\"inline\"><msub><mrow><mi class=\"qopname\">limsup<\/mi><mo>  <\/mo><\/mrow><mrow><mi>n<\/mi><mo class=\"MathClass-rel\">\u2192<\/mo><mi>\u221e<\/mi><\/mrow><\/msub><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> <mo class=\"MathClass-rel\">&lt;<\/mo> <mn>1<\/mn><\/math><\/span><span class=\"period\">,<\/span> <span class=\"ecti-1095\">dass die Reihe <\/span><math display=\"inline\"><msubsup><mrow><mi class=\"MathClass-op\">\u2211<\/mi><mo> <\/mo> <\/mrow><mrow><mi>n<\/mi><mo class=\"MathClass-rel\">=<\/mo><mn>1<\/mn><\/mrow><mrow><mi>\u221e<\/mi><\/mrow><\/msubsup><\/math> <span class=\"ecti-1095\">absolut konvergiert, was ebenso <\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">hnlich wie im Beweis von Satz <\/span><a href=\"..\/..\/chapter\/absolute-konvergenz#x1-193002r30\"><span class=\"ecti-1095\">7.30<\/span><\/a> <span class=\"ecti-1095\">gezeigt werden kann. Doch kann<\/span> <span class=\"ecti-1095\">aus <\/span><math display=\"inline\"><msub><mrow><mi class=\"qopname\"> limsup<\/mi><mo>  <\/mo> <\/mrow><mrow><mi>n<\/mi><mo class=\"MathClass-rel\">\u2192<\/mo><mi>\u221e<\/mi><\/mrow><\/msub><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> <mo class=\"MathClass-rel\">&gt;<\/mo> <mn>1<\/mn><\/math> <span class=\"ecti-1095\">keine Aussage <\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">ber die Konvergenz der Reihe getroffen werden. Als Beispiel hierzu betrachten wir<\/span> <span class=\"ecti-1095\">die Folge<\/span> <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub> <mo class=\"MathClass-rel\">=<\/mo> <mrow class=\"cases\"> <mrow><mo fence=\"true\" form=\"prefix\"> {<\/mo><mrow> <mtable align=\"axis\" class=\"array\" columnlines=\"none\" equalcolumns=\"false\" equalrows=\"false\"> <mtr><mtd class=\"array\" columnalign=\"left\"><msup><mrow><mn>2<\/mn><\/mrow><mrow><mo class=\"MathClass-bin\">\u2212<\/mo><mi>n<\/mi><\/mrow><\/msup><mspace class=\"quad\" width=\"1em\" \/><mstyle class=\"text\"><mtext>f\u00fcr&nbsp;ungerades&nbsp;<\/mtext><\/mstyle><mi>n<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2115<\/mi><mo class=\"MathClass-punc\">,<\/mo><mspace class=\"quad\" width=\"1em\" \/><\/mtd> <\/mtr> <mtr><mtd class=\"array\" columnalign=\"left\"><msup><mrow><mn>3<\/mn><\/mrow><mrow><mo class=\"MathClass-bin\">\u2212<\/mo><mi>n<\/mi><\/mrow><\/msup><mspace class=\"quad\" width=\"1em\" \/><mstyle class=\"text\"><mtext>f\u00fcr&nbsp;gerades&nbsp;<\/mtext><\/mstyle><mi>n<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2115<\/mi><mo class=\"MathClass-punc\">.<\/mo> <mspace class=\"quad\" width=\"1em\" \/><\/mtd><\/mtr> <\/mtable> <\/mrow><mo fence=\"true\" form=\"postfix\" \/><\/mrow><\/mrow><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\"><span class=\"ecti-1095\">F<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r diese Folge gilt<\/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\">limsup<\/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> <mo class=\"MathClass-rel\">=<\/mo><munder class=\"msub\"><mrow><mi class=\"qopname\"> lim<\/mi><mo>  <\/mo><\/mrow><mrow><mi>k<\/mi><mo class=\"MathClass-rel\">\u2192<\/mo><mi>\u221e<\/mi><\/mrow><\/munder><mfrac><mrow><msup><mrow><mn>2<\/mn><\/mrow><mrow><mo class=\"MathClass-bin\">\u2212<\/mo><mo class=\"MathClass-open\">(<\/mo><mn>2<\/mn><mi>k<\/mi><mo class=\"MathClass-bin\">+<\/mo><mn>1<\/mn><mo class=\"MathClass-close\">)<\/mo><\/mrow><\/msup><\/mrow> <mrow><msup><mrow><mn>3<\/mn><\/mrow><mrow><mo class=\"MathClass-bin\">\u2212<\/mo><mn>2<\/mn><mi>k<\/mi><\/mrow><\/msup><\/mrow><\/mfrac> <munder class=\"msub\"><mrow><mi class=\"qopname\"> lim<\/mi><mo>  <\/mo><\/mrow><mrow><mi>k<\/mi><mo class=\"MathClass-rel\">\u2192<\/mo><mi>\u221e<\/mi><\/mrow><\/munder><mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mn>2<\/mn><\/mrow><\/mfrac><msup><mrow> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mfrac><mrow><mn>3<\/mn><\/mrow> <mrow><mn>2<\/mn><\/mrow><\/mfrac> <\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <\/mrow><mrow><mn>2<\/mn><mi>k<\/mi><\/mrow><\/msup> <mo class=\"MathClass-rel\">=<\/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> <p class=\"noindent\"><span class=\"ecti-1095\">Trotzdem ist aber 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>1<\/mn><\/mrow><mrow><mi>\u221e<\/mi><\/mrow><\/msubsup><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><\/math> <span class=\"ecti-1095\">konvergent, was unter Verwendung von zwei geometrischen Reihen oder dem Wurzelkriterium schnell<\/span> <span class=\"ecti-1095\">folgt. <\/span><\/p><\/details>  <\/div> <a id=\"x1-193007r193\"><\/a> <h4 id=\"z964006478370\" class=\"subsectionHead\"><span class=\"titlemark\">7.2.2 <\/span> <a id=\"x1-1940002\"><\/a>Umordnen von Reihen<\/h4> <p class=\"noindent\">Im Gegensatz zur bedingten Konvergenz in Teilabschnitt <a href=\"..\/..\/chapter\/reihen#x1-1890002\">7.1.2<\/a> ist absolute Konvergenz sehr robust. Der erste dieser Robustheitss\u00e4tze ist folgender positive Umordnungssatz. <\/p> <div class=\"me metheorem\"> <p class=\"indent\"><\/p><h4 id=\"z11f682330900\"> <a id=\"x1-194001r35\"><\/a> <span class=\"ecbx-1095\">Satz 7.35 <\/span>(Umordnen absolut konvergenter Reihen)<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>1<\/mn><\/mrow><mrow><mi>\u221e<\/mi><\/mrow><\/msubsup><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><\/math> <span class=\"ecti-1095\">eine absolut konvergente Reihe mit komplexen Gliedern. Sei<\/span> <math display=\"inline\"><mi>\u03c6<\/mi> <mo class=\"MathClass-punc\">:<\/mo> <mi>\u2115<\/mi> <mo class=\"MathClass-rel\">\u2192<\/mo> <mi>\u2115<\/mi><\/math> <span class=\"ecti-1095\">eine Bijektion.<\/span> <span class=\"ecti-1095\">Dann ist <\/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><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>\u03c6<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>n<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow><\/msub><\/math> <span class=\"ecti-1095\">ebenso absolut konvergent und es gilt<\/span> <\/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>1<\/mn><\/mrow><mrow><mi>\u221e<\/mi><\/mrow><\/munderover><msub><mrow><mi>a<\/mi><\/mrow><mrow> <mi>n<\/mi><\/mrow><\/msub> <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>1<\/mn><\/mrow><mrow><mi>\u221e<\/mi><\/mrow><\/munderover><msub><mrow><mi>a<\/mi><\/mrow><mrow> <mi>\u03c6<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>n<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow><\/msub><mo class=\"MathClass-punc\">.<\/mo><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"><mstyle class=\"label\" id=\"x1-194002r2\" \/><mstyle class=\"maketag\"><mtext>(7.2)<\/mtext><\/mstyle><mspace class=\"nbsp\" width=\"0.33em\" \/> <\/mtd><\/mtr><\/mtable><\/math> <\/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\">Sei <math display=\"inline\"><mi>\u03c6<\/mi> <mo class=\"MathClass-punc\">:<\/mo> <mi>\u2115<\/mi> <mo class=\"MathClass-rel\">\u2192<\/mo> <mi>\u2115<\/mi><\/math> eine Bijektion und <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> Nach dem Cauchy-Kriterium (Satz <a href=\"..\/..\/chapter\/reihen#x1-191001r26\">7.26<\/a>) gibt es ein <span class=\"maperiod\"><math display=\"inline\"><mi>N<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2115<\/mi><\/math><\/span><span class=\"period\">,<\/span> so dass <math display=\"inline\"><msubsup><mrow><mi class=\"MathClass-op\"> \u2211<\/mi><mo> <\/mo> <\/mrow><mrow><mi>k<\/mi><mo class=\"MathClass-rel\">=<\/mo><mi>m<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msubsup><mo class=\"MathClass-rel\">|<\/mo><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>k<\/mi><\/mrow><\/msub><mo class=\"MathClass-rel\">|<\/mo> <mo class=\"MathClass-rel\">&lt;<\/mo> <mi>\ud835\udf00<\/mi><\/math> f\u00fcr alle nat\u00fcrliche Zahlen <span class=\"maperiod\"><math display=\"inline\"><mi>n<\/mi> <mo class=\"MathClass-rel\">\u2265<\/mo> <mi>m<\/mi> <mo class=\"MathClass-rel\">\u2265<\/mo> <mi>N<\/mi><\/math><\/span><span class=\"period\">.<\/span> Daher gilt auch nach der Dreiecksungleichung in Proposition <a href=\"..\/..\/chapter\/absolute-konvergenz#x1-192001r28\">7.28<\/a>, dass <\/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><munderover accent=\"false\" accentunder=\"false\"><mrow><mo> \u2211<\/mo> <\/mrow><mrow><mi>k<\/mi><mo class=\"MathClass-rel\">=<\/mo><mi>m<\/mi><\/mrow><mrow><mi>\u221e<\/mi><\/mrow><\/munderover><msub><mrow><mi>a<\/mi><\/mrow><mrow> <mi>k<\/mi><\/mrow><\/msub><mstyle><mrow><mo fence=\"true\" form=\"prefix\"> |<\/mo><mrow \/><mo fence=\"true\" form=\"postfix\" \/><\/mrow><\/mstyle> <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>m<\/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><mo class=\"MathClass-rel\">\u2264<\/mo> <mi>\ud835\udf00<\/mi><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">f\u00fcr alle&nbsp;<span class=\"maperiod\"><math display=\"inline\"><mi>m<\/mi> <mo class=\"MathClass-rel\">\u2265<\/mo> <mi>N<\/mi><\/math><\/span><span class=\"period\">.<\/span> (Es hilft vielleicht f\u00fcr das Folgende diese Absch\u00e4tzung als \u201eDie Summanden&nbsp;<math display=\"inline\"><msub><mrow><mi>a<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><mo class=\"MathClass-punc\">,<\/mo><mi class=\"MathClass-op\">\u2026<\/mi><mo> <\/mo><mo class=\"MathClass-punc\">,<\/mo><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>N<\/mi><\/mrow><\/msub><\/math> der urspr\u00fcnglichen Reihe sind wichtig, aber die restlichen Summanden sind weniger wichtig.\u201c zu interpretieren.) <\/p><p class=\"indent\">Wir definieren <math display=\"inline\"><mi>M<\/mi> <mo class=\"MathClass-rel\">=<\/mo><mi class=\"qopname\"> max<\/mi><mo>  <\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> {<\/mo><mrow><msup><mrow><mi>\u03c6<\/mi><\/mrow><mrow><mo class=\"MathClass-bin\">\u2212<\/mo><mn>1<\/mn><\/mrow><\/msup><mo class=\"MathClass-open\">(<\/mo><mi>k<\/mi><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-rel\">\u2223<\/mo><mi>k<\/mi> <mo class=\"MathClass-rel\">\u2264<\/mo> <mi>N<\/mi> <\/mrow><mo fence=\"true\" form=\"postfix\">}<\/mo><\/mrow><\/math> und w\u00e4hlen ein <span class=\"maperiod\"><math display=\"inline\"><mi>n<\/mi> <mo class=\"MathClass-rel\">\u2265<\/mo> <mi>M<\/mi><\/math><\/span><span class=\"period\">.<\/span> Dann gilt <\/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>\u2113<\/mi><mo class=\"MathClass-rel\">=<\/mo><mn>1<\/mn><\/mrow><mrow><mi>n<\/mi><\/mrow><\/munderover><msub><mrow><mi>a<\/mi><\/mrow><mrow> <mi>\u03c6<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>\u2113<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow><\/msub> <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>1<\/mn><\/mrow><mrow><mi>\u221e<\/mi><\/mrow><\/munderover><msub><mrow><mi>a<\/mi><\/mrow><mrow> <mi>k<\/mi><\/mrow><\/msub><\/mrow><mo fence=\"true\" form=\"postfix\">|<\/mo><\/mrow><\/mtd> <mtd class=\"align-even\"> <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>\u2113<\/mi><mo class=\"MathClass-rel\">=<\/mo><mn>1<\/mn><\/mrow><mrow><mi>n<\/mi><\/mrow><\/munderover><msub><mrow><mi>a<\/mi><\/mrow><mrow> <mi>\u03c6<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>\u2113<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow><\/msub> <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>1<\/mn><\/mrow><mrow><mi>N<\/mi><\/mrow><\/munderover><msub><mrow><mi>a<\/mi><\/mrow><mrow> <mi>k<\/mi><\/mrow><\/msub> <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><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><\/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><munder class=\"msub\"><mrow><mo>\u2211<\/mo> <\/mrow><mrow><mi>\u2113<\/mi><mo class=\"MathClass-rel\">\u2264<\/mo><mi>n<\/mi><mo class=\"MathClass-punc\">,<\/mo><mspace class=\"nbsp\" width=\"0.33em\" \/><mi>\u03c6<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>\u2113<\/mi><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-rel\">&gt;<\/mo><mi>N<\/mi><\/mrow><\/munder><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>\u03c6<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>\u2113<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow><\/msub><\/mrow><mo fence=\"true\" form=\"postfix\">|<\/mo><\/mrow> <mo class=\"MathClass-bin\">+<\/mo><munder class=\"msub\"><mrow><munder accentunder=\"false\"><mrow> <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><\/mrow><mo fence=\"true\" form=\"postfix\">|<\/mo><\/mrow><\/mrow><mo>\ufe38<\/mo><\/munder> <\/mrow><mrow><mo class=\"MathClass-rel\">\u2264<\/mo><mi>\ud835\udf00<\/mi><\/mrow><\/munder><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><munder class=\"msub\"><mrow><mo>\u2211<\/mo> <\/mrow><mrow><mi>\u2113<\/mi><mo class=\"MathClass-rel\">\u2264<\/mo><mi>n<\/mi><mo class=\"MathClass-punc\">,<\/mo><mspace class=\"nbsp\" width=\"0.33em\" \/><mi>\u03c6<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>\u2113<\/mi><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-rel\">&gt;<\/mo><mi>N<\/mi><\/mrow><\/munder> <mrow><mo fence=\"true\" form=\"prefix\"> |<\/mo><mrow><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>\u03c6<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>\u2113<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow><\/msub><\/mrow><mo fence=\"true\" form=\"postfix\">|<\/mo><\/mrow> <mo class=\"MathClass-bin\">+<\/mo> <mi>\ud835\udf00<\/mi> <mo class=\"MathClass-rel\">&lt;<\/mo> <mi>\ud835\udf00<\/mi> <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\">wobei wir verwendet haben, dass&nbsp;<math display=\"inline\"><mi>\u03c6<\/mi><\/math> eine Bijektion ist. Insbesondere treten damit und wegen&nbsp;<math display=\"inline\"><mi>n<\/mi> <mo class=\"MathClass-rel\">\u2265<\/mo> <mi>M<\/mi><\/math> alle&nbsp;<math display=\"inline\"><mi>k<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mo class=\"MathClass-open\">{<\/mo><mn>1<\/mn><mo class=\"MathClass-punc\">,<\/mo><mi class=\"MathClass-op\">\u2026<\/mi><mo> <\/mo><mo class=\"MathClass-punc\">,<\/mo><mi>N<\/mi><mo class=\"MathClass-close\">}<\/mo><\/math> genau einmal als&nbsp;<math display=\"inline\"><mi>\u03c6<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>\u2113<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math> f\u00fcr&nbsp;<math display=\"inline\"><mi>\u2113<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mo class=\"MathClass-open\">{<\/mo><mn>1<\/mn><mo class=\"MathClass-punc\">,<\/mo><mi class=\"MathClass-op\">\u2026<\/mi><mo> <\/mo><mo class=\"MathClass-punc\">,<\/mo><mi>n<\/mi><mo class=\"MathClass-close\">}<\/mo><\/math> auf und die Differenz&nbsp;<math display=\"inline\"><msubsup><mrow><mi class=\"MathClass-op\"> \u2211<\/mi><mo> <\/mo> <\/mrow><mrow><mi>\u2113<\/mi><mo class=\"MathClass-rel\">=<\/mo><mn>1<\/mn><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msubsup><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>\u03c6<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>\u2113<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow><\/msub> <mo class=\"MathClass-bin\">\u2212<\/mo><msubsup><mrow><mi class=\"MathClass-op\">\u2211<\/mi><mo> <\/mo> <\/mrow><mrow><mi>k<\/mi><mo class=\"MathClass-rel\">=<\/mo><mn>1<\/mn><\/mrow><mrow><mi>N<\/mi><\/mrow><\/msubsup><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>k<\/mi><\/mrow><\/msub><\/math> enth\u00e4lt nach Wegstreichen dieser Terme nur mehr eine Summe \u00fcber gewisse&nbsp;<math display=\"inline\"><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>k<\/mi> <\/mrow> <\/msub> <\/math> mit&nbsp;<math display=\"inline\"><mi>k<\/mi> <mo class=\"MathClass-rel\">\u2265<\/mo> <mi>N<\/mi><\/math> (welche wir im Sinne obiger Interpretation als \u201eweniger wichtig\u201c betrachten und formal eben insgesamt durch ein <math display=\"inline\"><mi>\ud835\udf00<\/mi><\/math> absch\u00e4tzen k\u00f6nnen). Da&nbsp;<math display=\"inline\"><mi>\ud835\udf00<\/mi> <mo class=\"MathClass-rel\">&gt;<\/mo> <mn>0<\/mn><\/math> beliebig war, zeigt dies die Gleichung&nbsp;(<a href=\"..\/..\/chapter\/absolute-konvergenz#x1-194002r2\">7.2<\/a>). <\/p><p class=\"indent\">Wenden wir dasselbe Argument wie oben auf 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>1<\/mn><\/mrow><mrow><mi>\u221e<\/mi><\/mrow><\/msubsup><mo class=\"MathClass-rel\">|<\/mo><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>\u03c6<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>n<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow><\/msub><mo class=\"MathClass-rel\">|<\/mo><\/math> an, ergibt sich auch die absolute Konvergenz von <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><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>\u03c6<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>n<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow><\/msub><\/math><\/span><span class=\"period\">.