{"id":98,"date":"2021-12-15T09:53:29","date_gmt":"2021-12-15T09:53:29","guid":{"rendered":"https:\/\/wp-prd.let.ethz.ch\/analysis19\/chapter\/drei-asymptotische-formeln\/"},"modified":"2021-12-15T09:53:29","modified_gmt":"2021-12-15T09:53:29","slug":"drei-asymptotische-formeln","status":"publish","type":"chapter","link":"https:\/\/wp-prd.let.ethz.ch\/analysis19\/chapter\/drei-asymptotische-formeln\/","title":{"raw":"Drei asymptotische Formeln","rendered":"Drei asymptotische Formeln"},"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: 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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=\"ze5d76fd3a8ac\" class=\"sectionHead\"><span class=\"titlemark\">9.6 <\/span> <a id=\"x1-2810006\"><\/a>Drei asymptotische Formeln<\/h3> <a id=\"x1-281001r280\"><\/a> <h4 id=\"z003f5dd8e139\" class=\"subsectionHead\"><span class=\"titlemark\">9.6.1 <\/span> <a id=\"x1-2820001\"><\/a>Das Wallissche Produkt<\/h4> <p class=\"noindent\">Die Formel <\/p><math display=\"block\"><mtable class=\"align\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mi>\u03c0<\/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><mn>1<\/mn><\/mrow> <mrow><mi>n<\/mi><\/mrow><\/mfrac> <mfrac><mrow><msup><mrow><mo class=\"MathClass-open\">(<\/mo><msup><mrow><mn>2<\/mn><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msup><mi>n<\/mi><mo class=\"MathClass-punc\">!<\/mo><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mn>4<\/mn><\/mrow><\/msup><\/mrow> <mrow><msup><mrow><mo class=\"MathClass-open\">(<\/mo><mo class=\"MathClass-open\">(<\/mo><mn>2<\/mn><mi>n<\/mi><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-punc\">!<\/mo><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><\/mrow><\/mfrac><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"><mstyle class=\"label\" id=\"x1-282001r17\" \/><mstyle class=\"maketag\"><mtext>(9.17)<\/mtext><\/mstyle><mspace class=\"nbsp\" width=\"0.33em\" \/> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">nach John Wallis (1616-1703) wird das <span class=\"ecbx-1095\">Wallissche Produkt <\/span>genannt. <\/p><p class=\"indent\">Um Gleichung (<a href=\"..\/..\/chapter\/drei-asymptotische-formeln#x1-282001r17\">9.17<\/a>) zu beweisen, wollen wir zuerst f\u00fcr jedes <math display=\"inline\"><mi>k<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <msub><mrow><mi>\u2115<\/mi><\/mrow><mrow><mn>0<\/mn> <\/mrow> <\/msub> <\/math> das Integral <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><msub><mrow><mi>I<\/mi><\/mrow><mrow><mi>k<\/mi><\/mrow><\/msub> <mo class=\"MathClass-rel\">=<\/mo><msubsup><mrow><mo> \u222b  <\/mo><\/mrow><mrow><mn>0<\/mn><\/mrow><mrow><mi>\u03c0<\/mi><\/mrow><\/msubsup><msup><mrow><mi class=\"qopname\"> sin<\/mi><mo>  <\/mo><\/mrow><mrow><mi>k<\/mi><\/mrow><\/msup> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">mit vollst\u00e4ndiger Induktion berechnen. <\/p><p class=\"indent\">F\u00fcr <math display=\"inline\"><mi>k<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mn>0<\/mn><\/math> gilt <math display=\"inline\"><msub><mrow><mi>I<\/mi><\/mrow><mrow><mn>0<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-rel\">=<\/mo> <mi>\u03c0<\/mi><\/math> und f\u00fcr <math display=\"inline\"><mi>k<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mn>1<\/mn><\/math> gilt <span class=\"maperiod\"><math display=\"inline\"><msub><mrow><mi>I<\/mi><\/mrow><mrow><mn>1<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-rel\">=<\/mo><msubsup><mrow><mi class=\"MathClass-op\"> \u222b  <\/mi><mo> <\/mo><\/mrow><mrow><mn>0<\/mn><\/mrow><mrow><mi>\u03c0<\/mi><\/mrow><\/msubsup><mi class=\"qopname\"> sin<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mn>2<\/mn><\/math><\/span><span class=\"period\">.<\/span> Mit partieller Integration                                                                                                                                                                           erhalten wir f\u00fcr alle <math display=\"inline\"><mi>k<\/mi> <mo class=\"MathClass-rel\">\u2265<\/mo> <mn>2<\/mn><\/math> <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><msub><mrow><mi>I<\/mi><\/mrow><mrow><mi>k<\/mi><\/mrow><\/msub><\/mtd> <mtd class=\"align-even\"> <mo class=\"MathClass-rel\">=<\/mo><msubsup><mrow><mo> \u222b  <\/mo><\/mrow><mrow><mn>0<\/mn><\/mrow><mrow><mi>\u03c0<\/mi><\/mrow><\/msubsup><msup><mrow><mi class=\"qopname\"> sin<\/mi><mo>  <\/mo><\/mrow><mrow><mi>k<\/mi><\/mrow><\/msup> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi> <mo class=\"MathClass-rel\">=<\/mo><msubsup><mrow><mo> \u222b  <\/mo><\/mrow><mrow><mn>0<\/mn><\/mrow><mrow><mi>\u03c0<\/mi><\/mrow><\/msubsup><msup><mrow><mi class=\"qopname\"> sin<\/mi><mo>  <\/mo><\/mrow><mrow><mi>k<\/mi><mo class=\"MathClass-bin\">\u2212<\/mo><mn>1<\/mn><\/mrow><\/msup> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><mi class=\"qopname\">sin<\/mi><mo>  <\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\" \/> <mtd class=\"align-label\"> <mspace width=\"2em\" \/><\/mtd><\/mtr><mtr><mtd class=\"align-odd\" columnalign=\"right\" \/> <mtd class=\"align-even\"> <mo class=\"MathClass-rel\">=<\/mo><msubsup><mrow> <mrow><mo fence=\"true\" form=\"prefix\"> [<\/mo><mrow><msup><mrow><mi class=\"qopname\">sin<\/mi><mo>  <\/mo><\/mrow><mrow><mi>k<\/mi><mo class=\"MathClass-bin\">\u2212<\/mo><mn>1<\/mn><\/mrow><\/msup> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><mstyle><mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow \/><mo fence=\"true\" form=\"postfix\" \/><\/mrow><\/mstyle> <mo class=\"MathClass-bin\">\u2212<\/mo><mi class=\"qopname\"> cos<\/mi><mo>  <\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><mstyle><mrow><mo fence=\"true\" form=\"prefix\"> )<\/mo><mrow \/><mo fence=\"true\" form=\"postfix\" \/><\/mrow><\/mstyle><\/mrow><mo fence=\"true\" form=\"postfix\">]<\/mo><\/mrow><\/mrow><mrow> <mn>0<\/mn><\/mrow><mrow><mi>\u03c0<\/mi><\/mrow><\/msubsup> <mo class=\"MathClass-bin\">+<\/mo><msubsup><mrow><mo> \u222b  <\/mo><\/mrow><mrow><mn>0<\/mn><\/mrow><mrow><mi>\u03c0<\/mi><\/mrow><\/msubsup> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>k<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <mn>1<\/mn><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><msup><mrow><mi class=\"qopname\">sin<\/mi><mo>  <\/mo><\/mrow><mrow><mi>k<\/mi><mo class=\"MathClass-bin\">\u2212<\/mo><mn>2<\/mn><\/mrow><\/msup> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><msup><mrow><mi class=\"qopname\">cos<\/mi><mo>  <\/mo><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\" \/> <mtd class=\"align-label\"> <mspace width=\"2em\" \/><\/mtd><\/mtr><mtr><mtd class=\"align-odd\" columnalign=\"right\" \/> <mtd class=\"align-even\"> <mo class=\"MathClass-rel\">=<\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>k<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <mn>1<\/mn><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><msubsup><mrow><mo>\u222b  <\/mo><\/mrow><mrow><mn>0<\/mn><\/mrow><mrow><mi>\u03c0<\/mi><\/mrow><\/msubsup><msup><mrow><mi class=\"qopname\"> sin<\/mi><mo>  <\/mo><\/mrow><mrow><mi>k<\/mi><mo