<\/span> <span>&nbsp;&nbsp;<\/span><\/p><div class=\"qed\">\u25a0<\/div><\/details><\/div> <a id=\"x1-194003r194\"><\/a> <h4 id=\"z130ee20529f2\" class=\"subsectionHead\"><span class=\"titlemark\">7.2.3 <\/span> <a id=\"x1-1950003\"><\/a>Produkte<\/h4> <p class=\"noindent\">Wir zeigen nun, dass wir absolut konvergente Reihen gliedweise ausmultiplizieren k\u00f6nnen. <\/p> <div class=\"me metheorem\"> <p class=\"indent\"><\/p><h4 id=\"z1e1953f850fc\"> <a id=\"x1-195001r36\"><\/a> <span class=\"ecbx-1095\">Satz 7.36 <\/span>(Produktsatz)<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>1<\/mn><\/mrow><mrow><mi>\u221e<\/mi><\/mrow><\/msubsup><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><\/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>1<\/mn><\/mrow><mrow><mi>\u221e<\/mi><\/mrow><\/msubsup><msub><mrow><mi>b<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><\/math> <span class=\"ecti-1095\">zwei absolut<\/span> <span class=\"ecti-1095\">konvergente Reihen und <\/span><math display=\"inline\"><mi>\u03c6<\/mi> <mo class=\"MathClass-punc\">:<\/mo> <mi>\u2115<\/mi> <mo class=\"MathClass-rel\">\u2192<\/mo> <mi>\u2115<\/mi> <mo class=\"MathClass-bin\">\u00d7<\/mo> <mi>\u2115<\/mi><\/math> <span class=\"ecti-1095\">eine bijektive Abbildung. Dann ist<\/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><msub><mrow><mi>a<\/mi><\/mrow><mrow> <mi>\u03c6<\/mi><msub><mrow><mo class=\"MathClass-open\">(<\/mo><mi>n<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><\/mrow><\/msub><msub><mrow><mi>b<\/mi><\/mrow><mrow><mi>\u03c6<\/mi><msub><mrow><mo class=\"MathClass-open\">(<\/mo><mi>n<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><\/mrow><\/msub><\/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 absolut konvergente Reihe, wobei <\/span><math display=\"inline\"><mi>\u03c6<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>n<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <mo class=\"MathClass-open\">(<\/mo><mi>\u03c6<\/mi><msub><mrow><mo class=\"MathClass-open\">(<\/mo><mi>n<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><mo class=\"MathClass-punc\">,<\/mo><mi>\u03c6<\/mi><msub><mrow><mo class=\"MathClass-open\">(<\/mo><mi>n<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><\/math> <span class=\"ecti-1095\">f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r alle <\/span><span class=\"maperiod\"><math display=\"inline\"><mi>n<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2115<\/mi><\/math><\/span><span class=\"period\">.<\/span> <span class=\"ecti-1095\">Weiters gilt<\/span> <\/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>1<\/mn><\/mrow><mrow><mi>\u221e<\/mi><\/mrow><\/munderover><msub><mrow><mi>a<\/mi><\/mrow><mrow> <mi>\u03c6<\/mi><msub><mrow><mo class=\"MathClass-open\">(<\/mo><mi>n<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><\/mrow><\/msub><msub><mrow><mi>b<\/mi><\/mrow><mrow><mi>\u03c6<\/mi><msub><mrow><mo class=\"MathClass-open\">(<\/mo><mi>n<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><\/mrow><\/msub> <mo class=\"MathClass-rel\">=<\/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>n<\/mi><mo class=\"MathClass-rel\">=<\/mo><mn>1<\/mn><\/mrow><mrow><mi>\u221e<\/mi><\/mrow><\/munderover><msub><mrow><mi>a<\/mi><\/mrow><mrow> <mi>n<\/mi><\/mrow><\/msub><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>1<\/mn><\/mrow><mrow><mi>\u221e<\/mi><\/mrow><\/munderover><msub><mrow><mi>b<\/mi><\/mrow><mrow> <mi>n<\/mi><\/mrow><\/msub><mstyle><mrow><mo fence=\"true\" form=\"prefix\"> )<\/mo><mrow \/><mo fence=\"true\" form=\"postfix\" \/><\/mrow><\/mstyle><mo class=\"MathClass-punc\">.<\/mo><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"><mstyle class=\"label\" id=\"x1-195002r3\" \/><mstyle class=\"maketag\"><mtext>(7.3)<\/mtext><\/mstyle><mspace class=\"nbsp\" width=\"0.33em\" \/> <\/mtd><\/mtr><\/mtable><\/math> <\/div> <p class=\"indent\">Informell ausgedr\u00fcckt kann man schreiben <\/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><munderover accent=\"false\" accentunder=\"false\"><mrow><mo> \u2211<\/mo> <\/mrow><mrow><mi>m<\/mi><mo class=\"MathClass-rel\">=<\/mo><mn>1<\/mn><\/mrow><mrow><mi>\u221e<\/mi><\/mrow><\/munderover><msub><mrow><mi>a<\/mi><\/mrow><mrow> <mi>m<\/mi><\/mrow><\/msub><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>1<\/mn><\/mrow><mrow><mi>\u221e<\/mi><\/mrow><\/munderover><msub><mrow><mi>b<\/mi><\/mrow><mrow> <mi>n<\/mi><\/mrow><\/msub><mstyle><mrow><mo fence=\"true\" form=\"prefix\"> )<\/mo><mrow \/><mo fence=\"true\" form=\"postfix\" \/><\/mrow><\/mstyle> <mo class=\"MathClass-rel\">=<\/mo><munderover accent=\"false\" accentunder=\"false\"><mrow><mo> \u2211<\/mo> <\/mrow><mrow><mi>m<\/mi><mo class=\"MathClass-rel\">=<\/mo><mn>1<\/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>n<\/mi><mo class=\"MathClass-rel\">=<\/mo><mn>1<\/mn><\/mrow><mrow><mi>\u221e<\/mi><\/mrow><\/munderover><msub><mrow><mi>b<\/mi><\/mrow><mrow> <mi>n<\/mi><\/mrow><\/msub><mstyle><mrow><mo fence=\"true\" form=\"prefix\"> )<\/mo><mrow \/><mo fence=\"true\" form=\"postfix\" \/><\/mrow><\/mstyle><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>m<\/mi><\/mrow><\/msub> <mo class=\"MathClass-rel\">=<\/mo><munder class=\"msub\"><mrow><mo> \u2211<\/mo> <\/mrow><mrow><mo class=\"MathClass-open\">(<\/mo><mi>m<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>n<\/mi><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-rel\">\u2208<\/mo><msup><mrow><mi>\u2115<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><\/mrow><\/munder><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>m<\/mi><\/mrow><\/msub><msub><mrow><mi>b<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><mo class=\"MathClass-punc\">.<\/mo><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">Die beiden (internen) Indices <math display=\"inline\"><mo class=\"MathClass-open\">(<\/mo><mi>m<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>n<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math> w\u00fcrde man nun gerne anders ausdr\u00fccken, damit aus der Doppelsumme auf der rechten Seite (die wir eigentlich nicht definiert haben) eine einfache Summe wird. W\u00e4hlt man eine Bijektion <span class=\"maperiod\"><math display=\"inline\"><mi>\u03c6<\/mi> <mo class=\"MathClass-punc\">:<\/mo> <mi>\u2115<\/mi> <mo class=\"MathClass-rel\">\u2192<\/mo> <mi>\u2115<\/mi> <mo class=\"MathClass-bin\">\u00d7<\/mo> <mi>\u2115<\/mi><\/math><\/span><span class=\"period\">,<\/span> so durchl\u00e4uft <math display=\"inline\"><mi>\u03c6<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>k<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math> alle <math display=\"inline\"><mo class=\"MathClass-open\">(<\/mo><mi>m<\/mi><mo class=\"MathClass-punc\">,<\/mo> <mi>n<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math> und somit wird aus der Doppelsumme eine einfache Summe <span class=\"maperiod\"><math display=\"inline\"><msubsup><mrow><mi class=\"MathClass-op\">\u2211<\/mi><mo> <\/mo> <\/mrow><mrow><mi>k<\/mi><mo class=\"MathClass-rel\">=<\/mo><mn>1<\/mn><\/mrow><mrow><mi>\u221e<\/mi><\/mrow><\/msubsup><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>\u03c6<\/mi><msub><mrow><mo class=\"MathClass-open\">(<\/mo><mi>k<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><\/mrow><\/msub><msub><mrow><mi>b<\/mi><\/mrow><mrow><mi>\u03c6<\/mi><msub><mrow><mo class=\"MathClass-open\">(<\/mo><mi>k<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><\/mrow><\/msub><\/math><\/span><span class=\"period\">.<\/span> Satz <a href=\"..\/..\/chapter\/absolute-konvergenz#x1-195001r36\">7.36<\/a> besagt nun, dass diese Reihe effektiv konvergiert und gleich dem gew\u00fcnschten Produkt ist. <\/p><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 w\u00e4hlen zuerst die Bijektion <math display=\"inline\"><mi>\u03c6<\/mi> <mo class=\"MathClass-punc\">:<\/mo> <mi>\u2115<\/mi> <mo class=\"MathClass-rel\">\u2192<\/mo> <msup><mrow><mi>\u2115<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><\/math> so dass                                                                                                                                                                           <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mo class=\"MathClass-open\">{<\/mo><mi>\u03c6<\/mi><mo class=\"MathClass-open\">(<\/mo><mn>1<\/mn><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-punc\">,<\/mo><mi>\u03c6<\/mi><mo class=\"MathClass-open\">(<\/mo><mn>2<\/mn><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-punc\">,<\/mo><mi class=\"MathClass-op\">\u2026<\/mi><mo> <\/mo><mo class=\"MathClass-punc\">,<\/mo><mi>\u03c6<\/mi><mo class=\"MathClass-open\">(<\/mo><msup><mrow><mi>n<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-close\">}<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <mo class=\"MathClass-open\">{<\/mo><mn>1<\/mn><mo class=\"MathClass-punc\">,<\/mo><mn>2<\/mn><mo class=\"MathClass-punc\">,<\/mo><mi class=\"MathClass-op\">\u2026<\/mi><mo> <\/mo><mo class=\"MathClass-punc\">,<\/mo><mi>n<\/mi><mo class=\"MathClass-close\">}<\/mo><mo class=\"MathClass-bin\">\u00d7<\/mo><mo class=\"MathClass-open\">{<\/mo><mn>1<\/mn><mo class=\"MathClass-punc\">,<\/mo><mn>2<\/mn><mo class=\"MathClass-punc\">,<\/mo><mi class=\"MathClass-op\">\u2026<\/mi><mo> <\/mo><mo class=\"MathClass-punc\">,<\/mo><mi>n<\/mi><mo class=\"MathClass-close\">}<\/mo><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">f\u00fcr alle <span class=\"maperiod\"><math display=\"inline\"><mi>n<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2115<\/mi><\/math><\/span><span class=\"period\">.<\/span> Zum Beispiel k\u00f6nnte <math display=\"inline\"><mi>\u03c6<\/mi><\/math> wie im folgenden Bild definiert sein.<button class=\"hover-trigger\" style=\"vertical-align: super;font: smaller\">\u2020<\/button><span class=\"hover-text\"><span class=\"marginpar\">\u2020 Wir verwenden hier das Bild um uns eine sonst eher langweilige formale Definition der Abbildung <math display=\"inline\"><mi>\u03c6<\/mi><\/math> zu ersparen. Sie sollten sich aber davon \u00fcberzeugen, dass man diese Definition durchaus formal machen kann oder einfach formal mittels Induktion nach <math display=\"inline\"><mi>n<\/mi><\/math> die Existenz einer Bijektion mit der gew\u00fcnschten Eigenschaft zeigen kann. Wir verwenden hier also das Bild als eine Abk\u00fcrzung f\u00fcr einen formalen Beweis und nicht als einen Ersatz.<\/span><\/span> <\/p> <div class=\"center\"> <p class=\"noindent\"> <\/p><p class=\"noindent\"><\/p><div class=\"mefigcentered\" id=\"wpsize=326&amp;url=Pictures\/reihen\/prodsatz2.pdf\"><img id=\"zfc9887eb0071\" alt=\"PIC\" src=\"https:\/\/people.math.ethz.ch\/~einsiedl\/Pictures\/reihen\/prodsatz2.svg\" width=\"326\"><\/div>  <\/div> <p class=\"indent\">F\u00fcr jedes <math display=\"inline\"><mi>n<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2115<\/mi><\/math> gilt dann f\u00fcr die Partialsumme bis <math display=\"inline\"><msup><mrow><mi>n<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><\/math> der gliedweise multiplizierten Reihe der Absolutbetr\u00e4ge das (endliche verallgemeinerte) Distributivgesetz <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><munderover accent=\"false\" accentunder=\"false\"><mrow><mo> \u2211<\/mo> <\/mrow><mrow><mi>k<\/mi><mo class=\"MathClass-rel\">=<\/mo><mn>1<\/mn><\/mrow><mrow><msup><mrow><mi>n<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup> <\/mrow><\/munderover><mo class=\"MathClass-rel\">|<\/mo><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>\u03c6<\/mi><msub><mrow><mo class=\"MathClass-open\">(<\/mo><mi>k<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><\/mrow><\/msub><mo class=\"MathClass-rel\">|<\/mo><mo class=\"MathClass-rel\">|<\/mo><msub><mrow><mi>b<\/mi><\/mrow><mrow><mi>\u03c6<\/mi><msub><mrow><mo class=\"MathClass-open\">(<\/mo><mi>k<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><\/mrow><\/msub><mo class=\"MathClass-rel\">|<\/mo> <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>\u2113<\/mi><mo class=\"MathClass-rel\">=<\/mo><mn>1<\/mn><\/mrow><mrow><mi>n<\/mi><\/mrow><\/munderover><mo class=\"MathClass-rel\">|<\/mo><msub><mrow><mi>a<\/mi><\/mrow><mrow> <mi>\u2113<\/mi><\/mrow><\/msub><mo class=\"MathClass-rel\">|<\/mo><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><munderover accent=\"false\" accentunder=\"false\"><mrow><mo>\u2211<\/mo> <\/mrow><mrow><mi>m<\/mi><mo class=\"MathClass-rel\">=<\/mo><mn>1<\/mn><\/mrow><mrow><mi>n<\/mi><\/mrow><\/munderover><mo class=\"MathClass-rel\">|<\/mo><msub><mrow><mi>b<\/mi><\/mrow><mrow> <mi>m<\/mi><\/mrow><\/msub><mo class=\"MathClass-rel\">|<\/mo><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><mo class=\"MathClass-punc\">.<\/mo><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">Insbesondere folgt also                                                                                                                                                                           <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><munderover accent=\"false\" accentunder=\"false\"><mrow><mo> \u2211<\/mo> <\/mrow><mrow><mi>k<\/mi><mo class=\"MathClass-rel\">=<\/mo><mn>1<\/mn><\/mrow><mrow><msup><mrow><mi>n<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup> <\/mrow><\/munderover><mo class=\"MathClass-rel\">|<\/mo><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>\u03c6<\/mi><msub><mrow><mo class=\"MathClass-open\">(<\/mo><mi>k<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><\/mrow><\/msub><mo class=\"MathClass-rel\">|<\/mo><mo class=\"MathClass-rel\">|<\/mo><msub><mrow><mi>b<\/mi><\/mrow><mrow><mi>\u03c6<\/mi><msub><mrow><mo class=\"MathClass-open\">(<\/mo><mi>k<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><\/mrow><\/msub><mo class=\"MathClass-rel\">|<\/mo><mo class=\"MathClass-rel\">\u2264<\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><munderover accent=\"false\" accentunder=\"false\"><mrow><mo>\u2211<\/mo> <\/mrow><mrow><mi>\u2113<\/mi><mo class=\"MathClass-rel\">=<\/mo><mn>1<\/mn><\/mrow><mrow><mi>\u221e<\/mi><\/mrow><\/munderover><mo class=\"MathClass-rel\">|<\/mo><msub><mrow><mi>a<\/mi><\/mrow><mrow> <mi>\u2113<\/mi><\/mrow><\/msub><mo class=\"MathClass-rel\">|<\/mo><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><munderover accent=\"false\" accentunder=\"false\"><mrow><mo>\u2211<\/mo> <\/mrow><mrow><mi>m<\/mi><mo class=\"MathClass-rel\">=<\/mo><mn>1<\/mn><\/mrow><mrow><mi>\u221e<\/mi><\/mrow><\/munderover><mo class=\"MathClass-rel\">|<\/mo><msub><mrow><mi>b<\/mi><\/mrow><mrow> <mi>m<\/mi><\/mrow><\/msub><mo class=\"MathClass-rel\">|<\/mo><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">f\u00fcr alle <span class=\"maperiod\"><math display=\"inline\"><mi>n<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2115<\/mi><\/math><\/span><span class=\"period\">.<\/span> Da aber f\u00fcr eine Reihe mit nicht-negativen Termen die Folge die Partialsummen monoton wachsend sind, folgt daraus dass die Reihe <math display=\"inline\"><msubsup><mrow><mi class=\"MathClass-op\"> \u2211<\/mi><mo> <\/mo> <\/mrow><mrow><mi>k<\/mi><mo class=\"MathClass-rel\">=<\/mo><mn>1<\/mn><\/mrow><mrow><mi>\u221e<\/mi><\/mrow><\/msubsup><mo class=\"MathClass-rel\">|<\/mo><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>\u03c6<\/mi><msub><mrow><mo class=\"MathClass-open\">(<\/mo><mi>k<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><\/mrow><\/msub><mo class=\"MathClass-rel\">|<\/mo><mo class=\"MathClass-rel\">|<\/mo><msub><mrow><mi>b<\/mi><\/mrow><mrow><mi>\u03c6<\/mi><msub><mrow><mo class=\"MathClass-open\">(<\/mo><mi>k<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><\/mrow><\/msub><mo class=\"MathClass-rel\">|<\/mo><\/math> konvergiert und damit die Reihe <math display=\"inline\"><msubsup><mrow><mi class=\"MathClass-op\"> \u2211<\/mi><mo> <\/mo> <\/mrow><mrow><mi>k<\/mi><mo class=\"MathClass-rel\">=<\/mo><mn>1<\/mn><\/mrow><mrow><mi>\u221e<\/mi><\/mrow><\/msubsup><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>\u03c6<\/mi><msub><mrow><mo class=\"MathClass-open\">(<\/mo><mi>k<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><\/mrow><\/msub><msub><mrow><mi>b<\/mi><\/mrow><mrow><mi>\u03c6<\/mi><msub><mrow><mo class=\"MathClass-open\">(<\/mo><mi>k<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><\/mrow><\/msub><\/math> absolut konvergent ist. <\/p><p class=\"indent\">Obiges Distributivgesetz gilt auch in der Form <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><munderover accent=\"false\" accentunder=\"false\"><mrow><mo> \u2211<\/mo> <\/mrow><mrow><mi>k<\/mi><mo class=\"MathClass-rel\">=<\/mo><mn>1<\/mn><\/mrow><mrow><msup><mrow><mi>n<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup> <\/mrow><\/munderover><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>\u03c6<\/mi><msub><mrow><mo class=\"MathClass-open\">(<\/mo><mi>k<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><\/mrow><\/msub><msub><mrow><mi>b<\/mi><\/mrow><mrow><mi>\u03c6<\/mi><msub><mrow><mo class=\"MathClass-open\">(<\/mo><mi>k<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><\/mrow><\/msub> <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>\u2113<\/mi><mo class=\"MathClass-rel\">=<\/mo><mn>1<\/mn><\/mrow><mrow><mi>n<\/mi><\/mrow><\/munderover><msub><mrow><mi>a<\/mi><\/mrow><mrow> <mi>\u2113<\/mi><\/mrow><\/msub><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><munderover accent=\"false\" accentunder=\"false\"><mrow><mo>\u2211<\/mo> <\/mrow><mrow><mi>m<\/mi><mo class=\"MathClass-rel\">=<\/mo><mn>1<\/mn><\/mrow><mrow><mi>n<\/mi><\/mrow><\/munderover><msub><mrow><mi>b<\/mi><\/mrow><mrow> <mi>m<\/mi><\/mrow><\/msub><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-punc\">.