class=\"MathClass-bin\">\u2212<\/mo><mn>2<\/mn><\/mrow><\/msup> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mn>1<\/mn> <mo class=\"MathClass-bin\">\u2212<\/mo><msup><mrow><mi class=\"qopname\"> sin<\/mi><mo>  <\/mo><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\" \/> <mtd class=\"align-label\"> <mspace width=\"2em\" \/><\/mtd><\/mtr><mtr><mtd class=\"align-odd\" columnalign=\"right\" \/> <mtd class=\"align-even\"> <mo class=\"MathClass-rel\">=<\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>k<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <mn>1<\/mn><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><msub><mrow><mi>I<\/mi><\/mrow><mrow><mi>k<\/mi><mo class=\"MathClass-bin\">\u2212<\/mo><mn>2<\/mn><\/mrow><\/msub> <mo class=\"MathClass-bin\">\u2212<\/mo> <msub><mrow><mi>I<\/mi><\/mrow><mrow><mi>k<\/mi><\/mrow><\/msub><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-punc\">,<\/mo><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\" \/> <mtd class=\"align-label\"> <mspace width=\"2em\" \/><\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">woraus sich die Rekursionsformel <\/p><table id=\"z030660109524\" class=\"equation\"><tr><td> <a id=\"x1-282002r18\"><\/a> <math class=\"equation\" display=\"block\"> <msub><mrow><mi>I<\/mi><\/mrow><mrow><mi>k<\/mi><\/mrow><\/msub> <mo class=\"MathClass-rel\">=<\/mo> <mfrac><mrow><mi>k<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <mn>1<\/mn><\/mrow> <mrow><mi>k<\/mi><\/mrow><\/mfrac> <msub><mrow><mi>I<\/mi><\/mrow><mrow><mi>k<\/mi><mo class=\"MathClass-bin\">\u2212<\/mo><mn>2<\/mn><\/mrow><\/msub> <\/math><\/td><td class=\"eq-no\">(9.18)<\/td><\/tr><\/table> <p class=\"indent\">f\u00fcr jedes ganze <math display=\"inline\"><mi>k<\/mi> <mo class=\"MathClass-rel\">\u2265<\/mo> <mn>2<\/mn><\/math> ergibt. <\/p><p class=\"indent\">Somit ist f\u00fcr <math display=\"inline\"><mi>n<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2115<\/mi><\/math> <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><msub><mrow><mi>I<\/mi><\/mrow><mrow><mn>2<\/mn><mi>n<\/mi><\/mrow><\/msub> <mo class=\"MathClass-rel\">=<\/mo> <mfrac><mrow><mn>2<\/mn><mi>n<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <mn>1<\/mn><\/mrow> <mrow><mn>2<\/mn><mi>n<\/mi><\/mrow><\/mfrac> <msub><mrow><mi>I<\/mi><\/mrow><mrow><mn>2<\/mn><mi>n<\/mi><mo class=\"MathClass-bin\">\u2212<\/mo><mn>2<\/mn><\/mrow><\/msub><\/mtd> <mtd class=\"align-even\"> <mo class=\"MathClass-rel\">=<\/mo> <mfrac><mrow><mn>2<\/mn><mi>n<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <mn>1<\/mn><\/mrow> <mrow><mn>2<\/mn><mi>n<\/mi><\/mrow><\/mfrac> <mo class=\"MathClass-bin\">\u22c5<\/mo><mfrac><mrow><mn>2<\/mn><mi>n<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <mn>3<\/mn><\/mrow> <mrow><mn>2<\/mn><mi>n<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <mn>2<\/mn><\/mrow><\/mfrac><mo class=\"MathClass-rel\">\u22ef<\/mo><mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mn>2<\/mn><\/mrow><\/mfrac><msub><mrow><mi>I<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><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> <mfrac><mrow><mo class=\"MathClass-open\">(<\/mo><mn>2<\/mn><mi>n<\/mi><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-open\">(<\/mo><mn>2<\/mn><mi>n<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <mn>1<\/mn><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-open\">(<\/mo><mn>2<\/mn><mi>n<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <mn>2<\/mn><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-rel\">\u22ef<\/mo><mn>1<\/mn><\/mrow> <mrow><mstyle><mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow \/><mo fence=\"true\" form=\"postfix\" \/><\/mrow><\/mstyle><mo class=\"MathClass-open\">(<\/mo><mn>2<\/mn><mi>n<\/mi><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-open\">(<\/mo><mn>2<\/mn><mi>n<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <mn>2<\/mn><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-rel\">\u22ef<\/mo><mn>2<\/mn><msup><mrow><mstyle><mrow><mo fence=\"true\" form=\"prefix\"> )<\/mo><mrow \/><mo fence=\"true\" form=\"postfix\" \/><\/mrow><\/mstyle><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><\/mrow><\/mfrac> <msub><mrow><mi>I<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub> <mo class=\"MathClass-rel\">=<\/mo> <mfrac><mrow><mo class=\"MathClass-open\">(<\/mo><mn>2<\/mn><mi>n<\/mi><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-punc\">!<\/mo><\/mrow> <mrow><msup><mrow><mo class=\"MathClass-open\">(<\/mo><msup><mrow><mn>2<\/mn><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msup><mi>n<\/mi><mo class=\"MathClass-punc\">!<\/mo><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><\/mrow><\/mfrac> <mo class=\"MathClass-bin\">\u22c5<\/mo> <mi>\u03c0<\/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 den Bruch mit&nbsp;<math display=\"inline\"><mn>2<\/mn><mi>n<\/mi><mo class=\"MathClass-open\">(<\/mo><mn>2<\/mn><mi>n<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <mn>2<\/mn><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-rel\">\u22ef<\/mo><mn>2<\/mn><\/math> erweitert haben um im Z\u00e4hler&nbsp;<math display=\"inline\"><mo class=\"MathClass-open\">(<\/mo><mn>2<\/mn><mi>n<\/mi><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-punc\">!<\/mo><\/math> zu erhalten. Auf dieselbe Weise ergibt sich <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><msub><mrow><mi>I<\/mi><\/mrow><mrow><mn>2<\/mn><mi>n<\/mi><mo class=\"MathClass-bin\">+<\/mo><mn>1<\/mn><\/mrow><\/msub> <mo class=\"MathClass-rel\">=<\/mo> <mfrac><mrow><mn>2<\/mn><mi>n<\/mi><\/mrow> <mrow><mn>2<\/mn><mi>n<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mn>1<\/mn><\/mrow><\/mfrac> <mo class=\"MathClass-bin\">\u22c5<\/mo><mfrac><mrow><mn>2<\/mn><mi>n<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <mn>2<\/mn><\/mrow> <mrow><mn>2<\/mn><mi>n<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <mn>1<\/mn><\/mrow><\/mfrac><mo class=\"MathClass-rel\">\u22ef<\/mo><mfrac><mrow><mn>2<\/mn><\/mrow> <mrow><mn>3<\/mn><\/mrow><\/mfrac><msub><mrow><mi>I<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub> <mo class=\"MathClass-rel\">=<\/mo> <mfrac><mrow><mstyle><mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow \/><mo fence=\"true\" form=\"postfix\" \/><\/mrow><\/mstyle><mo class=\"MathClass-open\">(<\/mo><mn>2<\/mn><mi>n<\/mi><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-open\">(<\/mo><mn>2<\/mn><mi>n<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <mn>2<\/mn><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-rel\">\u22ef<\/mo><mn>2<\/mn><msup><mrow><mstyle><mrow><mo fence=\"true\" form=\"prefix\"> )<\/mo><mrow \/><mo fence=\"true\" form=\"postfix\" \/><\/mrow><\/mstyle><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><\/mrow> <mrow><mo class=\"MathClass-open\">(<\/mo><mn>2<\/mn><mi>n<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mn>1<\/mn><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-open\">(<\/mo><mn>2<\/mn><mi>n<\/mi><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-rel\">\u22ef<\/mo><mn>1<\/mn><\/mrow><\/mfrac> <mo class=\"MathClass-bin\">\u22c5<\/mo> <mn>2<\/mn> <mo class=\"MathClass-rel\">=<\/mo> <mfrac><mrow><msup><mrow><mo class=\"MathClass-open\">(<\/mo><msup><mrow><mn>2<\/mn><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msup><mi>n<\/mi><mo class=\"MathClass-punc\">!