<\/mo><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">f\u00fcr alle <span class=\"maperiod\"><math display=\"inline\"><mi>n<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2115<\/mi><\/math><\/span><span class=\"period\">.<\/span> Mit Hilfe des Grenzwert\u00fcbergangs <math display=\"inline\"><mi>n<\/mi> <mo class=\"MathClass-rel\">\u2192<\/mo><mi>\u221e<\/mi><\/math> erhalten wir daraus                                                                                                                                                                           <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><munderover accent=\"false\" accentunder=\"false\"><mrow><mo> \u2211<\/mo> <\/mrow><mrow><mi>k<\/mi><mo class=\"MathClass-rel\">=<\/mo><mn>1<\/mn><\/mrow><mrow><mi>\u221e<\/mi><\/mrow><\/munderover><msub><mrow><mi>a<\/mi><\/mrow><mrow> <mi>\u03c6<\/mi><msub><mrow><mo class=\"MathClass-open\">(<\/mo><mi>k<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><\/mrow><\/msub><msub><mrow><mi>b<\/mi><\/mrow><mrow><mi>\u03c6<\/mi><msub><mrow><mo class=\"MathClass-open\">(<\/mo><mi>k<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><\/mrow><\/msub> <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>\u2113<\/mi><mo class=\"MathClass-rel\">=<\/mo><mn>1<\/mn><\/mrow><mrow><mi>\u221e<\/mi><\/mrow><\/munderover><msub><mrow><mi>a<\/mi><\/mrow><mrow> <mi>\u2113<\/mi><\/mrow><\/msub><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><munderover accent=\"false\" accentunder=\"false\"><mrow><mo>\u2211<\/mo> <\/mrow><mrow><mi>m<\/mi><mo class=\"MathClass-rel\">=<\/mo><mn>1<\/mn><\/mrow><mrow><mi>\u221e<\/mi><\/mrow><\/munderover><msub><mrow><mi>b<\/mi><\/mrow><mrow> <mi>m<\/mi><\/mrow><\/msub><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-punc\">.<\/mo><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">Betrachten wir eine beliebige Bijektion <span class=\"maperiod\"><math display=\"inline\"><mi>\u03c8<\/mi> <mo class=\"MathClass-punc\">:<\/mo> <mi>\u2115<\/mi> <mo class=\"MathClass-rel\">\u2192<\/mo> <msup><mrow><mi>\u2115<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><\/math><\/span><span class=\"period\">,<\/span> so ist <math display=\"inline\"><msup><mrow><mi>\u03c6<\/mi><\/mrow><mrow><mo class=\"MathClass-bin\">\u2212<\/mo><mn>1<\/mn> <\/mrow> <\/msup> <mo class=\"MathClass-bin\">\u2218<\/mo> <mi>\u03c8<\/mi> <mo class=\"MathClass-punc\">:<\/mo> <mi>\u2115<\/mi> <mo class=\"MathClass-rel\">\u2192<\/mo> <mi>\u2115<\/mi><\/math> eine Bijektion und die Formel <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><munderover accent=\"false\" accentunder=\"false\"><mrow><mo> \u2211<\/mo> <\/mrow><mrow><mi>k<\/mi><mo class=\"MathClass-rel\">=<\/mo><mn>1<\/mn><\/mrow><mrow><mi>\u221e<\/mi><\/mrow><\/munderover><msub><mrow><mi>a<\/mi><\/mrow><mrow> <mi>\u03c8<\/mi><msub><mrow><mo class=\"MathClass-open\">(<\/mo><mi>k<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><\/mrow><\/msub><msub><mrow><mi>b<\/mi><\/mrow><mrow><mi>\u03c8<\/mi><msub><mrow><mo class=\"MathClass-open\">(<\/mo><mi>k<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><\/mrow><\/msub> <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>\u2113<\/mi><mo class=\"MathClass-rel\">=<\/mo><mn>1<\/mn><\/mrow><mrow><mi>\u221e<\/mi><\/mrow><\/munderover><msub><mrow><mi>a<\/mi><\/mrow><mrow> <mi>\u2113<\/mi><\/mrow><\/msub><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><munderover accent=\"false\" accentunder=\"false\"><mrow><mo>\u2211<\/mo> <\/mrow><mrow><mi>m<\/mi><mo class=\"MathClass-rel\">=<\/mo><mn>1<\/mn><\/mrow><mrow><mi>\u221e<\/mi><\/mrow><\/munderover><msub><mrow><mi>b<\/mi><\/mrow><mrow> <mi>m<\/mi><\/mrow><\/msub><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-punc\">.<\/mo><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">folgt aus obigem und dem Umordnungssatz (Satz <a href=\"..\/..\/chapter\/absolute-konvergenz#x1-194001r35\">7.35<\/a>). <span>&nbsp;&nbsp;<\/span><\/p><div class=\"qed\">\u25a0<\/div><\/details><\/div> <p class=\"indent\">Wie wir in Abschnitt <a href=\"..\/..\/chapter\/polynome#x1-810002\">3.2<\/a> gesehen haben, lassen sich Polynome mittels der Regel <\/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><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>a<\/mi><\/mrow><mrow> <mi>n<\/mi><\/mrow><\/msub><msup><mrow><mi>x<\/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>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><mstyle><mrow><mo fence=\"true\" form=\"prefix\"> )<\/mo><mrow \/><mo fence=\"true\" form=\"postfix\" \/><\/mrow><\/mstyle> <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><mn>2<\/mn><mi>N<\/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>x<\/mi><\/mrow><mrow><mi>k<\/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\">multiplizieren, wobei wir&nbsp;<math display=\"inline\"><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> <mo class=\"MathClass-rel\">\u22ef<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <msub><mrow><mi>a<\/mi><\/mrow><mrow><mn>2<\/mn><mi>N<\/mi><\/mrow><\/msub> <mo class=\"MathClass-rel\">=<\/mo> <msub><mrow><mi>b<\/mi><\/mrow><mrow><mi>N<\/mi><mo class=\"MathClass-bin\">+<\/mo><mn>1<\/mn><\/mrow><\/msub> <mo class=\"MathClass-rel\">=<\/mo> <mo class=\"MathClass-rel\">\u22ef<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <msub><mrow><mi>b<\/mi><\/mrow><mrow><mn>2<\/mn><mi>N<\/mi><\/mrow><\/msub> <mo class=\"MathClass-rel\">=<\/mo> <mn>0<\/mn><\/math> setzen. Setzt man <span class=\"maperiod\"><math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mn>1<\/mn><\/math><\/span><span class=\"period\">,<\/span> erh\u00e4lt man insbesondere                                                                                                                                                                           <\/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><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>a<\/mi><\/mrow><mrow> <mi>n<\/mi><\/mrow><\/msub><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>N<\/mi><\/mrow><\/munderover><msub><mrow><mi>b<\/mi><\/mrow><mrow> <mi>n<\/mi><\/mrow><\/msub><mstyle><mrow><mo fence=\"true\" form=\"prefix\"> )<\/mo><mrow \/><mo fence=\"true\" form=\"postfix\" \/><\/mrow><\/mstyle> <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><mn>2<\/mn><mi>N<\/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><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\">Diese Identit\u00e4t trifft, wie sich herausstellt, analog f\u00fcr Reihen zu, was wir im folgenden Korollar des Produktsatzes (Satz <a href=\"..\/..\/chapter\/absolute-konvergenz#x1-195001r36\">7.36<\/a>) festhalten wollen. <\/p> <div class=\"me metheorem\"> <p class=\"indent\"><\/p><h4 id=\"z0a721819d754\"> <a id=\"x1-195003r37\"><\/a> <span class=\"ecbx-1095\">Korollar 7.37 <\/span>(Cauchy-Produkt)<span class=\"ecbx-1095\">.<\/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><\/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><\/math> <span class=\"ecti-1095\">absolut konvergente Reihen mit komplexen Gliedern sind, dann gilt<\/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><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> <mo class=\"MathClass-rel\">=<\/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>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><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><mstyle><mrow><mo fence=\"true\" form=\"prefix\"> )<\/mo><mrow \/><mo fence=\"true\" form=\"postfix\" \/><\/mrow><\/mstyle><mo class=\"MathClass-punc\">,<\/mo><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\"><span class=\"ecti-1095\">wobei 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><mstyle><mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow \/><mo fence=\"true\" form=\"postfix\" \/><\/mrow><\/mstyle><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>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><\/math> <span class=\"ecti-1095\">absolut konvergent ist.<\/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\">Dies folgt, indem wir die Abz\u00e4hlung <math display=\"inline\"><mi>\u03c6<\/mi><\/math> von <math display=\"inline\"><msub><mrow><mi>\u2115<\/mi><\/mrow><mrow><mn>0<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-bin\">\u00d7<\/mo> <msub><mrow><mi>\u2115<\/mi><\/mrow><mrow><mn>0<\/mn> <\/mrow> <\/msub> <\/math> aus dem Bild unten auf den Produktsatz (Satz <a href=\"..\/..\/chapter\/absolute-konvergenz#x1-195001r36\">7.36<\/a>) anwenden und dann Glieder zusammenfassen (Lemma <a href=\"..\/..\/chapter\/reihen#x1-187013r9\">7.9<\/a>). <\/p> <div class=\"center\"> <p class=\"noindent\"> <\/p><p class=\"noindent\"><\/p><div class=\"mefigcentered\" id=\"wpsize=320&amp;url=Pictures\/reihen\/cauchyprodv2.pdf\"><img id=\"zf598d69fa248\" alt=\"PIC\" src=\"https:\/\/people.math.ethz.ch\/~einsiedl\/Pictures\/reihen\/cauchyprodv2.svg\" width=\"320\"><\/div>  <\/div> <p class=\"noindent\">Die absolute Konvergenz folgt ebenso aus Satz&nbsp;<a href=\"..\/..\/chapter\/absolute-konvergenz#x1-195001r36\">7.36<\/a> und <\/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><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> <mo class=\"MathClass-rel\">\u2264<\/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><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><mo class=\"MathClass-rel\">|<\/mo><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><mo class=\"MathClass-rel\">|<\/mo> <mo class=\"MathClass-rel\">&lt;<\/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> <span>&nbsp;&nbsp;<\/span><div class=\"qed\">\u25a0<\/div><\/details><\/div> <div class=\"me meexample\"> <p class=\"indent\"><\/p><h4 id=\"zf2b8b908320c\"> <a id=\"x1-195004r38\"><\/a> <span class=\"ecbx-1095\">Beispiel 7.38.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Sei <\/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\">. Dann<\/span> <span class=\"ecti-1095\">konvergiert <\/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\">absolut. Wenden wir das Cauchy-Produkt auf diese Reihe und sich selbst an, so erhalten<\/span> <span class=\"ecti-1095\">wir<\/span> <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"> <mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><msup><mrow><mo class=\"MathClass-open\">(<\/mo><mn>1<\/mn> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>q<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><\/mrow><\/mfrac><\/mtd> <mtd class=\"align-even\"> <mo class=\"MathClass-rel\">=<\/mo><msup><mrow> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/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><msup><mrow><mi>q<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msup><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <\/mrow><mrow><mn>2<\/mn><\/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><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><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>q<\/mi><\/mrow><mrow><mi>n<\/mi><mo class=\"MathClass-bin\">\u2212<\/mo><mi>k<\/mi><\/mrow><\/msup><msup><mrow><mi>q<\/mi><\/mrow><mrow><mi>k<\/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><mi>n<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mn>1<\/mn><mo class=\"MathClass-close\">)<\/mo><msup><mrow><mi>q<\/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\">Auf diese Weise erhalten wir auch eine Summenformel f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r<\/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><mi>n<\/mi><msup><mrow><mi>q<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msup> <mo class=\"MathClass-rel\">=<\/mo> <mi>q<\/mi><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><mi>n<\/mi><msup><mrow><mi>q<\/mi><\/mrow><mrow><mi>n<\/mi><mo class=\"MathClass-bin\">\u2212<\/mo><mn>1<\/mn><\/mrow><\/msup> <mo class=\"MathClass-rel\">=<\/mo> <mi>q<\/mi><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><mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>k<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mn>1<\/mn><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><msup><mrow><mi>q<\/mi><\/mrow><mrow><mi>k<\/mi><\/mrow><\/msup> <mo class=\"MathClass-rel\">=<\/mo> <mfrac><mrow><mi>q<\/mi><\/mrow> <mrow><msup><mrow><mo class=\"MathClass-open\">(<\/mo><mn>1<\/mn> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>q<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><\/mrow><\/mfrac><mo class=\"MathClass-punc\">,<\/mo><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\"><span class=\"ecti-1095\">wobei wir die Indexverschiebung <\/span><math display=\"inline\"><mi>k<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mi>n<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <mn>1<\/mn><\/math> <span class=\"ecti-1095\">durchgef<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">hrt haben.<\/span> <\/p> <\/div> <div class=\"me meexample\"> <p class=\"indent\"><\/p><h4 id=\"z575fdb60534c\"> <a id=\"x1-195005r39\"><\/a> <span class=\"ecbx-1095\">\u00dc<\/span><span class=\"ecbx-1095\">bung 7.39.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Formal l<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">sst sich auch f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r bedingt konvergente Reihen<\/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><\/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><\/math> <span class=\"ecti-1095\">das<\/span> <span class=\"ecti-1095\">Cauchy-Produkt <\/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><mstyle><mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow \/><mo fence=\"true\" form=\"postfix\" \/><\/mrow><\/mstyle><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>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><\/math> <span class=\"ecti-1095\">bilden. Es muss jedoch nicht mehr konvergent sein: Zeigen Sie, dass das Cauchy-Produkt der bedingt<\/span> <span class=\"ecti-1095\">konvergenten Reihe<\/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><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><msqrt><mrow><mi>n<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mn>1<\/mn><\/mrow><\/msqrt><\/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\">mit sich selbst divergiert.<\/span> <\/p> <\/div> <a id=\"x1-195006r192\"><\/a> \n","rendered":"\n<style scoped=\"scoped\">.cmr-5{font-size:50%;}\n.cmr-7{font-size:70%;}\n.cmmi-5{font-size:50%;font-style: italic;}\n.cmmi-7{font-size:70%;font-style: italic;}\n.cmmi-10{font-style: italic;}\n.cmsy-5{font-size:50%;}\n.cmsy-7{font-size:70%;}\n.cmbx-10{ font-weight: bold;}\n.cmbsy-10{font-weight: bold;}\n.cmbsy-10{font-weight: bold;}\n.cmbsy-10{font-weight: bold;}\n.cmbsy-7{font-size:70%;font-weight: bold;}\n.cmbsy-7{font-weight: bold;}\n.cmbsy-7{font-weight: bold;}\n.cmbsy-5{font-size:50%;font-weight: bold;}\n.cmbsy-5{font-weight: bold;}\n.cmbsy-5{font-weight: bold;}\n.cmex-7{font-size:70%;}\n.cmex-7x-x-71{font-size:49%;}\n.msam-7{font-size:70%;}\n.msam-5{font-size:50%;}\n.msbm-7{font-size:70%;}\n.msbm-5{font-size:50%;}\n.cmr-17{font-size:170%;}\n.cmr-12{font-size:120%;}\n.cmti-10{ font-style: italic;}\np{margin-top:0;margin-bottom:0}\np.indent{text-indent:0;}\np + p{margin-top:1em;}\np + div, p + pre {margin-top:1em;}\ndiv + p, pre + p {margin-top:1em;}\n@media print {div.crosslinks {visibility:hidden;}}\na img { border-top: 0; 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width:125%;}\ndt {text-align:right; font-weight:bold; clear:left; float:left;}\ndd {width:100%; padding-left:1em; padding-top: 0px; clear:right;}\ndd + dd {float:right; clear:both;}\ndd + dt {clear:both;}\ndt + dt {width: 100%; float: none; padding: 0 70% 0 0;}\ndt + dt + dd {margin-top: -2em;}\ndt + dt + dd + dt {margin-top: 2em;}\n<\/style>\n<style scoped=\"scoped\">\n\/* CSS Analysis-Skript D-Math ETHZ *\/\n\n\/* Uniform Font, also for headers *\/\nh3 {\n\tfont-family: \"Times New Roman\", serif;\n\tmargin-bottom: 35px;\n}\nh4 {\n\tfont-family: \"Times New Roman\", serif;\n}\nh5 {\n\tfont-family: \"Times New Roman\", serif;\n}\n\n\/* Bold font, e.g. for definitions *\/\n.ecbx-1095 {font-weight: 550 ;}\n\n\n\/* Uniform spacing, indent: larger, noindent, enumerate, itemize *\/\np.indent {\n\tmargin: 25px 0px 0px 0px;\n\ttext-indent: 0px; \n}\np.noindent {\n\tmargin: 15px 0px 0px 0px;\n\ttext-indent: 0px; \n}\ndl.enumerate {\n\tmargin: 0px 0px 0px 0px;\n}\ndl.enumerate dt, dl.enumerate dd {\n\tmargin-top: 15px;\n\tmargin-bottom: 0px;\n}\ndiv.custom-itemize {\n\tmargin: 0px 0px 0px 0px;\n}\ndiv.custom-itemize div.item-head {\n\tmargin-top: 15px;\n\tmargin-bottom: 0px;\n\ttext-align: center;\n}\ndiv.custom-itemize div.item-head:first-of-type {\n\tmargin-top: 0px;\n} \ndiv.custom-itemize div.item-content {\n\tmargin-top: 15px;\n\tmargin-bottom: 0px;\n}\n.MJXc-display {\n\tmargin: 15px 0px 0px 0px;\n}\n\n\n\n\/* green metheorem\/melemma CSS class for more\/medium important latex-theorem-environments *\/\n\/* metheorem box+header *\/\ndiv.metheorem {\n    margin-bottom: 40px;\n    margin-top: 40px;\n\tpadding: 0px 15px 15px 15px;\n    border: 1px solid #333;\n    border-color: #4eb79e;\n    background: #c7e4da;\n}\ndiv.metheorem h4 {\n    background: #4eb79e;\n    color: white;\n\tmargin-top: 12px;\n\tmargin-left: -15px;\n\tmargin-right: -15px;\n\tpadding: 0px 15px 0px 15px;\n}\n\/* melemma box+header *\/\ndiv.melemma {\n    margin-bottom: 40px;\n    margin-top: 40px;\n\tpadding: 0px 15px 15px 15px;\n    border: 1px solid #333;\n    border-color: #4eb79e;\n    background: #F2F2F2;\n}\ndiv.melemma h4 {\n    background: #4eb79e;\n    color: white;\n\tmargin-top: 12px;\n\tmargin-left: -15px;\n\tmargin-right: -15px;\n\tpadding: 0px 15px 0px 15px;\n}\n\/* meexample box+header *\/\ndiv.meexample {\n    margin-bottom: 30px;\n    margin-top: 30px;\n\tpadding: 0px 15px 15px 15px;\n\tborder-color: gainsboro;\n\tborder-style: solid;\n\tborder-width: thin;\n}\ndiv.meexample h4 {\n\tfont-size: inherit;\n\tfont-weight: bold;\n    padding: 15px 0px 0px 0px;\n\tmargin-top: 0px;\n\tmargin-bottom: 5px;\n}\ndiv.meexample h4+p.noindent, div.meexample h4+p.indent {\n\tmargin-top: 5px;\n\ttext-indent: 0px;\n}\n\/* padding and margins for stuff inside these boxes, CSS-selector &gt; doesn't work in WP *\/\ndiv.me details {\n\tmargin: 10px 0px 0px 0px;\n}\ndiv.me dd {\n    width: calc(100% - 30px);\n}\t\n\n\n\/* fixing background of pictures *\/\nimg {\n\tbackground: white;\n}\n\n\/* div-container for centered geoapplet *\/\ndiv.geoapplet {\n\tmargin-left: auto;\n\tmargin-right: auto;\n\tmargin-top: 15px;\n\tmax-width: 100%;\n}\ndiv.geoapplet iframe {\n\tborder-style: none;\n\tmax-height: 110vw;\n}\n\n\/* div-container for centered squeezed tables *\/\ndiv.websqueeze {\n\tmargin-left: auto;\n\tmargin-right: auto;\n}\n\n\/* two containers for squeezing text sizes *\/\ndiv.mesmalltext, div.mesmalltext * {\n\tfont-size: 15px;\n}\nspan.metinytext, span.metinytext * {\n\tfont-size: 12px;\n}\n\n\n\/* removing grid lines in equations *\/\n#content table.equation tr td, #content table.equation tr th {\n    border: none;\n}\n#content table.equation {\n    border: none;\n}\n\n\/* hover\/click-solution for short inline explanations and footnotes *\/\n.hover-text {    \/* hidden part *\/\n    display: none;\n}\n.marginpar {     \/* style for footnote as marginpar *\/\n\ttext-decoration: none;\n\tborder: solid;\n\tborder-width: 1pt;\n\tpadding: 3pt;\t\n\twidth: 30%;\n\tbackground: white;\n}\n.hover-trigger { \/* style for hover\/click-trigger text\/symbol *\/\n\tbackground: none;\n\tborder: none;\n\tpadding: 0;\n\toutline: inherit;\t\n\ttext-transform: none;\n\tfont: inherit;\n\tposition: inherit;\n\tvertical-align: baseline;\n    color: #FF7F00;\n\tcursor: help;\n}\n.hover-trigger:hover +.hover-text{\n    display: inline;\n}\n.hover-trigger:active +.hover-text{\n    display: inline;\n}\n\n\/* simplifying style of details\/summary, removing triangle *\/\ndetails summary {\n  background: none;\n  list-style: none;\n  outline: none;\n  cursor: pointer;\n}\ndetails summary::-webkit-details-marker { \n  display: inline;\n  display: none;\n}\n\n\/* MC-True\/False as inline details\/summary *\/\ndetails.mcquest, div.me details.mcquest {\n\tdisplay: inline;\n\tmargin-top: 0px;\n}\nsummary.mcquest {\n\tdisplay: inline;\n\tcolor: #FF7F00;\n\tcursor: help;\n}\n\n\/* proof style: simple black box with gray background \n                little black square at the end on the right *\/\ndiv.proof {\n\tborder-color: black;\n\tborder-style: solid;\n\tborder-width: thin;\n\tbackground-color: #F2F2F2;\n\tpadding: 15px;\n\tmargin-top: 1em; \n}\ndiv.proof p:first-of-type {\n\tmargin: 0px;\n}\ndiv.qed {\n\tmargin-top: -25px;\n\tmargin-bottom: -7px;\n\ttext-align: right;\n}\ntable.equation+div.qed {\n\tmargin-top: -65px;\n}\n\n\/* The following is making also math-formulas inside the headers of Lemmas, etc., white. *\/\ndiv.melemma h4 span {\n    color: white;\n}\ndiv.metheorem h4 span {\n    color: white;\n}\n\n\/* The following are used to avoid fullstop, period, colon, semicolon, and endquote (broader) to move by itself to the next line after a formula.\n   The math-environment before needs to be wrapped in span.maperiod and the fullstop etc. in a span.period --- together they achieve what we want.  *\/\nspan.maperiod {\n       margin-right: 5px;\n}\nspan.period {\n       display: inline-block;\n       width: 0px;\n       margin-left: -5px;\n       margin-right: 4.9px;\n\t   text-indent: 0px;\n}\nspan.maendquote {\n       margin-right: 8px;\n}\nspan.endquote {\n       display: inline-block;\n       width: 0px;\n       margin-left: -8px;\n       margin-right: 7.9px;\n}\n\n\n\/* The following is removing an extra space left of the equation side in aligned equations *\/\nspan.mjx-mtd {\n    padding-left: 0em !important;\n}\n\n\/* The following fixes the weird problem that math appears smaller if it was rendered while the details tag was closed. *\/\ndetails span.mjx-chtml, details span.MathJax_CHTML {\n font-size: 100% !important;\n}\n\n\/* trying to fix line breaks in verbatim, new lines are missing *\/\npre.verbatim {\n\twhite-space: pre-wrap;\n\tfont-size: small;\n}\n<\/style><h3 id=\"z3583037af869\" class=\"sectionHead\"><span class=\"titlemark\">7.2 <\/span> <a id=\"x1-1920002\"><\/a>Absolute Konvergenz<\/h3> <p class=\"noindent\">In diesem Abschnitt wollen wir uns vor allem mit absolut konvergenten Reihen auseinandersetzen und einige Konvergenzkriterien beweisen. Auch m\u00f6chten wir zeigen, dass absolut konvergente Reihen im Gegensatz zu bedingt konvergenten Reihen stabilere Eigenschaften haben. <\/p> <div class=\"me metheorem\"> <div class=\"wp-nocaption \"><\/div><h4 id=\"z262506282f0e\"> <a id=\"x1-192001r28\"><\/a> <span class=\"ecbx-1095\">Proposition 7.28 <\/span>(Absolute Konvergenz)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Eine absolut konvergente 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>1<\/mn><\/mrow><mrow><mi>\u221e<\/mi><\/mrow><\/msubsup><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><\/math> <span class=\"ecti-1095\">ist auch konvergent und es gilt die verallgemeinerte Dreiecksungleichung<\/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><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><msub><mrow><mi>a<\/mi><\/mrow><mrow> <mi>n<\/mi><\/mrow><\/msub><mstyle><mrow><mo fence=\"true\" form=\"prefix\"> |<\/mo><mrow \/><mo fence=\"true\" form=\"postfix\" \/><\/mrow><\/mstyle> <mo class=\"MathClass-rel\">\u2264<\/mo><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><mo class=\"MathClass-rel\">|<\/mo><msub><mrow><mi>a<\/mi><\/mrow><mrow> <mi>n<\/mi><\/mrow><\/msub><mo class=\"MathClass-rel\">|<\/mo><mo class=\"MathClass-punc\">.<\/mo><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <\/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\">Der erste Teil folgt unmittelbar aus zweimaliger Anwendung des Cauchy-Kriteriums f\u00fcr Reihen (Satz <a href=\"..\/..\/chapter\/reihen#x1-191001r26\">7.26<\/a>): Da 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>1<\/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><\/math> konvergiert, gibt es f\u00fcr <math display=\"inline\"><mi>\ud835\udf00<\/mi> <mo class=\"MathClass-rel\">&gt;<\/mo> <mn>0<\/mn><\/math> nach dem Cauchy-Kriterium ein <span class=\"maperiod\"><math display=\"inline\"><mi>N<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2115<\/mi><\/math><\/span><span class=\"period\">,<\/span> so dass f\u00fcr <math display=\"inline\"><mi>n<\/mi> <mo class=\"MathClass-rel\">\u2265<\/mo> <mi>m<\/mi> <mo class=\"MathClass-rel\">\u2265<\/mo> <mi>N<\/mi><\/math> die Absch\u00e4tzung                                                                                                                                                                           <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><munderover accent=\"false\" accentunder=\"false\"><mrow><mo> \u2211<\/mo> <\/mrow><mrow><mi>k<\/mi><mo class=\"MathClass-rel\">=<\/mo><mi>m<\/mi><\/mrow><mrow><mi>n<\/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> <mo class=\"MathClass-rel\">&lt;<\/mo> <mi>\ud835\udf00<\/mi><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">gilt. Daraus folgt <\/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><munderover accent=\"false\" accentunder=\"false\"><mrow><mo> \u2211<\/mo> <\/mrow><mrow><mi>k<\/mi><mo class=\"MathClass-rel\">=<\/mo><mi>m<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/munderover><msub><mrow><mi>a<\/mi><\/mrow><mrow> <mi>k<\/mi><\/mrow><\/msub><mstyle><mrow><mo fence=\"true\" form=\"prefix\"> |<\/mo><mrow \/><mo fence=\"true\" form=\"postfix\" \/><\/mrow><\/mstyle> <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>m<\/mi><\/mrow><mrow><mi>n<\/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> <mo class=\"MathClass-rel\">&lt;<\/mo> <mi>\ud835\udf00<\/mi><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">mit der Dreiecksungleichung. Da&nbsp;<math display=\"inline\"><mi>\ud835\udf00<\/mi> <mo class=\"MathClass-rel\">&gt;<\/mo> <mn>0<\/mn><\/math> beliebig war, beweist dies nach dem Cauchy-Kriterium die Konvergenz der Reihe&nbsp;<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><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><\/math><\/span><span class=\"period\">.<\/span> <\/p><p class=\"indent\">Der zweite Teil folgt nun aus der Ungleichung <\/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><munderover accent=\"false\" accentunder=\"false\"><mrow><mo> \u2211<\/mo> <\/mrow><mrow><mi>k<\/mi><mo class=\"MathClass-rel\">=<\/mo><mn>1<\/mn><\/mrow><mrow><mi>n<\/mi><\/mrow><\/munderover><msub><mrow><mi>a<\/mi><\/mrow><mrow> <mi>k<\/mi><\/mrow><\/msub><mstyle><mrow><mo fence=\"true\" form=\"prefix\"> |<\/mo><mrow \/><mo fence=\"true\" form=\"postfix\" \/><\/mrow><\/mstyle> <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><mn>1<\/mn><\/mrow><mrow><mi>n<\/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><\/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 <math display=\"inline\"><mi>n<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2115<\/mi><\/math> und dem Grenz\u00fcbergang f\u00fcr <span class=\"maperiod\"><math display=\"inline\"><mi>n<\/mi> <mo class=\"MathClass-rel\">\u2192<\/mo><mi>\u221e<\/mi><\/math><\/span><span class=\"period\">.<\/span> <span>&nbsp;&nbsp;<\/span><\/p><div class=\"qed\">\u25a0<\/div><\/details><\/div> <a id=\"x1-192002r191\"><\/a> <h4 id=\"z6207f098ef60\" class=\"subsectionHead\"><span class=\"titlemark\">7.2.1 <\/span> <a id=\"x1-1930001\"><\/a>Hinreichende Kriterien f\u00fcr absolute Konvergenz<\/h4> <p class=\"noindent\">Falls sich die Glieder einer Reihe im Absolutbetrag durch die Glieder einer konvergenten Reihe absch\u00e4tzen lassen, so ist die Reihe konvergent, wie wir in folgendem Korollar des Vergleichssatzes (Korollar <a href=\"..\/..\/chapter\/reihen#x1-188002r12\">7.12<\/a>) zeigen. <\/p> <div class=\"me metheorem\"> <div class=\"wp-nocaption \"><\/div><h4 id=\"zd91bf72461cf\"> <a id=\"x1-193001r29\"><\/a> <span class=\"ecbx-1095\">Korollar 7.29 <\/span>(Majorantenkriterium von Weierstrass)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Sei <\/span><math display=\"inline\"><msub><mrow><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi> <\/mrow> <\/msub> <mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><\/math> <span class=\"ecti-1095\">eine komplexe und <\/span><math display=\"inline\"><msub><mrow><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>b<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><\/math> <span class=\"ecti-1095\">eine reelle Folge mit <\/span><math display=\"inline\"><mo class=\"MathClass-rel\">|<\/mo><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><mo class=\"MathClass-rel\">|<\/mo><mo class=\"MathClass-rel\">\u2264<\/mo> <msub><mrow><mi>b<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><\/math> <span class=\"ecti-1095\">f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r alle hinreichend grossen <\/span><span class=\"maperiod\"><math display=\"inline\"><mi>n<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2115<\/mi><\/math><\/span><span class=\"period\">.<\/span> <span class=\"ecti-1095\">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>1<\/mn><\/mrow><mrow><mi>\u221e<\/mi><\/mrow><\/msubsup><msub><mrow><mi>b<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><\/math> <span class=\"ecti-1095\">konvergiert, dann ist <\/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><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><\/math> <span class=\"ecti-1095\">absolut konvergent und daher auch konvergent.<\/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\">Da endlich viele Startglieder vernachl\u00e4ssigt werden k\u00f6nnen, d\u00fcrfen wir annehmen, dass die Ungleichung <math display=\"inline\"><mo class=\"MathClass-rel\">|<\/mo><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><mo class=\"MathClass-rel\">|<\/mo><mo class=\"MathClass-rel\">\u2264<\/mo> <msub><mrow><mi>b<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><\/math> f\u00fcr alle <math display=\"inline\"><mi>n<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2115<\/mi><\/math> gilt. Nach dem Vergleichssatz (Korollar <a href=\"..\/..\/chapter\/reihen#x1-188002r12\">7.12<\/a>) ist <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><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><\/math> dann absolut konvergent und die Konvergenz folgt aus Proposition&nbsp;<a href=\"..\/..\/chapter\/absolute-konvergenz#x1-192001r28\">7.28<\/a>. <span>&nbsp;&nbsp;<\/span><\/p><div class=\"qed\">\u25a0<\/div><\/details><\/div> <p class=\"indent\">Wir m\u00f6chten nun zwei Korollare des Majorantenkriteriums diskutieren. <\/p> <div class=\"me metheorem\"> <div class=\"wp-nocaption \"><\/div><h4 id=\"za08d8396415e\"> <a id=\"x1-193002r30\"><\/a> <span class=\"ecbx-1095\">Korollar 7.30 <\/span>(Cauchy-Wurzelkriterium)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Sei <\/span><math display=\"inline\"><msub><mrow><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi> <\/mrow> <\/msub> <mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><\/math> <span class=\"ecti-1095\">eine Folge komplexer Zahlen und<\/span> <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mi>\u03b1<\/mi> <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> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u211d<\/mi> <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><mo class=\"MathClass-punc\">.