<\/mo><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><\/mrow> <mrow><mo class=\"MathClass-open\">(<\/mo><mn>2<\/mn><mi>n<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mn>1<\/mn><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-punc\">!<\/mo><\/mrow><\/mfrac> <mo class=\"MathClass-bin\">\u22c5<\/mo> <mn>2<\/mn><mo class=\"MathClass-punc\">.<\/mo><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">Die Folge&nbsp;<math display=\"inline\"><msub><mrow><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>I<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><\/math> erf\u00fcllt noch eine weitere Eigenschaft: Denn f\u00fcr <math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> [<\/mo><mrow><mn>0<\/mn><mo class=\"MathClass-punc\">,<\/mo><mi>\u03c0<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">]<\/mo><\/mrow><\/math> ist <math display=\"inline\"><mi class=\"qopname\"> sin<\/mi><mo>  <\/mo> <mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">\u2208<\/mo> <mo class=\"MathClass-open\">[<\/mo><mn>0<\/mn><mo class=\"MathClass-punc\">,<\/mo><mn>1<\/mn><mo class=\"MathClass-close\">]<\/mo><\/math> und somit <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><msup><mrow><mi class=\"qopname\"> sin<\/mi><mo>  <\/mo><\/mrow><mrow><mn>2<\/mn><mi>n<\/mi><mo class=\"MathClass-bin\">+<\/mo><mn>2<\/mn><\/mrow><\/msup><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">\u2264<\/mo><msup><mrow><mi class=\"qopname\"> sin<\/mi><mo>  <\/mo><\/mrow><mrow><mn>2<\/mn><mi>n<\/mi><mo class=\"MathClass-bin\">+<\/mo><mn>1<\/mn><\/mrow><\/msup><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">\u2264<\/mo><msup><mrow><mi class=\"qopname\"> sin<\/mi><mo>  <\/mo><\/mrow><mrow><mn>2<\/mn><mi>n<\/mi><\/mrow><\/msup><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">f\u00fcr alle <math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mo class=\"MathClass-open\">[<\/mo><mn>0<\/mn><mo class=\"MathClass-punc\">,<\/mo><mi>\u03c0<\/mi><mo class=\"MathClass-close\">]<\/mo><\/math> and <span class=\"maperiod\"><math display=\"inline\"><mi>n<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2115<\/mi><\/math><\/span><span class=\"period\">.<\/span> Durch Integration folgt die Monotonieungleichung <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><msub><mrow><mi>I<\/mi><\/mrow><mrow><mn>2<\/mn><mi>n<\/mi><mo class=\"MathClass-bin\">+<\/mo><mn>2<\/mn><\/mrow><\/msub> <mo class=\"MathClass-rel\">=<\/mo><msubsup><mrow><mo> \u222b  <\/mo><\/mrow><mrow><mn>0<\/mn><\/mrow><mrow><mi>\u03c0<\/mi><\/mrow><\/msubsup><msup><mrow><mi class=\"qopname\"> sin<\/mi><mo>  <\/mo><\/mrow><mrow><mn>2<\/mn><mi>n<\/mi><mo class=\"MathClass-bin\">+<\/mo><mn>2<\/mn><\/mrow><\/msup><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2264<\/mo> <msub><mrow><mi>I<\/mi><\/mrow><mrow> <mn>2<\/mn><mi>n<\/mi><mo class=\"MathClass-bin\">+<\/mo><mn>1<\/mn><\/mrow><\/msub> <mo class=\"MathClass-rel\">=<\/mo><msubsup><mrow><mo> \u222b  <\/mo><\/mrow><mrow><mn>0<\/mn><\/mrow><mrow><mi>\u03c0<\/mi><\/mrow><\/msubsup><msup><mrow><mi class=\"qopname\"> sin<\/mi><mo>  <\/mo><\/mrow><mrow><mn>2<\/mn><mi>n<\/mi><mo class=\"MathClass-bin\">+<\/mo><mn>1<\/mn><\/mrow><\/msup><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2264<\/mo> <msub><mrow><mi>I<\/mi><\/mrow><mrow> <mn>2<\/mn><mi>n<\/mi><\/mrow><\/msub> <mo class=\"MathClass-rel\">=<\/mo><msubsup><mrow><mo> \u222b  <\/mo><\/mrow><mrow><mn>0<\/mn><\/mrow><mrow><mi>\u03c0<\/mi><\/mrow><\/msubsup><msup><mrow><mi class=\"qopname\"> sin<\/mi><mo>  <\/mo><\/mrow><mrow><mn>2<\/mn><mi>n<\/mi><\/mrow><\/msup><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">und mittels (<a href=\"..\/..\/chapter\/drei-asymptotische-formeln#x1-282002r18\">9.18<\/a>) auch <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><msub><mrow><mi>I<\/mi><\/mrow><mrow><mn>2<\/mn><mi>n<\/mi><mo class=\"MathClass-bin\">+<\/mo><mn>2<\/mn><\/mrow><\/msub> <mo class=\"MathClass-rel\">=<\/mo> <mfrac><mrow><mn>2<\/mn><mi>n<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mn>1<\/mn><\/mrow> <mrow><mn>2<\/mn><mi>n<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mn>2<\/mn><\/mrow><\/mfrac><munder class=\"msub\"><mrow><munder accentunder=\"false\"><mrow> <mfrac><mrow><mo class=\"MathClass-open\">(<\/mo><mn>2<\/mn><mi>n<\/mi><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-punc\">!<\/mo><\/mrow> <mrow><msup><mrow><mo class=\"MathClass-open\">(<\/mo><msup><mrow><mn>2<\/mn><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msup><mi>n<\/mi><mo class=\"MathClass-punc\">!<\/mo><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><\/mrow><\/mfrac><mi>\u03c0<\/mi><\/mrow><mo>\ufe38<\/mo><\/munder> <\/mrow><mrow><msub><mrow><mi>I<\/mi><\/mrow><mrow><mn>2<\/mn><mi>n<\/mi><\/mrow><\/msub><\/mrow><\/munder> <mo class=\"MathClass-rel\">\u2264<\/mo> <mfrac><mrow><msup><mrow><mo class=\"MathClass-open\">(<\/mo><msup><mrow><mn>2<\/mn><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msup><mi>n<\/mi><mo class=\"MathClass-punc\">!<\/mo><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><\/mrow> <mrow><mo class=\"MathClass-open\">(<\/mo><mn>2<\/mn><mi>n<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mn>1<\/mn><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-punc\">!<\/mo><\/mrow><\/mfrac><mn>2<\/mn> <mo class=\"MathClass-rel\">\u2264<\/mo> <msub><mrow><mi>I<\/mi><\/mrow><mrow><mn>2<\/mn><mi>n<\/mi><\/mrow><\/msub> <mo class=\"MathClass-rel\">=<\/mo> <mfrac><mrow><mo class=\"MathClass-open\">(<\/mo><mn>2<\/mn><mi>n<\/mi><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-punc\">!<\/mo><\/mrow> <mrow><msup><mrow><mo class=\"MathClass-open\">(<\/mo><msup><mrow><mn>2<\/mn><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msup><mi>n<\/mi><mo class=\"MathClass-punc\">!<\/mo><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><\/mrow><\/mfrac><mi>\u03c0<\/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\">Wir multiplizieren diese Ungleichungskette mit <math display=\"inline\"><mfrac><mrow><msup><mrow><mo class=\"MathClass-open\">(<\/mo><msup><mrow><mn>2<\/mn><\/mrow><mrow><mi>n<\/mi> <\/mrow> <\/msup> <mi>n<\/mi><mo class=\"MathClass-punc\">!<\/mo><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mn>2<\/mn> <\/mrow> <\/msup> <\/mrow> <mrow><mo class=\"MathClass-open\">(<\/mo><mn>2<\/mn><mi>n<\/mi><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-punc\">!