<\/mo><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\"><span class=\"ecti-1095\">Dann gilt<\/span> <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mi>\u03b1<\/mi> <mo class=\"MathClass-rel\">&lt;<\/mo> <mn>1<\/mn><\/mtd> <mtd class=\"align-even\"><mspace class=\"thickpace\" width=\"0.28em\" \/><mo class=\"MathClass-rel\">\u21d2<\/mo><mspace class=\"thickpace\" width=\"0.28em\" \/><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><msub><mrow><mi>a<\/mi><\/mrow><mrow> <mi>n<\/mi><\/mrow><\/msub><mstyle class=\"text\"><mtext>&nbsp;ist&nbsp;absolut&nbsp;konvergent<\/mtext><\/mstyle><mo class=\"MathClass-punc\">,<\/mo><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\" \/> <mtd class=\"align-label\"> <mspace width=\"2em\" \/><\/mtd><\/mtr><mtr><mtd class=\"align-odd\" columnalign=\"right\"><mi>\u03b1<\/mi> <mo class=\"MathClass-rel\">&gt;<\/mo> <mn>1<\/mn><\/mtd> <mtd class=\"align-even\"><mspace class=\"thickpace\" width=\"0.28em\" \/><mo class=\"MathClass-rel\">\u21d2<\/mo><mspace class=\"thickpace\" width=\"0.28em\" \/><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><msub><mrow><mi>a<\/mi><\/mrow><mrow> <mi>n<\/mi><\/mrow><\/msub><mstyle class=\"text\"><mtext>&nbsp;ist&nbsp;divergent&nbsp;und&nbsp;<\/mtext><\/mstyle><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><\/mrow><\/msub><mstyle class=\"text\"><mtext>&nbsp;ist&nbsp;keine&nbsp;Nullfolge<\/mtext><\/mstyle><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> <\/div> <p class=\"indent\">Sehr schwammig ausgedr\u00fcckt l\u00e4sst sich eine Folge <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> <\/mrow> <\/msub> <\/math> f\u00fcr <math display=\"inline\"><mi>\u03b1<\/mi> <mo class=\"MathClass-rel\">&lt;<\/mo> <mn>1<\/mn><\/math> wie in Korollar <a href=\"..\/..\/chapter\/absolute-konvergenz#x1-193002r30\">7.30<\/a> bis auf endlich viele Glieder und einen kleinen Fehler <math display=\"inline\"><mi>\ud835\udf00<\/mi> <mo class=\"MathClass-rel\">&gt;<\/mo> <mn>0<\/mn><\/math> von oben durch die geometrische Folge <math display=\"inline\"><msup><mrow><mo class=\"MathClass-open\">(<\/mo><mi>\u03b1<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mi>\ud835\udf00<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msup><\/math> absch\u00e4tzen. Somit kann man Korollar <a href=\"..\/..\/chapter\/absolute-konvergenz#x1-193001r29\">7.29<\/a> anwenden. Nun aber genauer. <\/p><div class=\"wp-nocaption \"><\/div> <div class=\"proof\"> <p class=\"indent\"><span class=\"head\"><\/span><\/p><details open=\"open\"><summary><b>Beweis.<\/b><\/summary><p class=\"indent\" style=\"margin-top: 10\">Angenommen <span class=\"maperiod\"><math display=\"inline\"><mi>\u03b1<\/mi> <mo class=\"MathClass-rel\">&lt;<\/mo> <mn>1<\/mn><\/math><\/span><span class=\"period\">.<\/span> Dann gibt es ein <math display=\"inline\"><mi>n<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2115<\/mi><\/math> mit <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><munder class=\"msub\"><mrow><mi class=\"qopname\"> sup<\/mi><mo>  <\/mo><\/mrow><mrow><mi>k<\/mi><mo class=\"MathClass-rel\">\u2265<\/mo><mi>n<\/mi><\/mrow><\/munder><mroot><mrow><mo class=\"MathClass-rel\">|<\/mo><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>k<\/mi><\/mrow><\/msub><mo class=\"MathClass-rel\">|<\/mo><\/mrow><mrow><mi>k<\/mi><\/mrow><\/mroot> <mo class=\"MathClass-rel\">&lt;<\/mo> <mi>q<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mfrac><mrow><mn>1<\/mn> <mo class=\"MathClass-bin\">+<\/mo> <mi>\u03b1<\/mi><\/mrow> <mrow><mn>2<\/mn><\/mrow><\/mfrac> <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\">und somit <math display=\"inline\"><mo class=\"MathClass-rel\">|<\/mo><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>k<\/mi><\/mrow><\/msub><mo class=\"MathClass-rel\">|<\/mo> <mo class=\"MathClass-rel\">&lt;<\/mo> <msup><mrow><mi>q<\/mi><\/mrow><mrow><mi>k<\/mi><\/mrow><\/msup><\/math> f\u00fcr alle <span class=\"maperiod\"><math display=\"inline\"><mi>k<\/mi> <mo class=\"MathClass-rel\">\u2265<\/mo> <mi>n<\/mi><\/math><\/span><span class=\"period\">.<\/span> Die Reihe <math display=\"inline\"><msubsup><mrow><mi class=\"MathClass-op\"> \u2211<\/mi><mo> <\/mo> <\/mrow><mrow><mi>k<\/mi><mo class=\"MathClass-rel\">=<\/mo><mn>1<\/mn><\/mrow><mrow><mi>\u221e<\/mi><\/mrow><\/msubsup><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>k<\/mi><\/mrow><\/msub><\/math> konvergiert somit absolut wegen dem Majorantenkriterium (Korollar <a href=\"..\/..\/chapter\/absolute-konvergenz#x1-193001r29\">7.29<\/a>) und der geometrischen Reihe in Beispiel <a href=\"..\/..\/chapter\/reihen#x1-187003r3\">7.3<\/a>. <\/p><p class=\"indent\">Falls <math display=\"inline\"><mi>\u03b1<\/mi> <mo class=\"MathClass-rel\">&gt;<\/mo> <mn>1<\/mn><\/math> gilt, gibt es nach Satz <a href=\"..\/..\/chapter\/reelle-folgen#x1-160001r15\">6.15<\/a> eine Teilfolge <math display=\"inline\"><msub><mrow><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>a<\/mi><\/mrow><mrow><msub><mrow><mi>n<\/mi><\/mrow><mrow><mi>k<\/mi><\/mrow><\/msub><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mi>k<\/mi><\/mrow><\/msub><\/math> mit <math display=\"inline\"><mroot><mrow><mo class=\"MathClass-rel\">|<\/mo><msub><mrow><mi>a<\/mi><\/mrow><mrow><msub><mrow><mi>n<\/mi><\/mrow><mrow><mi>k<\/mi> <\/mrow> <\/msub> <\/mrow> <\/msub> <mo class=\"MathClass-rel\">|<\/mo><\/mrow> <mrow> <msub><mrow><mi>n<\/mi><\/mrow><mrow><mi>k<\/mi> <\/mrow> <\/msub> <\/mrow> <\/mroot> <mo class=\"MathClass-rel\">&gt;<\/mo> <mn>1<\/mn><\/math> f\u00fcr alle <span class=\"maperiod\"><math display=\"inline\"><mi>k<\/mi><\/math><\/span><span class=\"period\">.<\/span> Daraus folgt aber <span class=\"maperiod\"><math display=\"inline\"><mo class=\"MathClass-rel\">|<\/mo><msub><mrow><mi>a<\/mi><\/mrow><mrow><msub><mrow><mi>n<\/mi><\/mrow><mrow><mi>k<\/mi> <\/mrow> <\/msub> <\/mrow> <\/msub> <mo class=\"MathClass-rel\">|<\/mo> <mo class=\"MathClass-rel\">&gt;<\/mo> <mn>1<\/mn><\/math><\/span><span class=\"period\">.<\/span> Insbesondere ist <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> <\/mrow> <\/msub> <\/math> keine Nullfolge 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>1<\/mn><\/mrow><mrow><mi>\u221e<\/mi><\/mrow><\/msubsup><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><\/math> divergiert nach Proposition <a href=\"..\/..\/chapter\/reihen#x1-187002r2\">7.2<\/a>. <span>&nbsp;&nbsp;<\/span><\/p><div class=\"qed\">\u25a0<\/div><\/details><\/div> <div class=\"me meexample\"> <div class=\"wp-nocaption \"><\/div><h4 id=\"z18d1e762804c\"> <a id=\"x1-193003r31\"><\/a> <span class=\"ecbx-1095\">Beispiel 7.31 <\/span>(Der Fall <math display=\"inline\"><mi>\u03b1<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mn>1<\/mn><\/math> in Korollar <a href=\"..\/..\/chapter\/absolute-konvergenz#x1-193002r30\">7.30<\/a>)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Sei <\/span><math display=\"inline\"><msub><mrow><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi> <\/mrow> <\/msub> <mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><\/math> <span class=\"ecti-1095\">eine Folge komplexer Zahlen<\/span> <span class=\"ecti-1095\">und <\/span><math display=\"inline\"><mi>\u03b1<\/mi> <mo class=\"MathClass-rel\">=<\/mo><msub><mrow><mi class=\"qopname\"> limsup<\/mi><mo>  <\/mo><\/mrow><mrow><mi>n<\/mi><mo class=\"MathClass-rel\">\u2192<\/mo><mi>\u221e<\/mi><\/mrow><\/msub><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><\/math> <span class=\"ecti-1095\">wie im Wurzelkriterium<\/span> <span class=\"ecti-1095\">(Korollar <\/span><a href=\"..\/..\/chapter\/absolute-konvergenz#x1-193002r30\"><span class=\"ecti-1095\">7.30<\/span><\/a><span class=\"ecti-1095\">). Falls <\/span><span class=\"maperiod\"><math display=\"inline\"><mi>\u03b1<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mn>1<\/mn><\/math><\/span><span class=\"period\">,<\/span> <span class=\"ecti-1095\">dann kann anhand des Wurzelkriteriums keine Entscheidung <\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">ber Konvergenz oder Divergenz der<\/span> <span class=\"ecti-1095\">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>1<\/mn><\/mrow><mrow><mi>\u221e<\/mi><\/mrow><\/msubsup><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><\/math> <span class=\"ecti-1095\">getroffen werden.<\/span> <\/p> <div class=\"custom-itemize\"><div class=\"item-head\"> <span class=\"tcrm-1095\">\u2022<\/span><\/div><div class=\"item-content\"><span class=\"ecti-1095\">Ist <\/span><math display=\"inline\"><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi> <\/mrow> <\/msub> <mo class=\"MathClass-rel\">=<\/mo> <mfrac> <mrow> <mn>1<\/mn><\/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><span class=\"maperiod\"><math display=\"inline\"><mi>n<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2115<\/mi><\/math><\/span><span class=\"period\">,<\/span> <span class=\"ecti-1095\">dann gilt <\/span><math display=\"inline\"><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><mn>1<\/mn><\/mrow> <mrow><mroot><mrow><mi>n<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/mroot><\/mrow><\/mfrac> <mo class=\"MathClass-rel\">\u2192<\/mo> <mn>1<\/mn><\/math> <span class=\"ecti-1095\">f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r <\/span><math display=\"inline\"><mi>n<\/mi> <mo class=\"MathClass-rel\">\u2192<\/mo> <mi>\u221e<\/mi><\/math> <span class=\"ecti-1095\">wegen Beispiel <\/span><a href=\"..\/..\/chapter\/reelle-folgen#x1-157006r4\"><span class=\"ecti-1095\">6.4<\/span><\/a> <span class=\"ecti-1095\">und wegen Beispiel <\/span><a href=\"..\/..\/chapter\/reihen#x1-187004r4\"><span class=\"ecti-1095\">7.4<\/span><\/a> <span class=\"ecti-1095\">divergiert 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>1<\/mn><\/mrow><mrow><mi>\u221e<\/mi><\/mrow><\/msubsup><mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mi>n<\/mi><\/mrow><\/mfrac><\/math> <span class=\"ecti-1095\">gegen Unendlich.<\/span> <\/div><div class=\"item-head\"> <span class=\"tcrm-1095\">\u2022<\/span><\/div><div class=\"item-content\"><span class=\"ecti-1095\">Ist <\/span><math display=\"inline\"><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi> <\/mrow> <\/msub> <mo class=\"MathClass-rel\">=<\/mo> <mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><msup><mrow><mi>n<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><\/mrow><\/mfrac><\/math> <span class=\"ecti-1095\">f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r alle <\/span><span class=\"maperiod\"><math display=\"inline\"><mi>n<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2115<\/mi><\/math><\/span><span class=\"period\">,<\/span> <span class=\"ecti-1095\">dann gilt <\/span><math display=\"inline\"><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><mn>1<\/mn><\/mrow> <mrow><msup><mrow><mroot><mrow><mi>n<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/mroot><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><\/mrow><\/mfrac> <mo class=\"MathClass-rel\">\u2192<\/mo> <mn>1<\/mn><\/math> <span class=\"ecti-1095\">f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r <\/span><math display=\"inline\"><mi>n<\/mi> <mo class=\"MathClass-rel\">\u2192<\/mo> <mi>\u221e<\/mi><\/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>1<\/mn><\/mrow><mrow><mi>\u221e<\/mi><\/mrow><\/msubsup> <mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><msup><mrow><mi>n<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><\/mrow><\/mfrac><\/math> <span class=\"ecti-1095\">konvergiert nach Beispiel <\/span><a href=\"..\/..\/chapter\/reihen#x1-188004r14\"><span class=\"ecti-1095\">7.14<\/span><\/a><span class=\"ecti-1095\">.<\/span><\/div><\/div> <\/div> <div class=\"me metheorem\"> <div class=\"wp-nocaption \"><\/div><h4 id=\"zd87963d5bdaf\"> <a id=\"x1-193004r32\"><\/a> <span class=\"ecbx-1095\">Korollar 7.32 <\/span>(D\u2019Alemberts Quotientenkriterium)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Sei <\/span><math display=\"inline\"><msub><mrow><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi> <\/mrow> <\/msub> <mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><\/math> <span class=\"ecti-1095\">eine Folge<\/span> <span class=\"ecti-1095\">komplexer Zahlen 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><span class=\"maperiod\"><math display=\"inline\"><mi>n<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2115<\/mi><\/math><\/span><span class=\"period\">,<\/span> <span class=\"ecti-1095\">so dass<\/span> <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mi>\u03b1<\/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><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> <\/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\">existiert. Dann gilt<\/span> <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mi>\u03b1<\/mi> <mo class=\"MathClass-rel\">&lt;<\/mo> <mn>1<\/mn><\/mtd> <mtd class=\"align-even\"><mspace class=\"thickpace\" width=\"0.28em\" \/><mo class=\"MathClass-rel\">\u21d2<\/mo><mspace class=\"thickpace\" width=\"0.28em\" \/><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><msub><mrow><mi>a<\/mi><\/mrow><mrow> <mi>n<\/mi><\/mrow><\/msub><mstyle class=\"text\"><mtext>&nbsp;ist&nbsp;absolut&nbsp;konvergent.<\/mtext><\/mstyle><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\" \/> <mtd class=\"align-label\"> <mspace width=\"2em\" \/><\/mtd><\/mtr><mtr><mtd class=\"align-odd\" columnalign=\"right\"><mi>\u03b1<\/mi> <mo class=\"MathClass-rel\">&gt;<\/mo> <mn>1<\/mn><\/mtd> <mtd class=\"align-even\"><mspace class=\"thickpace\" width=\"0.28em\" \/><mo class=\"MathClass-rel\">\u21d2<\/mo><mspace class=\"thickpace\" width=\"0.28em\" \/><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><msub><mrow><mi>a<\/mi><\/mrow><mrow> <mi>n<\/mi><\/mrow><\/msub><mstyle class=\"text\"><mtext>&nbsp;ist&nbsp;divergent&nbsp;und&nbsp;<\/mtext><\/mstyle><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><\/mrow><\/msub><mstyle class=\"text\"><mtext>&nbsp;konvergiert&nbsp;nicht&nbsp;gegen&nbsp;Null.<\/mtext><\/mstyle><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\" \/> <mtd class=\"align-label\"> <mspace width=\"2em\" \/><\/mtd><\/mtr><\/mtable><\/math> <\/div> <div class=\"me melemma\"> <div class=\"wp-nocaption \"><\/div><h4 id=\"z6338726d86e3\"> <a id=\"x1-193005r33\"><\/a> <span class=\"ecbx-1095\">Wichtige <\/span><span class=\"ecbx-1095\">\u00dc<\/span><span class=\"ecbx-1095\">bung 7.33.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Beweisen Sie Korollar <\/span><a href=\"..\/..\/chapter\/absolute-konvergenz#x1-193004r32\"><span class=\"ecti-1095\">7.32<\/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\">Gehen Sie wie im Beweis von Korollar <\/span><a href=\"..\/..\/chapter\/absolute-konvergenz#x1-193002r30\"><span class=\"ecti-1095\">7.30<\/span><\/a> <span class=\"ecti-1095\">vor. F<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r <\/span><math display=\"inline\"><mi>\u03b1<\/mi> <mo class=\"MathClass-rel\">&lt;<\/mo> <mn>1<\/mn><\/math> <span class=\"ecti-1095\">finden Sie ein <\/span><span class=\"maperiod\"><math display=\"inline\"><mi>q<\/mi> <mo class=\"MathClass-rel\">&lt;<\/mo> <mn>1<\/mn><\/math><\/span><span class=\"period\">,<\/span> <span class=\"ecti-1095\">so dass <\/span><math display=\"inline\"><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> <mo class=\"MathClass-rel\">&lt;<\/mo> <mi>q<\/mi><\/math> <span class=\"ecti-1095\">f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r alle bis auf endlich viele <\/span><span class=\"maperiod\"><math display=\"inline\"><mi>n<\/mi><\/math><\/span><span class=\"period\">.<\/span><\/p><\/details>  <\/div> <div class=\"me meexample\"> <div class=\"wp-nocaption \"><\/div><h4 id=\"z0fe7afa80b15\"> <a id=\"x1-193006r34\"><\/a> <span class=\"ecbx-1095\">\u00dc<\/span><span class=\"ecbx-1095\">bung 7.34.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Wieso kann man im Quotientenkriterium (Korollar <\/span><a href=\"..\/..\/chapter\/absolute-konvergenz#x1-193004r32\"><span class=\"ecti-1095\">7.32<\/span><\/a><span class=\"ecti-1095\">) nicht auch den Limes superior<\/span> <span class=\"ecti-1095\">anstelle des Limes verwenden?<\/span> <\/p><div class=\"wp-nocaption \"><\/div><details><summary style=\"color:#FF7F00\"><span class=\"ecti-1095\">L<\/span><span class=\"ecti-1095\">\u00f6<\/span><span class=\"ecti-1095\">sung.<\/span><\/summary><p class=\"indent\" style=\"margin-top: 0\"> <span class=\"ecti-1095\">In der Tat folgt aus <\/span><span class=\"maperiod\"><math display=\"inline\"><msub><mrow><mi class=\"qopname\">limsup<\/mi><mo>  <\/mo><\/mrow><mrow><mi>n<\/mi><mo class=\"MathClass-rel\">\u2192<\/mo><mi>\u221e<\/mi><\/mrow><\/msub><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> <mo class=\"MathClass-rel\">&lt;<\/mo> <mn>1<\/mn><\/math><\/span><span class=\"period\">,<\/span> <span class=\"ecti-1095\">dass die Reihe <\/span><math display=\"inline\"><msubsup><mrow><mi class=\"MathClass-op\">\u2211<\/mi><mo> <\/mo> <\/mrow><mrow><mi>n<\/mi><mo class=\"MathClass-rel\">=<\/mo><mn>1<\/mn><\/mrow><mrow><mi>\u221e<\/mi><\/mrow><\/msubsup><\/math> <span class=\"ecti-1095\">absolut konvergiert, was ebenso <\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">hnlich wie im Beweis von Satz <\/span><a href=\"..