<\/mo><\/mrow><\/mfrac> <mfrac><mrow><mn>2<\/mn><mi>n<\/mi><mo class=\"MathClass-bin\">+<\/mo><mn>1<\/mn><\/mrow> <mrow><mn>2<\/mn><mi>n<\/mi><\/mrow><\/mfrac> <\/math> und erhalten                                                                                                                                                                           <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"> <mfrac><mrow><msup><mrow><mo class=\"MathClass-open\">(<\/mo><mn>2<\/mn><mi>n<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mn>1<\/mn><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><\/mrow> <mrow><mn>2<\/mn><mi>n<\/mi><mo class=\"MathClass-open\">(<\/mo><mn>2<\/mn><mi>n<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mn>2<\/mn><mo class=\"MathClass-close\">)<\/mo><\/mrow><\/mfrac><mi>\u03c0<\/mi> <mo class=\"MathClass-rel\">\u2264<\/mo> <mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mi>n<\/mi><\/mrow><\/mfrac> <mfrac><mrow><msup><mrow><mo class=\"MathClass-open\">(<\/mo><msup><mrow><mn>2<\/mn><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msup><mi>n<\/mi><mo class=\"MathClass-punc\">!<\/mo><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mn>4<\/mn><\/mrow><\/msup><\/mrow> <mrow><msup><mrow><mo class=\"MathClass-open\">(<\/mo><mo class=\"MathClass-open\">(<\/mo><mn>2<\/mn><mi>n<\/mi><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-punc\">!<\/mo><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><\/mrow><\/mfrac> <mo class=\"MathClass-rel\">\u2264<\/mo> <mfrac><mrow><mn>2<\/mn><mi>n<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mn>1<\/mn><\/mrow> <mrow><mn>2<\/mn><mi>n<\/mi><\/mrow><\/mfrac> <mi>\u03c0<\/mi><mo class=\"MathClass-punc\">.<\/mo><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">Dies beweist die Formel (<a href=\"..\/..\/chapter\/drei-asymptotische-formeln#x1-282001r17\">9.17<\/a>) nach Wallis unter Anwendung des Sandwich-Lemmas (Lemma&nbsp;<a href=\"..\/..\/chapter\/reelle-folgen#x1-157004r2\">6.2<\/a>). <a id=\"x1-282003r282\"><\/a> <\/p> <h4 id=\"zf481290825ce\" class=\"subsectionHead\"><span class=\"titlemark\">9.6.2 <\/span> <a id=\"x1-2830002\"><\/a>Stirling-Formel<\/h4> <p class=\"noindent\">Wir verwenden nun das Wallissche Produkt und Taylor-Approximationen, um das asymptotische Verhalten von <math display=\"inline\"><mi>n<\/mi><mo class=\"MathClass-punc\">!<\/mo><\/math> zu bestimmen. Genauer formuliert wollen wir die <span class=\"ecbx-1095\">Stirling-Formel<\/span> <\/p><math display=\"block\"><mtable class=\"align\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><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><mi>n<\/mi><mo class=\"MathClass-punc\">!<\/mo><\/mrow> <mrow><msqrt><mrow><mn>2<\/mn><mi>\u03c0<\/mi><mi>n<\/mi><\/mrow><\/msqrt><msup><mrow> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mfrac><mrow><mi>n<\/mi><\/mrow> <mrow><mi class=\"qopname\"> e<\/mi><mo>  <\/mo> <\/mrow><\/mfrac> <\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <\/mrow><mrow><mi>n<\/mi><\/mrow><\/msup><\/mrow><\/mfrac> <mo class=\"MathClass-rel\">=<\/mo> <mn>1<\/mn><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"><mstyle class=\"label\" id=\"x1-283001r19\" \/><mstyle class=\"maketag\"><mtext>(9.19)<\/mtext><\/mstyle><mspace class=\"nbsp\" width=\"0.33em\" \/> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">beweisen. Diese ist nach James Stirling (1692-1770) benannt und wird auch als <math display=\"inline\"><mi>n<\/mi><mo class=\"MathClass-punc\">!<\/mo> <mo class=\"MathClass-rel\">\u2248<\/mo> <msqrt> <mrow><mn>2<\/mn><mi>\u03c0<\/mi><mi>n<\/mi><\/mrow><\/msqrt><msup><mrow> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow> <mfrac> <mrow> <mi>n<\/mi><\/mrow> <mrow><mi class=\"qopname\"> e<\/mi><mo>  <\/mo> <\/mrow><\/mfrac> <\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <\/mrow><mrow><mi>n<\/mi><\/mrow><\/msup><\/math> geschrieben. <\/p><p class=\"indent\">Der Grundgedanke f\u00fcr den Beweis von (<a href=\"..\/..\/chapter\/drei-asymptotische-formeln#x1-283001r19\">9.19<\/a>) ist die Gleichung                                                                                                                                                                           <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mi class=\"qopname\"> log<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>n<\/mi><mo class=\"MathClass-punc\">!<\/mo><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">=<\/mo><mi class=\"qopname\"> log<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mn>1<\/mn><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">+<\/mo><mi class=\"qopname\"> log<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mn>2<\/mn><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">+<\/mo> <mi class=\"qopname\">\u2026<\/mi><mo>  <\/mo> <mo class=\"MathClass-bin\">+<\/mo><mi class=\"qopname\"> log<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/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 <math display=\"inline\"><mi>n<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2115<\/mi><\/math> genauer zu betrachten, wobei wir die Summe aber durch das Integral <math display=\"inline\"><msubsup><mrow><mi class=\"MathClass-op\">\u222b  <\/mi><mo> <\/mo><\/mrow><mrow><mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mn>2<\/mn><\/mrow><\/mfrac> <\/mrow><mrow><mi>n<\/mi><mo class=\"MathClass-bin\">+<\/mo><mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mn>2<\/mn><\/mrow><\/mfrac> <\/mrow><\/msubsup><mi class=\"qopname\"> log<\/mi><mo>  <\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi><\/math> ersetzen wollen, da dieses einfacher zu berechnen ist. <\/p><p class=\"indent\">F\u00fcr <math display=\"inline\"><mi>k<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2115<\/mi><\/math> und <math display=\"inline\"><mi>t<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> [<\/mo><mrow><mi>k<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo><mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mn>2<\/mn><\/mrow><\/mfrac><mo class=\"MathClass-punc\">,<\/mo><mi>k<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mn>2<\/mn><\/mrow><\/mfrac> <\/mrow><mo fence=\"true\" form=\"postfix\">]<\/mo><\/mrow><\/math> gilt nach Beispiel <a href=\"..\/..\/chapter\/numerische-integration#x1-280002r59\">9.59<\/a> <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mi class=\"qopname\"> log<\/mi><mo>  <\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>t<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-rel\">=<\/mo><mi class=\"qopname\"> log<\/mi><mo>  <\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>k<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-bin\">+<\/mo><mfrac><mrow> <mn>1<\/mn><\/mrow> <mrow><mi>k<\/mi><\/mrow><\/mfrac> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>t<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>k<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-bin\">+<\/mo> <mi>O<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mfrac><mrow> <mn>1<\/mn><\/mrow> <mrow><msup><mrow><mi>k<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><\/mrow><\/mfrac> <\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-punc\">.<\/mo><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">Wir erhalten <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><msubsup><mrow><mo> \u222b  <\/mo><\/mrow><mrow><mi>k<\/mi><mo class=\"MathClass-bin\">\u2212<\/mo><mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mn>2<\/mn><\/mrow><\/mfrac> <\/mrow><mrow><mi>k<\/mi><mo class=\"MathClass-bin\">+<\/mo><mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mn>2<\/mn><\/mrow><\/mfrac> <\/mrow><\/msubsup><mi class=\"qopname\"> log<\/mi><mo>  <\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>t<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>t<\/mi> <mo class=\"MathClass-rel\">=<\/mo><mi class=\"qopname\"> log<\/mi><mo>  <\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>k<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-bin\">+<\/mo> <mi>O<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mfrac><mrow> <mn>1<\/mn><\/mrow><mrow><msup><mrow><mi>k<\/mi><\/mrow><mrow><mn>2<\/mn> <\/mrow><\/msup><\/mrow><\/mfrac> <\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">f\u00fcr alle <span class=\"maperiod\"><math display=\"inline\"><mi>k<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2115<\/mi><\/math><\/span><span class=\"period\">,<\/span> da sich der lineare Term bei der Integration wegl\u00f6scht.                                                                                                                                                                           <\/p><p class=\"indent\">F\u00fcr ein festes&nbsp;<math display=\"inline\"><mi>n<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2115<\/mi><\/math> und die Summe \u00fcber <math display=\"inline\"><mi>k<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> {<\/mo><mrow><mn>1<\/mn><mo class=\"MathClass-punc\">,<\/mo><mi class=\"MathClass-op\">\u2026<\/mi><mo> <\/mo><mo class=\"MathClass-punc\">,<\/mo><mi>n<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">}<\/mo><\/mrow><\/math> folgt daraus <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mi class=\"qopname\"> log<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>n<\/mi><mo class=\"MathClass-punc\">!<\/mo><mo class=\"MathClass-close\">)<\/mo><\/mtd> <mtd class=\"align-even\"> <mo class=\"MathClass-rel\">=<\/mo><munderover accent=\"false\" accentunder=\"false\"><mrow><mo> \u2211<\/mo> <\/mrow><mrow><mi>k<\/mi><mo class=\"MathClass-rel\">=<\/mo><mn>1<\/mn><\/mrow><mrow><mi>n<\/mi><\/mrow><\/munderover><mi class=\"qopname\"> log<\/mi><mo>  <\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>k<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-rel\">=<\/mo><munderover accent=\"false\" accentunder=\"false\"><mrow><mo> \u2211<\/mo> <\/mrow><mrow><mi>k<\/mi><mo class=\"MathClass-rel\">=<\/mo><mn>1<\/mn><\/mrow><mrow><mi>n<\/mi><\/mrow><\/munderover><msubsup><mrow><mo> \u222b  <\/mo><\/mrow><mrow><mi>k<\/mi><mo class=\"MathClass-bin\">\u2212<\/mo><mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mn>2<\/mn><\/mrow><\/mfrac> <\/mrow><mrow><mi>k<\/mi><mo class=\"MathClass-bin\">+<\/mo><mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mn>2<\/mn><\/mrow><\/mfrac> <\/mrow><\/msubsup><mi class=\"qopname\"> log<\/mi><mo>  <\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>t<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>t<\/mi> <mo class=\"MathClass-bin\">+<\/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><mi>O<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mfrac><mrow> <mn>1<\/mn><\/mrow> <mrow><msup><mrow><mi>k<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><\/mrow><\/mfrac> <\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-rel\">=<\/mo><msubsup><mrow><mo> \u222b  <\/mo><\/mrow><mrow><mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mn>2<\/mn><\/mrow><\/mfrac> <\/mrow><mrow><mi>n<\/mi><mo class=\"MathClass-bin\">+<\/mo><mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mn>2<\/mn><\/mrow><\/mfrac> <\/mrow><\/msubsup><mi class=\"qopname\"> log<\/mi><mo>  <\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>t<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>t<\/mi> <mo class=\"MathClass-bin\">+<\/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><mi>O<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mfrac><mrow> <mn>1<\/mn><\/mrow> <mrow><msup><mrow><mi>k<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><\/mrow><\/mfrac> <\/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\">=<\/mo> <mo class=\"MathClass-open\">(<\/mo><mi>n<\/mi> <mo class=\"MathClass-bin\">+<\/mo><mfrac><mrow> <mn>1<\/mn><\/mrow> <mrow><mn>2<\/mn><\/mrow><\/mfrac><mo class=\"MathClass-close\">)<\/mo><mi class=\"qopname\">log<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>n<\/mi> <mo class=\"MathClass-bin\">+<\/mo><mfrac><mrow> <mn>1<\/mn><\/mrow> <mrow><mn>2<\/mn><\/mrow><\/mfrac><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">\u2212<\/mo> <mo class=\"MathClass-open\">(<\/mo><mi>n<\/mi> <mo class=\"MathClass-bin\">+<\/mo><mfrac><mrow> <mn>1<\/mn><\/mrow> <mrow><mn>2<\/mn><\/mrow><\/mfrac><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>c<\/mi> <mo class=\"MathClass-bin\">+<\/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><mi>O<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mfrac><mrow> <mn>1<\/mn><\/mrow> <mrow><msup><mrow><mi>k<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><\/mrow><\/mfrac> <\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-punc\">,<\/mo><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\" \/> <mtd class=\"align-label\"> <mspace width=\"2em\" \/><\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">wobei sich die Konstante <math display=\"inline\"><mi>c<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u211d<\/mi><\/math> durch Auswerten der Stammfunktion bei der unteren Grenze <math display=\"inline\"><mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mn>2<\/mn><\/mrow><\/mfrac><\/math> ergibt. Wiederum nach Taylorapproximation gilt <span class=\"maperiod\"><math display=\"inline\"><mi class=\"qopname\"> log<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>n<\/mi> <mo class=\"MathClass-bin\">+<\/mo><mfrac><mrow> <mn>1<\/mn><\/mrow> <mrow><mn>2<\/mn><\/mrow><\/mfrac><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">=<\/mo><mi class=\"qopname\"> log<\/mi><mo>  <\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>n<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-bin\">+<\/mo><mfrac><mrow> <mn>1<\/mn><\/mrow> <mrow><mi>n<\/mi><\/mrow><\/mfrac> <mo class=\"MathClass-bin\">\u22c5<\/mo><mfrac><mrow> <mn>1<\/mn><\/mrow> <mrow><mn>2<\/mn><\/mrow><\/mfrac> <mo class=\"MathClass-bin\">+<\/mo> <mi>O<\/mi><mo class=\"MathClass-open\">(<\/mo><mfrac><mrow> <mn>1<\/mn><\/mrow> <mrow><msup><mrow><mi>n<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><\/mrow><\/mfrac><mo class=\"MathClass-close\">)<\/mo><\/math><\/span><span class=\"period\">.<\/span> Wir multiplizieren diese Approximation mit <math display=\"inline\"><mo class=\"MathClass-open\">(<\/mo><mi>n<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mn>2<\/mn><\/mrow><\/mfrac><mo class=\"MathClass-close\">)<\/mo><\/math> und erhalten (mit&nbsp;<math display=\"inline\"><mo class=\"MathClass-open\">(<\/mo><mi>n<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mn>2<\/mn><\/mrow><\/mfrac><mo class=\"MathClass-close\">)<\/mo><mi>O<\/mi><mo class=\"MathClass-open\">(<\/mo> <mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><msup><mrow><mi>n<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><\/mrow><\/mfrac><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <mi>O<\/mi><mo class=\"MathClass-open\">(<\/mo><mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mi>n<\/mi><\/mrow><\/mfrac><mo class=\"MathClass-close\">)<\/mo><\/math>) somit <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mi class=\"qopname\"> log<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>n<\/mi><mo class=\"MathClass-punc\">!