\/..\/chapter\/absolute-konvergenz#x1-193002r30\"><span class=\"ecti-1095\">7.30<\/span><\/a> <span class=\"ecti-1095\">gezeigt werden kann. Doch kann<\/span> <span class=\"ecti-1095\">aus <\/span><math display=\"inline\"><msub><mrow><mi class=\"qopname\"> limsup<\/mi><mo>  <\/mo> <\/mrow><mrow><mi>n<\/mi><mo class=\"MathClass-rel\">\u2192<\/mo><mi>\u221e<\/mi><\/mrow><\/msub><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> <mo class=\"MathClass-rel\">&gt;<\/mo> <mn>1<\/mn><\/math> <span class=\"ecti-1095\">keine Aussage <\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">ber die Konvergenz der Reihe getroffen werden. Als Beispiel hierzu betrachten wir<\/span> <span class=\"ecti-1095\">die Folge<\/span> <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub> <mo class=\"MathClass-rel\">=<\/mo> <mrow class=\"cases\"> <mrow><mo fence=\"true\" form=\"prefix\"> {<\/mo><mrow> <mtable align=\"axis\" class=\"array\" columnlines=\"none\" equalcolumns=\"false\" equalrows=\"false\"> <mtr><mtd class=\"array\" columnalign=\"left\"><msup><mrow><mn>2<\/mn><\/mrow><mrow><mo class=\"MathClass-bin\">\u2212<\/mo><mi>n<\/mi><\/mrow><\/msup><mspace class=\"quad\" width=\"1em\" \/><mstyle class=\"text\"><mtext>f\u00fcr&nbsp;ungerades&nbsp;<\/mtext><\/mstyle><mi>n<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2115<\/mi><mo class=\"MathClass-punc\">,<\/mo><mspace class=\"quad\" width=\"1em\" \/><\/mtd> <\/mtr> <mtr><mtd class=\"array\" columnalign=\"left\"><msup><mrow><mn>3<\/mn><\/mrow><mrow><mo class=\"MathClass-bin\">\u2212<\/mo><mi>n<\/mi><\/mrow><\/msup><mspace class=\"quad\" width=\"1em\" \/><mstyle class=\"text\"><mtext>f\u00fcr&nbsp;gerades&nbsp;<\/mtext><\/mstyle><mi>n<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2115<\/mi><mo class=\"MathClass-punc\">.<\/mo> <mspace class=\"quad\" width=\"1em\" \/><\/mtd><\/mtr> <\/mtable> <\/mrow><mo fence=\"true\" form=\"postfix\" \/><\/mrow><\/mrow><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\"><span class=\"ecti-1095\">F<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r diese Folge gilt<\/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\">limsup<\/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> <mo class=\"MathClass-rel\">=<\/mo><munder class=\"msub\"><mrow><mi class=\"qopname\"> lim<\/mi><mo>  <\/mo><\/mrow><mrow><mi>k<\/mi><mo class=\"MathClass-rel\">\u2192<\/mo><mi>\u221e<\/mi><\/mrow><\/munder><mfrac><mrow><msup><mrow><mn>2<\/mn><\/mrow><mrow><mo class=\"MathClass-bin\">\u2212<\/mo><mo class=\"MathClass-open\">(<\/mo><mn>2<\/mn><mi>k<\/mi><mo class=\"MathClass-bin\">+<\/mo><mn>1<\/mn><mo class=\"MathClass-close\">)<\/mo><\/mrow><\/msup><\/mrow> <mrow><msup><mrow><mn>3<\/mn><\/mrow><mrow><mo class=\"MathClass-bin\">\u2212<\/mo><mn>2<\/mn><mi>k<\/mi><\/mrow><\/msup><\/mrow><\/mfrac> <munder class=\"msub\"><mrow><mi class=\"qopname\"> lim<\/mi><mo>  <\/mo><\/mrow><mrow><mi>k<\/mi><mo class=\"MathClass-rel\">\u2192<\/mo><mi>\u221e<\/mi><\/mrow><\/munder><mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mn>2<\/mn><\/mrow><\/mfrac><msup><mrow> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mfrac><mrow><mn>3<\/mn><\/mrow> <mrow><mn>2<\/mn><\/mrow><\/mfrac> <\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <\/mrow><mrow><mn>2<\/mn><mi>k<\/mi><\/mrow><\/msup> <mo class=\"MathClass-rel\">=<\/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> <p class=\"noindent\"><span class=\"ecti-1095\">Trotzdem ist aber 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>1<\/mn><\/mrow><mrow><mi>\u221e<\/mi><\/mrow><\/msubsup><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><\/math> <span class=\"ecti-1095\">konvergent, was unter Verwendung von zwei geometrischen Reihen oder dem Wurzelkriterium schnell<\/span> <span class=\"ecti-1095\">folgt. <\/span><\/p><\/details>  <\/div> <a id=\"x1-193007r193\"><\/a> <h4 id=\"z964006478370\" class=\"subsectionHead\"><span class=\"titlemark\">7.2.2 <\/span> <a id=\"x1-1940002\"><\/a>Umordnen von Reihen<\/h4> <p class=\"noindent\">Im Gegensatz zur bedingten Konvergenz in Teilabschnitt <a href=\"..\/..\/chapter\/reihen#x1-1890002\">7.1.2<\/a> ist absolute Konvergenz sehr robust. Der erste dieser Robustheitss\u00e4tze ist folgender positive Umordnungssatz. <\/p> <div class=\"me metheorem\"> <div class=\"wp-nocaption \"><\/div><h4 id=\"z11f682330900\"> <a id=\"x1-194001r35\"><\/a> <span class=\"ecbx-1095\">Satz 7.35 <\/span>(Umordnen absolut konvergenter Reihen)<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>1<\/mn><\/mrow><mrow><mi>\u221e<\/mi><\/mrow><\/msubsup><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><\/math> <span class=\"ecti-1095\">eine absolut konvergente Reihe mit komplexen Gliedern. Sei<\/span> <math display=\"inline\"><mi>\u03c6<\/mi> <mo class=\"MathClass-punc\">:<\/mo> <mi>\u2115<\/mi> <mo class=\"MathClass-rel\">\u2192<\/mo> <mi>\u2115<\/mi><\/math> <span class=\"ecti-1095\">eine Bijektion.<\/span> <span class=\"ecti-1095\">Dann ist <\/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><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>\u03c6<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>n<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow><\/msub><\/math> <span class=\"ecti-1095\">ebenso absolut konvergent und es gilt<\/span> <\/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>1<\/mn><\/mrow><mrow><mi>\u221e<\/mi><\/mrow><\/munderover><msub><mrow><mi>a<\/mi><\/mrow><mrow> <mi>n<\/mi><\/mrow><\/msub> <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>1<\/mn><\/mrow><mrow><mi>\u221e<\/mi><\/mrow><\/munderover><msub><mrow><mi>a<\/mi><\/mrow><mrow> <mi>\u03c6<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>n<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow><\/msub><mo class=\"MathClass-punc\">.<\/mo><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"><mstyle class=\"label\" id=\"x1-194002r2\" \/><mstyle class=\"maketag\"><mtext>(7.2)<\/mtext><\/mstyle><mspace class=\"nbsp\" width=\"0.33em\" \/> <\/mtd><\/mtr><\/mtable><\/math> <\/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\">Sei <math display=\"inline\"><mi>\u03c6<\/mi> <mo class=\"MathClass-punc\">:<\/mo> <mi>\u2115<\/mi> <mo class=\"MathClass-rel\">\u2192<\/mo> <mi>\u2115<\/mi><\/math> eine Bijektion und <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> Nach dem Cauchy-Kriterium (Satz <a href=\"..\/..\/chapter\/reihen#x1-191001r26\">7.26<\/a>) gibt es ein <span class=\"maperiod\"><math display=\"inline\"><mi>N<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2115<\/mi><\/math><\/span><span class=\"period\">,<\/span> so dass <math display=\"inline\"><msubsup><mrow><mi class=\"MathClass-op\"> \u2211<\/mi><mo> <\/mo> <\/mrow><mrow><mi>k<\/mi><mo class=\"MathClass-rel\">=<\/mo><mi>m<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msubsup><mo class=\"MathClass-rel\">|<\/mo><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>k<\/mi><\/mrow><\/msub><mo class=\"MathClass-rel\">|<\/mo> <mo class=\"MathClass-rel\">&lt;<\/mo> <mi>\ud835\udf00<\/mi><\/math> f\u00fcr alle nat\u00fcrliche Zahlen <span class=\"maperiod\"><math display=\"inline\"><mi>n<\/mi> <mo class=\"MathClass-rel\">\u2265<\/mo> <mi>m<\/mi> <mo class=\"MathClass-rel\">\u2265<\/mo> <mi>N<\/mi><\/math><\/span><span class=\"period\">.<\/span> Daher gilt auch nach der Dreiecksungleichung in Proposition <a href=\"..\/..\/chapter\/absolute-konvergenz#x1-192001r28\">7.28<\/a>, dass <\/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><munderover accent=\"false\" accentunder=\"false\"><mrow><mo> \u2211<\/mo> <\/mrow><mrow><mi>k<\/mi><mo class=\"MathClass-rel\">=<\/mo><mi>m<\/mi><\/mrow><mrow><mi>\u221e<\/mi><\/mrow><\/munderover><msub><mrow><mi>a<\/mi><\/mrow><mrow> <mi>k<\/mi><\/mrow><\/msub><mstyle><mrow><mo fence=\"true\" form=\"prefix\"> |<\/mo><mrow \/><mo fence=\"true\" form=\"postfix\" \/><\/mrow><\/mstyle> <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>m<\/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><mo class=\"MathClass-rel\">\u2264<\/mo> <mi>\ud835\udf00<\/mi><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">f\u00fcr alle&nbsp;<span class=\"maperiod\"><math display=\"inline\"><mi>m<\/mi> <mo class=\"MathClass-rel\">\u2265<\/mo> <mi>N<\/mi><\/math><\/span><span class=\"period\">.<\/span> (Es hilft vielleicht f\u00fcr das Folgende diese Absch\u00e4tzung als \u201eDie Summanden&nbsp;<math display=\"inline\"><msub><mrow><mi>a<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><mo class=\"MathClass-punc\">,<\/mo><mi class=\"MathClass-op\">\u2026<\/mi><mo> <\/mo><mo class=\"MathClass-punc\">,<\/mo><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>N<\/mi><\/mrow><\/msub><\/math> der urspr\u00fcnglichen Reihe sind wichtig, aber die restlichen Summanden sind weniger wichtig.\u201c zu interpretieren.) <\/p><p class=\"indent\">Wir definieren <math display=\"inline\"><mi>M<\/mi> <mo class=\"MathClass-rel\">=<\/mo><mi class=\"qopname\"> max<\/mi><mo>  <\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> {<\/mo><mrow><msup><mrow><mi>\u03c6<\/mi><\/mrow><mrow><mo class=\"MathClass-bin\">\u2212<\/mo><mn>1<\/mn><\/mrow><\/msup><mo class=\"MathClass-open\">(<\/mo><mi>k<\/mi><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-rel\">\u2223<\/mo><mi>k<\/mi> <mo class=\"MathClass-rel\">\u2264<\/mo> <mi>N<\/mi> <\/mrow><mo fence=\"true\" form=\"postfix\">}<\/mo><\/mrow><\/math> und w\u00e4hlen ein <span class=\"maperiod\"><math display=\"inline\"><mi>n<\/mi> <mo class=\"MathClass-rel\">\u2265<\/mo> <mi>M<\/mi><\/math><\/span><span class=\"period\">.<\/span> Dann gilt <\/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>\u2113<\/mi><mo class=\"MathClass-rel\">=<\/mo><mn>1<\/mn><\/mrow><mrow><mi>n<\/mi><\/mrow><\/munderover><msub><mrow><mi>a<\/mi><\/mrow><mrow> <mi>\u03c6<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>\u2113<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow><\/msub> <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>1<\/mn><\/mrow><mrow><mi>\u221e<\/mi><\/mrow><\/munderover><msub><mrow><mi>a<\/mi><\/mrow><mrow> <mi>k<\/mi><\/mrow><\/msub><\/mrow><mo fence=\"true\" form=\"postfix\">|<\/mo><\/mrow><\/mtd> <mtd class=\"align-even\"> <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>\u2113<\/mi><mo class=\"MathClass-rel\">=<\/mo><mn>1<\/mn><\/mrow><mrow><mi>n<\/mi><\/mrow><\/munderover><msub><mrow><mi>a<\/mi><\/mrow><mrow> <mi>\u03c6<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>\u2113<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow><\/msub> <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>1<\/mn><\/mrow><mrow><mi>N<\/mi><\/mrow><\/munderover><msub><mrow><mi>a<\/mi><\/mrow><mrow> <mi>k<\/mi><\/mrow><\/msub> <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><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><\/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><munder class=\"msub\"><mrow><mo>\u2211<\/mo> <\/mrow><mrow><mi>\u2113<\/mi><mo class=\"MathClass-rel\">\u2264<\/mo><mi>n<\/mi><mo class=\"MathClass-punc\">,<\/mo><mspace class=\"nbsp\" width=\"0.33em\" \/><mi>\u03c6<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>\u2113<\/mi><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-rel\">&gt;<\/mo><mi>N<\/mi><\/mrow><\/munder><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>\u03c6<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>\u2113<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow><\/msub><\/mrow><mo fence=\"true\" form=\"postfix\">|<\/mo><\/mrow> <mo class=\"MathClass-bin\">+<\/mo><munder class=\"msub\"><mrow><munder accentunder=\"false\"><mrow> <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><\/mrow><mo fence=\"true\" form=\"postfix\">|<\/mo><\/mrow><\/mrow><mo>\ufe38<\/mo><\/munder> <\/mrow><mrow><mo class=\"MathClass-rel\">\u2264<\/mo><mi>\ud835\udf00<\/mi><\/mrow><\/munder><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><munder class=\"msub\"><mrow><mo>\u2211<\/mo> <\/mrow><mrow><mi>\u2113<\/mi><mo class=\"MathClass-rel\">\u2264<\/mo><mi>n<\/mi><mo class=\"MathClass-punc\">,<\/mo><mspace class=\"nbsp\" width=\"0.33em\" \/><mi>\u03c6<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>\u2113<\/mi><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-rel\">&gt;<\/mo><mi>N<\/mi><\/mrow><\/munder> <mrow><mo fence=\"true\" form=\"prefix\"> |<\/mo><mrow><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>\u03c6<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>\u2113<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow><\/msub><\/mrow><mo fence=\"true\" form=\"postfix\">|<\/mo><\/mrow> <mo class=\"MathClass-bin\">+<\/mo> <mi>\ud835\udf00<\/mi> <mo class=\"MathClass-rel\">&lt;<\/mo> <mi>\ud835\udf00<\/mi> <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\">wobei wir verwendet haben, dass&nbsp;<math display=\"inline\"><mi>\u03c6<\/mi><\/math> eine Bijektion ist. Insbesondere treten damit und wegen&nbsp;<math display=\"inline\"><mi>n<\/mi> <mo class=\"MathClass-rel\">\u2265<\/mo> <mi>M<\/mi><\/math> alle&nbsp;<math display=\"inline\"><mi>k<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mo class=\"MathClass-open\">{<\/mo><mn>1<\/mn><mo class=\"MathClass-punc\">,<\/mo><mi class=\"MathClass-op\">\u2026<\/mi><mo> <\/mo><mo class=\"MathClass-punc\">,<\/mo><mi>N<\/mi><mo class=\"MathClass-close\">}<\/mo><\/math> genau einmal als&nbsp;<math display=\"inline\"><mi>\u03c6<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>\u2113<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math> f\u00fcr&nbsp;<math display=\"inline\"><mi>\u2113<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mo class=\"MathClass-open\">{<\/mo><mn>1<\/mn><mo class=\"MathClass-punc\">,<\/mo><mi class=\"MathClass-op\">\u2026<\/mi><mo> <\/mo><mo class=\"MathClass-punc\">,<\/mo><mi>n<\/mi><mo class=\"MathClass-close\">}<\/mo><\/math> auf und die Differenz&nbsp;<math display=\"inline\"><msubsup><mrow><mi class=\"MathClass-op\"> \u2211<\/mi><mo> <\/mo> <\/mrow><mrow><mi>\u2113<\/mi><mo class=\"MathClass-rel\">=<\/mo><mn>1<\/mn><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msubsup><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>\u03c6<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>\u2113<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow><\/msub> <mo class=\"MathClass-bin\">\u2212<\/mo><msubsup><mrow><mi class=\"MathClass-op\">\u2211<\/mi><mo> <\/mo> <\/mrow><mrow><mi>k<\/mi><mo class=\"MathClass-rel\">=<\/mo><mn>1<\/mn><\/mrow><mrow><mi>N<\/mi><\/mrow><\/msubsup><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>k<\/mi><\/mrow><\/msub><\/math> enth\u00e4lt nach Wegstreichen dieser Terme nur mehr eine Summe \u00fcber gewisse&nbsp;<math display=\"inline\"><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>k<\/mi> <\/mrow> <\/msub> <\/math> mit&nbsp;<math display=\"inline\"><mi>k<\/mi> <mo class=\"MathClass-rel\">\u2265<\/mo> <mi>N<\/mi><\/math> (welche wir im Sinne obiger Interpretation als \u201eweniger wichtig\u201c betrachten und formal eben insgesamt durch ein <math display=\"inline\"><mi>\ud835\udf00<\/mi><\/math> absch\u00e4tzen k\u00f6nnen). Da&nbsp;<math display=\"inline\"><mi>\ud835\udf00<\/mi> <mo class=\"MathClass-rel\">&gt;<\/mo> <mn>0<\/mn><\/math> beliebig war, zeigt dies die Gleichung&nbsp;(<a href=\"..\/..\/chapter\/absolute-konvergenz#x1-194002r2\">7.2<\/a>). <\/p><p class=\"indent\">Wenden wir dasselbe Argument wie oben auf 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>1<\/mn><\/mrow><mrow><mi>\u221e<\/mi><\/mrow><\/msubsup><mo class=\"MathClass-rel\">|<\/mo><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>\u03c6<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>n<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow><\/msub><mo class=\"MathClass-rel\">|<\/mo><\/math> an, ergibt sich auch die absolute Konvergenz von <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><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>\u03c6<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>n<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow><\/msub><\/math><\/span><span class=\"period\">.<\/span> <span>&nbsp;&nbsp;<\/span><\/p><div class=\"qed\">\u25a0<\/div><\/details><\/div> <a id=\"x1-194003r194\"><\/a> <h4 id=\"z130ee20529f2\" class=\"subsectionHead\"><span class=\"titlemark\">7.2.3 <\/span> <a id=\"x1-1950003\"><\/a>Produkte<\/h4> <p class=\"noindent\">Wir zeigen nun, dass wir absolut konvergente Reihen gliedweise ausmultiplizieren k\u00f6nnen. <\/p> <div class=\"me metheorem\"> <div class=\"wp-nocaption \"><\/div><h4 id=\"z1e1953f850fc\"> <a id=\"x1-195001r36\"><\/a> <span class=\"ecbx-1095\">Satz 7.36 <\/span>(Produktsatz)<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>1<\/mn><\/mrow><mrow><mi>\u221e<\/mi><\/mrow><\/msubsup><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><\/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>1<\/mn><\/mrow><mrow><mi>\u221e<\/mi><\/mrow><\/msubsup><msub><mrow><mi>b<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><\/math> <span class=\"ecti-1095\">zwei absolut<\/span> <span class=\"ecti-1095\">konvergente Reihen und <\/span><math display=\"inline\"><mi>\u03c6<\/mi> <mo class=\"MathClass-punc\">:<\/mo> <mi>\u2115<\/mi> <mo class=\"MathClass-rel\">\u2192<\/mo> <mi>\u2115<\/mi> <mo class=\"MathClass-bin\">\u00d7<\/mo> <mi>\u2115<\/mi><\/math> <span class=\"ecti-1095\">eine bijektive Abbildung. Dann ist<\/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><msub><mrow><mi>a<\/mi><\/mrow><mrow> <mi>\u03c6<\/mi><msub><mrow><mo class=\"MathClass-open\">(<\/mo><mi>n<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><\/mrow><\/msub><msub><mrow><mi>b<\/mi><\/mrow><mrow><mi>\u03c6<\/mi><msub><mrow><mo class=\"MathClass-open\">(<\/mo><mi>n<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><\/mrow><\/msub><\/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 absolut konvergente Reihe, wobei <\/span><math display=\"inline\"><mi>\u03c6<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>n<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <mo class=\"MathClass-open\">(<\/mo><mi>\u03c6<\/mi><msub><mrow><mo class=\"MathClass-open\">(<\/mo><mi>n<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><mo class=\"MathClass-punc\">,<\/mo><mi>\u03c6<\/mi><msub><mrow><mo class=\"MathClass-open\">(<\/mo><mi>n<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><\/math> <span class=\"ecti-1095\">f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r alle <\/span><span class=\"maperiod\"><math display=\"inline\"><mi>n<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2115<\/mi><\/math><\/span><span class=\"period\">.<\/span> <span class=\"ecti-1095\">Weiters gilt<\/span> <\/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>1<\/mn><\/mrow><mrow><mi>\u221e<\/mi><\/mrow><\/munderover><msub><mrow><mi>a<\/mi><\/mrow><mrow> <mi>\u03c6<\/mi><msub><mrow><mo class=\"MathClass-open\">(<\/mo><mi>n<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><\/mrow><\/msub><msub><mrow><mi>b<\/mi><\/mrow><mrow><mi>\u03c6<\/mi><msub><mrow><mo class=\"MathClass-open\">(<\/mo><mi>n<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><\/mrow><\/msub> <mo class=\"MathClass-rel\">=<\/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>n<\/mi><mo class=\"MathClass-rel\">=<\/mo><mn>1<\/mn><\/mrow><mrow><mi>\u221e<\/mi><\/mrow><\/munderover><msub><mrow><mi>a<\/mi><\/mrow><mrow> <mi>n<\/mi><\/mrow><\/msub><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>1<\/mn><\/mrow><mrow><mi>\u221e<\/mi><\/mrow><\/munderover><msub><mrow><mi>b<\/mi><\/mrow><mrow> <mi>n<\/mi><\/mrow><\/msub><mstyle><mrow><mo fence=\"true\" form=\"prefix\"> )<\/mo><mrow \/><mo fence=\"true\" form=\"postfix\" \/><\/mrow><\/mstyle><mo class=\"MathClass-punc\">.<\/mo><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"><mstyle class=\"label\" id=\"x1-195002r3\" \/><mstyle class=\"maketag\"><mtext>(7.3)<\/mtext><\/mstyle><mspace class=\"nbsp\" width=\"0.33em\" \/> <\/mtd><\/mtr><\/mtable><\/math> <\/div> <p class=\"indent\">Informell ausgedr\u00fcckt kann man schreiben <\/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><munderover accent=\"false\" accentunder=\"false\"><mrow><mo> \u2211<\/mo> <\/mrow><mrow><mi>m<\/mi><mo class=\"MathClass-rel\">=<\/mo><mn>1<\/mn><\/mrow><mrow><mi>\u221e<\/mi><\/mrow><\/munderover><msub><mrow><mi>a<\/mi><\/mrow><mrow> <mi>m<\/mi><\/mrow><\/msub><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>1<\/mn><\/mrow><mrow><mi>\u221e<\/mi><\/mrow><\/munderover><msub><mrow><mi>b<\/mi><\/mrow><mrow> <mi>n<\/mi><\/mrow><\/msub><mstyle><mrow><mo fence=\"true\" form=\"prefix\"> )<\/mo><mrow \/><mo fence=\"true\" form=\"postfix\" \/><\/mrow><\/mstyle> <mo class=\"MathClass-rel\">=<\/mo><munderover accent=\"false\" accentunder=\"false\"><mrow><mo> \u2211<\/mo> <\/mrow><mrow><mi>m<\/mi><mo class=\"MathClass-rel\">=<\/mo><mn>1<\/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>n<\/mi><mo class=\"MathClass-rel\">=<\/mo><mn>1<\/mn><\/mrow><mrow><mi>\u221e<\/mi><\/mrow><\/munderover><msub><mrow><mi>b<\/mi><\/mrow><mrow> <mi>n<\/mi><\/mrow><\/msub><mstyle><mrow><mo fence=\"true\" form=\"prefix\"> )<\/mo><mrow \/><mo fence=\"true\" form=\"postfix\" \/><\/mrow><\/mstyle><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>m<\/mi><\/mrow><\/msub> <mo class=\"MathClass-rel\">=<\/mo><munder class=\"msub\"><mrow><mo> \u2211<\/mo> <\/mrow><mrow><mo class=\"MathClass-open\">(<\/mo><mi>m<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>n<\/mi><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-rel\">\u2208<\/mo><msup><mrow><mi>\u2115<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><\/mrow><\/munder><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>m<\/mi><\/mrow><\/msub><msub><mrow><mi>b<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><mo class=\"MathClass-punc\">.<\/mo><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">Die beiden (internen) Indices <math display=\"inline\"><mo class=\"MathClass-open\">(<\/mo><mi>m<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>n<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math> w\u00fcrde man nun gerne anders ausdr\u00fccken, damit aus der Doppelsumme auf der rechten Seite (die wir eigentlich nicht definiert haben) eine einfache Summe wird. W\u00e4hlt man eine Bijektion <span class=\"maperiod\"><math display=\"inline\"><mi>\u03c6<\/mi> <mo class=\"MathClass-punc\">:<\/mo> <mi>\u2115<\/mi> <mo class=\"MathClass-rel\">\u2192<\/mo> <mi>\u2115<\/mi> <mo class=\"MathClass-bin\">\u00d7<\/mo> <mi>\u2115<\/mi><\/math><\/span><span class=\"period\">,<\/span> so durchl\u00e4uft <math display=\"inline\"><mi>\u03c6<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>k<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math> alle <math display=\"inline\"><mo class=\"MathClass-open\">(<\/mo><mi>m<\/mi><mo class=\"MathClass-punc\">,<\/mo> <mi>n<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math> und somit wird aus der Doppelsumme eine einfache Summe <span class=\"maperiod\"><math display=\"inline\"><msubsup><mrow><mi class=\"MathClass-op\">\u2211<\/mi><mo> <\/mo> <\/mrow><mrow><mi>k<\/mi><mo class=\"MathClass-rel\">=<\/mo><mn>1<\/mn><\/mrow><mrow><mi>\u221e<\/mi><\/mrow><\/msubsup><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>\u03c6<\/mi><msub><mrow><mo class=\"MathClass-open\">(<\/mo><mi>k<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><\/mrow><\/msub><msub><mrow><mi>b<\/mi><\/mrow><mrow><mi>\u03c6<\/mi><msub><mrow><mo class=\"MathClass-open\">(<\/mo><mi>k<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><\/mrow><\/msub><\/math><\/span><span class=\"period\">.<\/span> Satz <a href=\"..\/..\/chapter\/absolute-konvergenz#x1-195001r36\">7.36<\/a> besagt nun, dass diese Reihe effektiv konvergiert und gleich dem gew\u00fcnschten Produkt ist. <\/p><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 w\u00e4hlen zuerst die Bijektion <math display=\"inline\"><mi>\u03c6<\/mi> <mo class=\"MathClass-punc\">:<\/mo> <mi>\u2115<\/mi> <mo class=\"MathClass-rel\">\u2192<\/mo> <msup><mrow><mi>\u2115<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><\/math> so dass                                                                                                                                                                           <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mo class=\"MathClass-open\">{<\/mo><mi>\u03c6<\/mi><mo class=\"MathClass-open\">(<\/mo><mn>1<\/mn><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-punc\">,<\/mo><mi>\u03c6<\/mi><mo class=\"MathClass-open\">(<\/mo><mn>2<\/mn><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-punc\">,<\/mo><mi class=\"MathClass-op\">\u2026<\/mi><mo> <\/mo><mo class=\"MathClass-punc\">,<\/mo><mi>\u03c6<\/mi><mo class=\"MathClass-open\">(<\/mo><msup><mrow><mi>n<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-close\">}<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <mo class=\"MathClass-open\">{<\/mo><mn>1<\/mn><mo class=\"MathClass-punc\">,<\/mo><mn>2<\/mn><mo class=\"MathClass-punc\">,<\/mo><mi class=\"MathClass-op\">\u2026<\/mi><mo> <\/mo><mo class=\"MathClass-punc\">,<\/mo><mi>n<\/mi><mo class=\"MathClass-close\">}<\/mo><mo class=\"MathClass-bin\">\u00d7<\/mo><mo class=\"MathClass-open\">{<\/mo><mn>1<\/mn><mo class=\"MathClass-punc\">,<\/mo><mn>2<\/mn><mo class=\"MathClass-punc\">,<\/mo><mi class=\"MathClass-op\">\u2026<\/mi><mo> <\/mo><mo class=\"MathClass-punc\">,<\/mo><mi>n<\/mi><mo class=\"MathClass-close\">}<\/mo><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">f\u00fcr alle <span class=\"maperiod\"><math display=\"inline\"><mi>n<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2115<\/mi><\/math><\/span><span class=\"period\">.<\/span> Zum Beispiel k\u00f6nnte <math display=\"inline\"><mi>\u03c6<\/mi><\/math> wie im folgenden Bild definiert sein.<button class=\"hover-trigger\" style=\"vertical-align: super;font: smaller\">\u2020<\/button><span class=\"hover-text\"><span class=\"marginpar\">\u2020 Wir verwenden hier das Bild um uns eine sonst eher langweilige formale Definition der Abbildung <math display=\"inline\"><mi>\u03c6<\/mi><\/math> zu ersparen. Sie sollten sich aber davon \u00fcberzeugen, dass man diese Definition durchaus formal machen kann oder einfach formal mittels Induktion nach <math display=\"inline\"><mi>n<\/mi><\/math> die Existenz einer Bijektion mit der gew\u00fcnschten Eigenschaft zeigen kann. Wir verwenden hier also das Bild als eine Abk\u00fcrzung f\u00fcr einen formalen Beweis und nicht als einen Ersatz.<\/span><\/span> <\/p> <div class=\"center\"> <div class=\"wp-nocaption \"><\/div><div class=\"wp-nocaption \"><\/div><div class=\"mefigcentered\" id=\"wpsize=326&amp;url=Pictures\/reihen\/prodsatz2.pdf\"><img decoding=\"async\" id=\"zfc9887eb0071\" alt=\"PIC\" src=\"https:\/\/people.math.ethz.ch\/~einsiedl\/Pictures\/reihen\/prodsatz2.svg\" width=\"326\" \/><\/div>  <\/div> <p class=\"indent\">F\u00fcr jedes <math display=\"inline\"><mi>n<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2115<\/mi><\/math> gilt dann f\u00fcr die Partialsumme bis <math display=\"inline\"><msup><mrow><mi>n<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><\/math> der gliedweise multiplizierten Reihe der Absolutbetr\u00e4ge das (endliche verallgemeinerte) Distributivgesetz <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><munderover accent=\"false\" accentunder=\"false\"><mrow><mo> \u2211<\/mo> <\/mrow><mrow><mi>k<\/mi><mo class=\"MathClass-rel\">=<\/mo><mn>1<\/mn><\/mrow><mrow><msup><mrow><mi>n<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup> <\/mrow><\/munderover><mo class=\"MathClass-rel\">|<\/mo><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>\u03c6<\/mi><msub><mrow><mo class=\"MathClass-open\">(<\/mo><mi>k<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><\/mrow><\/msub><mo class=\"MathClass-rel\">|<\/mo><mo class=\"MathClass-rel\">|<\/mo><msub><mrow><mi>b<\/mi><\/mrow><mrow><mi>\u03c6<\/mi><msub><mrow><mo class=\"MathClass-open\">(<\/mo><mi>k<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><\/mrow><\/msub><mo class=\"MathClass-rel\">|<\/mo> <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>\u2113<\/mi><mo class=\"MathClass-rel\">=<\/mo><mn>1<\/mn><\/mrow><mrow><mi>n<\/mi><\/mrow><\/munderover><mo class=\"MathClass-rel\">|<\/mo><msub><mrow><mi>a<\/mi><\/mrow><mrow> <mi>\u2113<\/mi><\/mrow><\/msub><mo class=\"MathClass-rel\">|<\/mo><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><munderover accent=\"false\" accentunder=\"false\"><mrow><mo>\u2211<\/mo> <\/mrow><mrow><mi>m<\/mi><mo class=\"MathClass-rel\">=<\/mo><mn>1<\/mn><\/mrow><mrow><mi>n<\/mi><\/mrow><\/munderover><mo class=\"MathClass-rel\">|<\/mo><msub><mrow><mi>b<\/mi><\/mrow><mrow> <mi>m<\/mi><\/mrow><\/msub><mo class=\"MathClass-rel\">|<\/mo><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><mo class=\"MathClass-punc\">.<\/mo><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">Insbesondere folgt also                                                                                                                                                                           <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><munderover accent=\"false\" accentunder=\"false\"><mrow><mo> \u2211<\/mo> <\/mrow><mrow><mi>k<\/mi><mo class=\"MathClass-rel\">=<\/mo><mn>1<\/mn><\/mrow><mrow><msup><mrow><mi>n<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup> <\/mrow><\/munderover><mo class=\"MathClass-rel\">|<\/mo><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>\u03c6<\/mi><msub><mrow><mo class=\"MathClass-open\">(<\/mo><mi>k<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><\/mrow><\/msub><mo class=\"MathClass-rel\">|<\/mo><mo class=\"MathClass-rel\">|<\/mo><msub><mrow><mi>b<\/mi><\/mrow><mrow><mi>\u03c6<\/mi><msub><mrow><mo class=\"MathClass-open\">(<\/mo><mi>k<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><\/mrow><\/msub><mo class=\"MathClass-rel\">|<\/mo><mo class=\"MathClass-rel\">\u2264<\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><munderover accent=\"false\" accentunder=\"false\"><mrow><mo>\u2211<\/mo> <\/mrow><mrow><mi>\u2113<\/mi><mo class=\"MathClass-rel\">=<\/mo><mn>1<\/mn><\/mrow><mrow><mi>\u221e<\/mi><\/mrow><\/munderover><mo class=\"MathClass-rel\">|<\/mo><msub><mrow><mi>a<\/mi><\/mrow><mrow> <mi>\u2113<\/mi><\/mrow><\/msub><mo class=\"MathClass-rel\">|<\/mo><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><munderover accent=\"false\" accentunder=\"false\"><mrow><mo>\u2211<\/mo> <\/mrow><mrow><mi>m<\/mi><mo class=\"MathClass-rel\">=<\/mo><mn>1<\/mn><\/mrow><mrow><mi>\u221e<\/mi><\/mrow><\/munderover><mo class=\"MathClass-rel\">|<\/mo><msub><mrow><mi>b<\/mi><\/mrow><mrow> <mi>m<\/mi><\/mrow><\/msub><mo class=\"MathClass-rel\">|<\/mo><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">f\u00fcr alle <span class=\"maperiod\"><math display=\"inline\"><mi>n<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2115<\/mi><\/math><\/span><span class=\"period\">.<\/span> Da aber f\u00fcr eine Reihe mit nicht-negativen Termen die Folge die Partialsummen monoton wachsend sind, folgt daraus dass die Reihe <math display=\"inline\"><msubsup><mrow><mi class=\"MathClass-op\"> \u2211<\/mi><mo> <\/mo> <\/mrow><mrow><mi>k<\/mi><mo class=\"MathClass-rel\">=<\/mo><mn>1<\/mn><\/mrow><mrow><mi>\u221e<\/mi><\/mrow><\/msubsup><mo class=\"MathClass-rel\">|<\/mo><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>\u03c6<\/mi><msub><mrow><mo class=\"MathClass-open\">(<\/mo><mi>k<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><\/mrow><\/msub><mo class=\"MathClass-rel\">|<\/mo><mo class=\"MathClass-rel\">|<\/mo><msub><mrow><mi>b<\/mi><\/mrow><mrow><mi>\u03c6<\/mi><msub><mrow><mo class=\"MathClass-open\">(<\/mo><mi>k<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><\/mrow><\/msub><mo class=\"MathClass-rel\">|<\/mo><\/math> konvergiert und damit die Reihe <math display=\"inline\"><msubsup><mrow><mi class=\"MathClass-op\"> \u2211<\/mi><mo> <\/mo> <\/mrow><mrow><mi>k<\/mi><mo class=\"MathClass-rel\">=<\/mo><mn>1<\/mn><\/mrow><mrow><mi>\u221e<\/mi><\/mrow><\/msubsup><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>\u03c6<\/mi><msub><mrow><mo class=\"MathClass-open\">(<\/mo><mi>k<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><\/mrow><\/msub><msub><mrow><mi>b<\/mi><\/mrow><mrow><mi>\u03c6<\/mi><msub><mrow><mo class=\"MathClass-open\">(<\/mo><mi>k<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><\/mrow><\/msub><\/math> absolut konvergent ist. <\/p><p class=\"indent\">Obiges Distributivgesetz gilt auch in der Form <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><munderover accent=\"false\" accentunder=\"false\"><mrow><mo> \u2211<\/mo> <\/mrow><mrow><mi>k<\/mi><mo class=\"MathClass-rel\">=<\/mo><mn>1<\/mn><\/mrow><mrow><msup><mrow><mi>n<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup> <\/mrow><\/munderover><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>\u03c6<\/mi><msub><mrow><mo class=\"MathClass-open\">(<\/mo><mi>k<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><\/mrow><\/msub><msub><mrow><mi>b<\/mi><\/mrow><mrow><mi>\u03c6<\/mi><msub><mrow><mo class=\"MathClass-open\">(<\/mo><mi>k<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><\/mrow><\/msub> <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>\u2113<\/mi><mo class=\"MathClass-rel\">=<\/mo><mn>1<\/mn><\/mrow><mrow><mi>n<\/mi><\/mrow><\/munderover><msub><mrow><mi>a<\/mi><\/mrow><mrow> <mi>\u2113<\/mi><\/mrow><\/msub><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><munderover accent=\"false\" accentunder=\"false\"><mrow><mo>\u2211<\/mo> <\/mrow><mrow><mi>m<\/mi><mo class=\"MathClass-rel\">=<\/mo><mn>1<\/mn><\/mrow><mrow><mi>n<\/mi><\/mrow><\/munderover><msub><mrow><mi>b<\/mi><\/mrow><mrow> <mi>m<\/mi><\/mrow><\/msub><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-punc\">.