<\/mo><mo class=\"MathClass-close\">)<\/mo><\/mtd> <mtd class=\"align-even\"> <mo class=\"MathClass-rel\">=<\/mo> <mo class=\"MathClass-open\">(<\/mo><mi>n<\/mi> <mo class=\"MathClass-bin\">+<\/mo><mfrac><mrow> <mn>1<\/mn><\/mrow> <mrow><mn>2<\/mn><\/mrow><\/mfrac><mo class=\"MathClass-close\">)<\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi class=\"qopname\">log<\/mi><mo>  <\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>n<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-bin\">+<\/mo><mfrac><mrow> <mn>1<\/mn><\/mrow> <mrow><mn>2<\/mn><mi>n<\/mi><\/mrow><\/mfrac> <mo class=\"MathClass-bin\">+<\/mo> <mi>O<\/mi><mo class=\"MathClass-open\">(<\/mo><mfrac><mrow> <mn>1<\/mn><\/mrow> <mrow><msup><mrow><mi>n<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><\/mrow><\/mfrac><mo class=\"MathClass-close\">)<\/mo><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-bin\">\u2212<\/mo> <mo class=\"MathClass-open\">(<\/mo><mi>n<\/mi> <mo class=\"MathClass-bin\">+<\/mo><mfrac><mrow> <mn>1<\/mn><\/mrow> <mrow><mn>2<\/mn><\/mrow><\/mfrac><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>c<\/mi> <mo class=\"MathClass-bin\">+<\/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><mi>O<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mfrac><mrow> <mn>1<\/mn><\/mrow> <mrow><msup><mrow><mi>k<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><\/mrow><\/mfrac> <\/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\">=<\/mo> <mo class=\"MathClass-open\">(<\/mo><mi>n<\/mi> <mo class=\"MathClass-bin\">+<\/mo><mfrac><mrow> <mn>1<\/mn><\/mrow> <mrow><mn>2<\/mn><\/mrow><\/mfrac><mo class=\"MathClass-close\">)<\/mo><mi class=\"qopname\">log<\/mi><mo>  <\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>n<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-bin\">+<\/mo><mfrac><mrow> <mn>1<\/mn><\/mrow> <mrow><mn>2<\/mn><\/mrow><\/mfrac> <mo class=\"MathClass-bin\">+<\/mo> <mi>O<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mfrac><mrow> <mn>1<\/mn><\/mrow> <mrow><mi>n<\/mi><\/mrow><\/mfrac> <\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>n<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo><mfrac><mrow> <mn>1<\/mn><\/mrow> <mrow><mn>2<\/mn><\/mrow><\/mfrac> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>c<\/mi> <mo class=\"MathClass-bin\">+<\/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><mi>O<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mfrac><mrow> <mn>1<\/mn><\/mrow> <mrow><msup><mrow><mi>k<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><\/mrow><\/mfrac> <\/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\">=<\/mo> <mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mn>2<\/mn><\/mrow><\/mfrac><mi class=\"qopname\">log<\/mi><mo>  <\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>n<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-bin\">+<\/mo> <mi>n<\/mi><mi class=\"qopname\">log<\/mi><mo>  <\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>n<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>n<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\" \/> <mtd class=\"align-label\"> <mspace width=\"2em\" \/><\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">f\u00fcr eine konvergente 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> in <math display=\"inline\"><mi>\u211d<\/mi><\/math> (wobei wir Proposition&nbsp;<a href=\"..\/..\/chapter\/absolute-konvergenz#x1-192001r28\">7.28<\/a> verwendet haben um die absolut konvergenten Fehlerterme&nbsp;<math display=\"inline\"><msub><mrow><mi>\ud835\udf00<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub> <mo class=\"MathClass-rel\">=<\/mo> <mi>O<\/mi><mo class=\"MathClass-open\">(<\/mo> <mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><msup><mrow><mi>n<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><\/mrow><\/mfrac><mo class=\"MathClass-close\">)<\/mo><\/math> aufzusummieren). Wenden wir die Exponentialabbildung an, so erhalten wir f\u00fcr alle <math display=\"inline\"><mi>n<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2115<\/mi><\/math> <\/p><math display=\"block\"><mtable class=\"align\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mi>n<\/mi><mo class=\"MathClass-punc\">!<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <msqrt><mrow><mi>n<\/mi><\/mrow><\/msqrt><msup><mrow><mi>n<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msup><msup><mrow><mi class=\"qopname\"> e<\/mi><mo>  <\/mo><\/mrow><mrow><mo class=\"MathClass-bin\">\u2212<\/mo><mi>n<\/mi><\/mrow><\/msup><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\"><mstyle class=\"label\" id=\"x1-283002r20\" \/><mstyle class=\"maketag\"><mtext>(9.20)<\/mtext><\/mstyle><mspace class=\"nbsp\" width=\"0.33em\" \/> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">wobei <math display=\"inline\"><msub><mrow><mi>b<\/mi><\/mrow><mrow><mi>n<\/mi> <\/mrow> <\/msub> <mo class=\"MathClass-rel\">=<\/mo><mi class=\"qopname\"> exp<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><\/math> ist. Da <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> konvergent ist, ist auch <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> konvergent und der Grenzwert <math display=\"inline\"><mi>B<\/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><mi class=\"qopname\">exp<\/mi><mo>  <\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-rel\">=<\/mo><mi class=\"qopname\"> exp<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><munder class=\"msub\"><mrow><mi class=\"qopname\">lim<\/mi><mo>  <\/mo><\/mrow><mrow><mi>n<\/mi><mo class=\"MathClass-rel\">\u2192<\/mo><mi>\u221e<\/mi><\/mrow><\/munder><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><\/math> ist positiv. <\/p><p class=\"indent\">Um <math display=\"inline\"><mi>B<\/mi><\/math> zu berechnen, verwenden wir (<a href=\"..\/..\/chapter\/drei-asymptotische-formeln#x1-283002r20\">9.20<\/a>) gemeinsam mit dem Wallisschen Produkt (<a href=\"..\/..\/chapter\/drei-asymptotische-formeln#x1-282001r17\">9.17<\/a>). Es gilt <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mi>\u03c0<\/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><mn>1<\/mn><\/mrow> <mrow><mi>n<\/mi><\/mrow><\/mfrac> <mfrac><mrow><msup><mrow><mo class=\"MathClass-open\">(<\/mo><msup><mrow><mn>2<\/mn><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msup><mi>n<\/mi><mo class=\"MathClass-punc\">!<\/mo><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mn>4<\/mn><\/mrow><\/msup><\/mrow> <mrow><msup><mrow><mo class=\"MathClass-open\">(<\/mo><mo class=\"MathClass-open\">(<\/mo><mn>2<\/mn><mi>n<\/mi><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-punc\">!