<\/mo><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">f\u00fcr alle <span class=\"maperiod\"><math display=\"inline\"><mi>n<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2115<\/mi><\/math><\/span><span class=\"period\">.<\/span> Mit Hilfe des Grenzwert\u00fcbergangs <math display=\"inline\"><mi>n<\/mi> <mo class=\"MathClass-rel\">\u2192<\/mo><mi>\u221e<\/mi><\/math> erhalten wir daraus                                                                                                                                                                           <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><munderover accent=\"false\" accentunder=\"false\"><mrow><mo> \u2211<\/mo> <\/mrow><mrow><mi>k<\/mi><mo class=\"MathClass-rel\">=<\/mo><mn>1<\/mn><\/mrow><mrow><mi>\u221e<\/mi><\/mrow><\/munderover><msub><mrow><mi>a<\/mi><\/mrow><mrow> <mi>\u03c6<\/mi><msub><mrow><mo class=\"MathClass-open\">(<\/mo><mi>k<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><\/mrow><\/msub><msub><mrow><mi>b<\/mi><\/mrow><mrow><mi>\u03c6<\/mi><msub><mrow><mo class=\"MathClass-open\">(<\/mo><mi>k<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><\/mrow><\/msub> <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>\u2113<\/mi><mo class=\"MathClass-rel\">=<\/mo><mn>1<\/mn><\/mrow><mrow><mi>\u221e<\/mi><\/mrow><\/munderover><msub><mrow><mi>a<\/mi><\/mrow><mrow> <mi>\u2113<\/mi><\/mrow><\/msub><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><munderover accent=\"false\" accentunder=\"false\"><mrow><mo>\u2211<\/mo> <\/mrow><mrow><mi>m<\/mi><mo class=\"MathClass-rel\">=<\/mo><mn>1<\/mn><\/mrow><mrow><mi>\u221e<\/mi><\/mrow><\/munderover><msub><mrow><mi>b<\/mi><\/mrow><mrow> <mi>m<\/mi><\/mrow><\/msub><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-punc\">.<\/mo><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">Betrachten wir eine beliebige Bijektion <span class=\"maperiod\"><math display=\"inline\"><mi>\u03c8<\/mi> <mo class=\"MathClass-punc\">:<\/mo> <mi>\u2115<\/mi> <mo class=\"MathClass-rel\">\u2192<\/mo> <msup><mrow><mi>\u2115<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><\/math><\/span><span class=\"period\">,<\/span> so ist <math display=\"inline\"><msup><mrow><mi>\u03c6<\/mi><\/mrow><mrow><mo class=\"MathClass-bin\">\u2212<\/mo><mn>1<\/mn> <\/mrow> <\/msup> <mo class=\"MathClass-bin\">\u2218<\/mo> <mi>\u03c8<\/mi> <mo class=\"MathClass-punc\">:<\/mo> <mi>\u2115<\/mi> <mo class=\"MathClass-rel\">\u2192<\/mo> <mi>\u2115<\/mi><\/math> eine Bijektion und die Formel <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><munderover accent=\"false\" accentunder=\"false\"><mrow><mo> \u2211<\/mo> <\/mrow><mrow><mi>k<\/mi><mo class=\"MathClass-rel\">=<\/mo><mn>1<\/mn><\/mrow><mrow><mi>\u221e<\/mi><\/mrow><\/munderover><msub><mrow><mi>a<\/mi><\/mrow><mrow> <mi>\u03c8<\/mi><msub><mrow><mo class=\"MathClass-open\">(<\/mo><mi>k<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><\/mrow><\/msub><msub><mrow><mi>b<\/mi><\/mrow><mrow><mi>\u03c8<\/mi><msub><mrow><mo class=\"MathClass-open\">(<\/mo><mi>k<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><\/mrow><\/msub> <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>\u2113<\/mi><mo class=\"MathClass-rel\">=<\/mo><mn>1<\/mn><\/mrow><mrow><mi>\u221e<\/mi><\/mrow><\/munderover><msub><mrow><mi>a<\/mi><\/mrow><mrow> <mi>\u2113<\/mi><\/mrow><\/msub><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><munderover accent=\"false\" accentunder=\"false\"><mrow><mo>\u2211<\/mo> <\/mrow><mrow><mi>m<\/mi><mo class=\"MathClass-rel\">=<\/mo><mn>1<\/mn><\/mrow><mrow><mi>\u221e<\/mi><\/mrow><\/munderover><msub><mrow><mi>b<\/mi><\/mrow><mrow> <mi>m<\/mi><\/mrow><\/msub><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-punc\">.<\/mo><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">folgt aus obigem und dem Umordnungssatz (Satz <a href=\"..\/..\/chapter\/absolute-konvergenz#x1-194001r35\">7.35<\/a>). <span>&nbsp;&nbsp;<\/span><\/p><div class=\"qed\">\u25a0<\/div><\/details><\/div> <p class=\"indent\">Wie wir in Abschnitt <a href=\"..\/..\/chapter\/polynome#x1-810002\">3.2<\/a> gesehen haben, lassen sich Polynome mittels der Regel <\/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><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>a<\/mi><\/mrow><mrow> <mi>n<\/mi><\/mrow><\/msub><msup><mrow><mi>x<\/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>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><mstyle><mrow><mo fence=\"true\" form=\"prefix\"> )<\/mo><mrow \/><mo fence=\"true\" form=\"postfix\" \/><\/mrow><\/mstyle> <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><mn>2<\/mn><mi>N<\/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>x<\/mi><\/mrow><mrow><mi>k<\/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\">multiplizieren, wobei wir&nbsp;<math display=\"inline\"><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> <mo class=\"MathClass-rel\">\u22ef<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <msub><mrow><mi>a<\/mi><\/mrow><mrow><mn>2<\/mn><mi>N<\/mi><\/mrow><\/msub> <mo class=\"MathClass-rel\">=<\/mo> <msub><mrow><mi>b<\/mi><\/mrow><mrow><mi>N<\/mi><mo class=\"MathClass-bin\">+<\/mo><mn>1<\/mn><\/mrow><\/msub> <mo class=\"MathClass-rel\">=<\/mo> <mo class=\"MathClass-rel\">\u22ef<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <msub><mrow><mi>b<\/mi><\/mrow><mrow><mn>2<\/mn><mi>N<\/mi><\/mrow><\/msub> <mo class=\"MathClass-rel\">=<\/mo> <mn>0<\/mn><\/math> setzen. Setzt man <span class=\"maperiod\"><math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mn>1<\/mn><\/math><\/span><span class=\"period\">,<\/span> erh\u00e4lt man insbesondere                                                                                                                                                                           <\/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><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>a<\/mi><\/mrow><mrow> <mi>n<\/mi><\/mrow><\/msub><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>N<\/mi><\/mrow><\/munderover><msub><mrow><mi>b<\/mi><\/mrow><mrow> <mi>n<\/mi><\/mrow><\/msub><mstyle><mrow><mo fence=\"true\" form=\"prefix\"> )<\/mo><mrow \/><mo fence=\"true\" form=\"postfix\" \/><\/mrow><\/mstyle> <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><mn>2<\/mn><mi>N<\/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><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\">Diese Identit\u00e4t trifft, wie sich herausstellt, analog f\u00fcr Reihen zu, was wir im folgenden Korollar des Produktsatzes (Satz <a href=\"..\/..\/chapter\/absolute-konvergenz#x1-195001r36\">7.36<\/a>) festhalten wollen. <\/p> <div class=\"me metheorem\"> <div class=\"wp-nocaption \"><\/div><h4 id=\"z0a721819d754\"> <a id=\"x1-195003r37\"><\/a> <span class=\"ecbx-1095\">Korollar 7.37 <\/span>(Cauchy-Produkt)<span class=\"ecbx-1095\">.<\/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><\/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><\/math> <span class=\"ecti-1095\">absolut konvergente Reihen mit komplexen Gliedern sind, dann gilt<\/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><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> <mo class=\"MathClass-rel\">=<\/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>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><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><mstyle><mrow><mo fence=\"true\" form=\"prefix\"> )<\/mo><mrow \/><mo fence=\"true\" form=\"postfix\" \/><\/mrow><\/mstyle><mo class=\"MathClass-punc\">,<\/mo><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\"><span class=\"ecti-1095\">wobei 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><mstyle><mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow \/><mo fence=\"true\" form=\"postfix\" \/><\/mrow><\/mstyle><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>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><\/math> <span class=\"ecti-1095\">absolut konvergent ist.<\/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\">Dies folgt, indem wir die Abz\u00e4hlung <math display=\"inline\"><mi>\u03c6<\/mi><\/math> von <math display=\"inline\"><msub><mrow><mi>\u2115<\/mi><\/mrow><mrow><mn>0<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-bin\">\u00d7<\/mo> <msub><mrow><mi>\u2115<\/mi><\/mrow><mrow><mn>0<\/mn> <\/mrow> <\/msub> <\/math> aus dem Bild unten auf den Produktsatz (Satz <a href=\"..\/..\/chapter\/absolute-konvergenz#x1-195001r36\">7.36<\/a>) anwenden und dann Glieder zusammenfassen (Lemma <a href=\"..\/..\/chapter\/reihen#x1-187013r9\">7.9<\/a>). <\/p> <div class=\"center\"> <div class=\"wp-nocaption \"><\/div><div class=\"wp-nocaption \"><\/div><div class=\"mefigcentered\" id=\"wpsize=320&amp;url=Pictures\/reihen\/cauchyprodv2.pdf\"><img decoding=\"async\" id=\"zf598d69fa248\" alt=\"PIC\" src=\"https:\/\/people.math.ethz.ch\/~einsiedl\/Pictures\/reihen\/cauchyprodv2.svg\" width=\"320\" \/><\/div>  <\/div> <p class=\"noindent\">Die absolute Konvergenz folgt ebenso aus Satz&nbsp;<a href=\"..\/..\/chapter\/absolute-konvergenz#x1-195001r36\">7.36<\/a> und <\/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><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> <mo class=\"MathClass-rel\">\u2264<\/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><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><mo class=\"MathClass-rel\">|<\/mo><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><mo class=\"MathClass-rel\">|<\/mo> <mo class=\"MathClass-rel\">&lt;<\/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> <span>&nbsp;&nbsp;<\/span><div class=\"qed\">\u25a0<\/div><\/details><\/div> <div class=\"me meexample\"> <div class=\"wp-nocaption \"><\/div><h4 id=\"zf2b8b908320c\"> <a id=\"x1-195004r38\"><\/a> <span class=\"ecbx-1095\">Beispiel 7.38.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Sei <\/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\">. Dann<\/span> <span class=\"ecti-1095\">konvergiert <\/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\">absolut. Wenden wir das Cauchy-Produkt auf diese Reihe und sich selbst an, so erhalten<\/span> <span class=\"ecti-1095\">wir<\/span> <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"> <mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><msup><mrow><mo class=\"MathClass-open\">(<\/mo><mn>1<\/mn> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>q<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><\/mrow><\/mfrac><\/mtd> <mtd class=\"align-even\"> <mo class=\"MathClass-rel\">=<\/mo><msup><mrow> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/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><msup><mrow><mi>q<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msup><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <\/mrow><mrow><mn>2<\/mn><\/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><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><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>q<\/mi><\/mrow><mrow><mi>n<\/mi><mo class=\"MathClass-bin\">\u2212<\/mo><mi>k<\/mi><\/mrow><\/msup><msup><mrow><mi>q<\/mi><\/mrow><mrow><mi>k<\/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><mi>n<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mn>1<\/mn><mo class=\"MathClass-close\">)<\/mo><msup><mrow><mi>q<\/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\">Auf diese Weise erhalten wir auch eine Summenformel f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r<\/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><mi>n<\/mi><msup><mrow><mi>q<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msup> <mo class=\"MathClass-rel\">=<\/mo> <mi>q<\/mi><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><mi>n<\/mi><msup><mrow><mi>q<\/mi><\/mrow><mrow><mi>n<\/mi><mo class=\"MathClass-bin\">\u2212<\/mo><mn>1<\/mn><\/mrow><\/msup> <mo class=\"MathClass-rel\">=<\/mo> <mi>q<\/mi><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><mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>k<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mn>1<\/mn><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><msup><mrow><mi>q<\/mi><\/mrow><mrow><mi>k<\/mi><\/mrow><\/msup> <mo class=\"MathClass-rel\">=<\/mo> <mfrac><mrow><mi>q<\/mi><\/mrow> <mrow><msup><mrow><mo class=\"MathClass-open\">(<\/mo><mn>1<\/mn> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>q<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><\/mrow><\/mfrac><mo class=\"MathClass-punc\">,<\/mo><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\"><span class=\"ecti-1095\">wobei wir die Indexverschiebung <\/span><math display=\"inline\"><mi>k<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mi>n<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <mn>1<\/mn><\/math> <span class=\"ecti-1095\">durchgef<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">hrt haben.<\/span> <\/p> <\/div> <div class=\"me meexample\"> <div class=\"wp-nocaption \"><\/div><h4 id=\"z575fdb60534c\"> <a id=\"x1-195005r39\"><\/a> <span class=\"ecbx-1095\">\u00dc<\/span><span class=\"ecbx-1095\">bung 7.39.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Formal l<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">sst sich auch f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r bedingt konvergente Reihen<\/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><\/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><\/math> <span class=\"ecti-1095\">das<\/span> <span class=\"ecti-1095\">Cauchy-Produkt <\/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><mstyle><mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow \/><mo fence=\"true\" form=\"postfix\" \/><\/mrow><\/mstyle><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>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><\/math> <span class=\"ecti-1095\">bilden. Es muss jedoch nicht mehr konvergent sein: Zeigen Sie, dass das Cauchy-Produkt der bedingt<\/span> <span class=\"ecti-1095\">konvergenten Reihe<\/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><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><msqrt><mrow><mi>n<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mn>1<\/mn><\/mrow><\/msqrt><\/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\">mit sich selbst divergiert.<\/span> <\/p> <\/div> <a id=\"x1-195006r192\"><\/a> \n","protected":false},"author":1089,"menu_order":2,"template":"","meta":{"pb_show_title":"","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-77","chapter","type-chapter","status-publish","hentry"],"part":75,"_links":{"self":[{"href":"https:\/\/wp-prd.let.ethz.ch\/analysis19\/wp-json\/pressbooks\/v2\/chapters\/77","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/wp-prd.let.ethz.ch\/analysis19\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/wp-prd.let.ethz.ch\/analysis19\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/wp-prd.let.ethz.ch\/analysis19\/wp-json\/wp\/v2\/users\/1089"}],"version-history":[{"count":0,"href":"https:\/\/wp-prd.let.ethz.ch\/analysis19\/wp-json\/pressbooks\/v2\/chapters\/77\/revisions"}],"part":[{"href":"https:\/\/wp-prd.let.ethz.ch\/analysis19\/wp-json\/pressbooks\/v2\/parts\/75"}],"metadata":[{"href":"https:\/\/wp-prd.let.ethz.ch\/analysis19\/wp-json\/pressbooks\/v2\/chapters\/77\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/wp-prd.let.ethz.ch\/analysis19\/wp-json\/wp\/v2\/media?parent=77"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/wp-prd.let.ethz.ch\/analysis19\/wp-json\/pressbooks\/v2\/chapter-type?post=77"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/wp-prd.let.ethz.ch\/analysis19\/wp-json\/wp\/v2\/contributor?post=77"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/wp-prd.let.ethz.ch\/analysis19\/wp-json\/wp\/v2\/license?post=77"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}