<\/mo><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><\/mrow><\/mfrac> <mo class=\"MathClass-rel\">=<\/mo><munder class=\"msub\"><mrow><mi class=\"qopname\"> lim<\/mi><mo>  <\/mo><\/mrow><mrow><mi>n<\/mi><mo class=\"MathClass-rel\">\u2192<\/mo><mi>\u221e<\/mi><\/mrow><\/munder><mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mi>n<\/mi><\/mrow><\/mfrac> <mfrac><mrow><msup><mrow><mrow><mo class=\"MathClass-open\" fence=\"true\" mathsize=\"1.19em\">(<\/mo><mrow><msup><mrow><mn>2<\/mn><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msup><msqrt><mrow><mi>n<\/mi><\/mrow><\/msqrt><msup><mrow><mi>n<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msup><msup><mrow><mi class=\"qopname\"> e<\/mi><mo>  <\/mo><\/mrow><mrow><mo class=\"MathClass-bin\">\u2212<\/mo><mi>n<\/mi><\/mrow><\/msup><msub><mrow><mi>b<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><\/mrow><mo class=\"MathClass-close\" fence=\"true\" mathsize=\"1.19em\">)<\/mo><\/mrow><\/mrow><mrow><mn>4<\/mn><\/mrow><\/msup><\/mrow> <mrow><msup><mrow><mrow><mo class=\"MathClass-open\" fence=\"true\" mathsize=\"1.19em\">(<\/mo><mrow><msqrt><mrow><mn>2<\/mn><mi>n<\/mi><\/mrow><\/msqrt><msup><mrow><mo class=\"MathClass-open\">(<\/mo><mn>2<\/mn><mi>n<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mn>2<\/mn><mi>n<\/mi><\/mrow><\/msup><msup><mrow><mi>e<\/mi><\/mrow><mrow><mo class=\"MathClass-bin\">\u2212<\/mo><mn>2<\/mn><mi>n<\/mi><\/mrow><\/msup><msub><mrow><mi>b<\/mi><\/mrow><mrow><mn>2<\/mn><mi>n<\/mi><\/mrow><\/msub><\/mrow><mo class=\"MathClass-close\" fence=\"true\" mathsize=\"1.19em\">)<\/mo><\/mrow><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><\/mrow><\/mfrac> <mo class=\"MathClass-rel\">=<\/mo><munder class=\"msub\"><mrow><mi class=\"qopname\"> lim<\/mi><mo>  <\/mo><\/mrow><mrow><mi>n<\/mi><mo class=\"MathClass-rel\">\u2192<\/mo><mi>\u221e<\/mi><\/mrow><\/munder> <mfrac><mrow><msubsup><mrow><mi>b<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><mrow><mn>4<\/mn><\/mrow><\/msubsup><\/mrow> <mrow><mn>2<\/mn><msubsup><mrow><mi>b<\/mi><\/mrow><mrow><mn>2<\/mn><mi>n<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msubsup><\/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\">da die Potenzen von <span class=\"maperiod\"><math display=\"inline\"><mn>2<\/mn><\/math><\/span><span class=\"period\">,<\/span> von&nbsp;<math display=\"inline\"><mi>n<\/mi><\/math> und von&nbsp;<math display=\"inline\"><mi>e<\/mi><\/math> sich jeweils direkt k\u00fcrzen, und die Wurzelterme gemeinsam mit dem&nbsp;<math display=\"inline\"> <mfrac> <mrow> <mn>1<\/mn><\/mrow> <mrow><mi>n<\/mi><\/mrow><\/mfrac><\/math> vor dem Bruch sich zu einem&nbsp;<math display=\"inline\"><mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mn>2<\/mn><\/mrow><\/mfrac><\/math> vereinfachen. Daraus folgt <\/p><table id=\"z45543ab021a2\" class=\"equation-star\"><tr><td> <math class=\"equation\" display=\"block\"> <mi>\u03c0<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mfrac><mrow><msup><mrow><mi>B<\/mi><\/mrow><mrow><mn>4<\/mn><\/mrow><\/msup><\/mrow> <mrow><mn>2<\/mn><msup><mrow><mi>B<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><\/mrow><\/mfrac> <mo class=\"MathClass-rel\">=<\/mo><mfrac><mrow> <mn>1<\/mn><\/mrow> <mrow><mn>2<\/mn><\/mrow><\/mfrac><msup><mrow><mi>B<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup> <\/math><\/td><\/tr><\/table> <p class=\"indent\">und somit muss <math display=\"inline\"><mi>B<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <msqrt><mrow><mn>2<\/mn><mi>\u03c0<\/mi><\/mrow><\/msqrt><\/math> gelten. Aus Gleichung (<a href=\"..\/..\/chapter\/drei-asymptotische-formeln#x1-283002r20\">9.20<\/a>) erhalten wir <\/p><table id=\"z6dbf5813031d\" class=\"equation-star\"><tr><td> <math class=\"equation\" display=\"block\"> <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><mi>n<\/mi><mo class=\"MathClass-punc\">!<\/mo><\/mrow> <mrow><msqrt><mrow><mi>n<\/mi><\/mrow><\/msqrt><msup><mrow> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mfrac><mrow><mi>n<\/mi><\/mrow> <mrow><mi class=\"qopname\"> e<\/mi><mo>  <\/mo> <\/mrow><\/mfrac> <\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <\/mrow><mrow><mi>n<\/mi><\/mrow><\/msup><\/mrow><\/mfrac> <mo class=\"MathClass-rel\">=<\/mo><munder class=\"msub\"><mrow><mi class=\"qopname\"> lim<\/mi><mo>  <\/mo><\/mrow><mrow><mi>n<\/mi><mo class=\"MathClass-rel\">\u2192<\/mo><mi>\u221e<\/mi><\/mrow><\/munder><msub><mrow><mi>b<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub> <mo class=\"MathClass-rel\">=<\/mo> <msqrt><mrow><mn>2<\/mn><mi>\u03c0<\/mi><\/mrow><\/msqrt><mo class=\"MathClass-punc\">,<\/mo> <\/math><\/td><\/tr><\/table> <p class=\"indent\">was zur Stirling-Formel (<a href=\"..\/..\/chapter\/drei-asymptotische-formeln#x1-283001r19\">9.19<\/a>) \u00e4quivalent ist. <\/p><p class=\"indent\">Auf Grund der Bedeutung von der Fakult\u00e4tsfunktion, zum Beispiel in kombinatorischen \u00dcberlegungen, ist es nicht verwunderlich, dass die Sterling-Formel vielfache Anwendungen besitzt. In den folgenden \u00dcbungen wollen wir einige wenige solcher Anwendungen pr\u00e4sentieren. <\/p> <div class=\"me meexample\"> <p class=\"indent\"><\/p><h4 id=\"z3d03330a7bbf\"> <a id=\"x1-283003r60\"><\/a> <span class=\"ecbx-1095\">\u00dc<\/span><span class=\"ecbx-1095\">bung 9.60 <\/span>(Besuche bei der Irrfahrt auf <math display=\"inline\"><mi>\u2124<\/mi><\/math>)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Wir betrachten eine (diskrete) zuf<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">llige Bewegung auf den ganzen Zahlen ausgehend von<\/span> <span class=\"ecti-1095\">Null. Also beginnen wir zur Zeit <\/span><math display=\"inline\"><mi>t<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mn>0<\/mn><\/math> <span class=\"ecti-1095\">im Ursprung, h<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">pfen dann mit Wahrscheinlichkeit <\/span><math display=\"inline\"><mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mn>2<\/mn><\/mrow><\/mfrac><\/math> <span class=\"ecti-1095\">entweder nach links auf die Position <\/span><math display=\"inline\"> <mo class=\"MathClass-bin\">\u2212<\/mo> <mn>1<\/mn><\/math> <span class=\"ecti-1095\">oder nach rechts auf die Position <\/span><span class=\"maperiod\"><math display=\"inline\"><mn>1<\/mn><\/math><\/span><span class=\"period\">.<\/span> <span class=\"ecti-1095\">Bezeichnet im Allgemeinen <\/span><math display=\"inline\"><msub><mrow><mi>x<\/mi><\/mrow><mrow><mi>t<\/mi><\/mrow><\/msub> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2124<\/mi><\/math> <span class=\"ecti-1095\">die Position zur Zeit <\/span><span class=\"maperiod\"><math display=\"inline\"><mi>t<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2115<\/mi><\/math><\/span><span class=\"period\">,<\/span> <span class=\"ecti-1095\">so bewegen wir uns auf die Zeit <\/span><math display=\"inline\"><mi>t<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mn>1<\/mn><\/math> <span class=\"ecti-1095\">hin wieder entweder links oder rechts mit gleicher Wahrscheinlichkeit von <\/span><span class=\"maperiod\"><math display=\"inline\"><mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mn>2<\/mn><\/mrow><\/mfrac><\/math><\/span><span class=\"period\">.<\/span> <\/p><p class=\"indent\"><span class=\"ecti-1095\">Berechnen     Sie     die     Wahrscheinlichkeit,     dass     man     sich     zur     Zeit<\/span> <math display=\"inline\"><mi>t<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2115<\/mi><\/math> <span class=\"ecti-1095\">bei Null befindet und approximieren Sie diese mit der Stirling-Formel.<\/span> <\/p> <\/div> <div class=\"me meexample\"> <p class=\"indent\"><\/p><h4 id=\"zc173ee8030df\"> <a id=\"x1-283004r61\"><\/a> <span class=\"ecbx-1095\">\u00dc<\/span><span class=\"ecbx-1095\">bung 9.61.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Eine M<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">nze wird <\/span><math display=\"inline\"><mi>n<\/mi><\/math><span class=\"ecti-1095\">-mal<\/span> <span class=\"ecti-1095\">geworfen. Sei <\/span><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> <span class=\"ecti-1095\">Zeigen Sie, dass die Wahrscheinlichkeit, dass die M<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">nze h<\/span><span class=\"ecti-1095\">\u00f6<\/span><span class=\"ecti-1095\">chstens <\/span><math display=\"inline\"><mo class=\"MathClass-open\">(<\/mo><mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mn>2<\/mn><\/mrow><\/mfrac> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>\ud835\udf00<\/mi><mo class=\"MathClass-close\">)<\/mo><mi>n<\/mi><\/math><span class=\"ecti-1095\">-mal<\/span> <span class=\"ecti-1095\">auf Kopf f<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">llt, 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\">gegen Null geht.<\/span> <\/p><p class=\"indent\"><span class=\"ecti-1095\">Hinweis: Sie d<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">rfen dabei annehmen, dass <\/span><math display=\"inline\"><msup><mrow><mn>2<\/mn><\/mrow><mrow><mo class=\"MathClass-bin\">\u2212<\/mo><mi>n<\/mi><\/mrow><\/msup><mfenced close=\")\" open=\"(\" separators><mfrac linethickness=\"0.0pt\"><mrow><mi>n<\/mi><\/mrow> <mrow><mi>k<\/mi><\/mrow><\/mfrac><\/mfenced> <\/math> <span class=\"ecti-1095\">die Wahrscheinlichkeit ist, dass bei <\/span><math display=\"inline\"><mi>n<\/mi><\/math><span class=\"ecti-1095\">-maligem<\/span> <span class=\"ecti-1095\">Wurf die M<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">nze <\/span><math display=\"inline\"><mi>k<\/mi><\/math><span class=\"ecti-1095\">-mal<\/span> <span class=\"ecti-1095\">auf Kopf f<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">llt. Zeigen Sie auch, dass die Funktion <\/span><math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mo class=\"MathClass-open\">(<\/mo><mn>0<\/mn><mo class=\"MathClass-punc\">,<\/mo><mn>1<\/mn><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-rel\">\u21a6<\/mo><mi>x<\/mi><mi class=\"qopname\">log<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">+<\/mo> <mo class=\"MathClass-open\">(<\/mo><mn>1<\/mn> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><mi class=\"qopname\">log<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mn>1<\/mn> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math> <span class=\"ecti-1095\">ein striktes lokales Maximum bei <\/span><math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mn>2<\/mn><\/mrow><\/mfrac><\/math> <span class=\"ecti-1095\">hat.<\/span> <\/p> <\/div> <div class=\"me meexample\"> <p class=\"indent\"><\/p><h4 id=\"z8b5c72abd994\"> <a id=\"x1-283005r62\"><\/a> <span class=\"ecbx-1095\">\u00dc<\/span><span class=\"ecbx-1095\">bung 9.62 <\/span>(Ein Ausblick auf Algorithmik)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">In dieser <\/span><span class=\"ecti-1095\">\u00dc<\/span><span class=\"ecti-1095\">bung wollen wir die worst case Geschwindigkeit eines Sortieralgorithmus absch<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">tzen.<\/span> <span class=\"ecti-1095\">Sei also gegeben ein Algorithmus, der zu einem Datensatz <\/span><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> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2124<\/mi><\/math> <span class=\"ecti-1095\">diesen umordnet, das heisst, eine Abbildung <\/span><math display=\"inline\"><mi>j<\/mi> <mo class=\"MathClass-punc\">:<\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> {<\/mo><mrow><mn>1<\/mn><mo class=\"MathClass-punc\">,<\/mo><mi class=\"MathClass-op\">\u2026<\/mi><mo> <\/mo><mo class=\"MathClass-punc\">,<\/mo><mi>n<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">}<\/mo><\/mrow> <mo class=\"MathClass-rel\">\u2192<\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> {<\/mo><mrow><mn>1<\/mn><mo class=\"MathClass-punc\">,<\/mo><mi class=\"MathClass-op\">\u2026<\/mi><mo> <\/mo><mo class=\"MathClass-punc\">,<\/mo><mi>n<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">}<\/mo><\/mrow><\/math> <span class=\"ecti-1095\">findet, so dass <\/span><span class=\"maperiod\"><math display=\"inline\"><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>j<\/mi><mo class=\"MathClass-open\">(<\/mo><mn>1<\/mn><mo class=\"MathClass-close\">)<\/mo><\/mrow><\/msub> <mo class=\"MathClass-rel\">\u2264<\/mo><mi class=\"MathClass-op\">\u2026<\/mi><mo> <\/mo> <mo class=\"MathClass-rel\">\u2264<\/mo> <msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>j<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>n<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow><\/msub><\/math><\/span><span class=\"period\">.<\/span> <span class=\"ecti-1095\">Wir interessieren uns nun f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r die Zahl <\/span><math display=\"inline\"><mi>W<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>n<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math> <span class=\"ecti-1095\">der Anzahl Vergleiche zweier Zahlen des Datensatzes, die der Algorithmus schlimmstenfalls<\/span> <span class=\"ecti-1095\">durchf<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">hren muss, um den Datensatz zu sortieren. Zeigen Sie, dass eine von <\/span><math display=\"inline\"><mi>n<\/mi><\/math> <span class=\"ecti-1095\">unabh<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">ngige Konstante <\/span><math display=\"inline\"><mi>C<\/mi> <mo class=\"MathClass-rel\">&gt;<\/mo> <mn>0<\/mn><\/math> <span class=\"ecti-1095\">existiert, so dass <\/span><span class=\"maperiod\"><math display=\"inline\"><mi>W<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>n<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">\u2265<\/mo> <mi>C<\/mi><mi>n<\/mi><mi class=\"qopname\">log<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>n<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math><\/span><span class=\"period\">.<\/span> <\/p><p class=\"indent\"><\/p><details><summary style=\"color:#FF7F00\"><span class=\"ecti-1095\">Hinweis.<\/span><\/summary><p class=\"indent\" style=\"margin-top: 0\"><span class=\"ecti-1095\">Z<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">hlen         Sie         die         Anzahl         Datens<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">tze,         die         mit<\/span> <math display=\"inline\"><mi>A<\/mi><\/math> <span class=\"ecti-1095\">Umsortierungen aus einem fixen Datensatz konstruiert werden k<\/span><span class=\"ecti-1095\">\u00f6<\/span><span class=\"ecti-1095\">nnen.<\/span><\/p><\/details>  <\/div> <a id=\"x1-283006r283\"><\/a> <h4 id=\"z717d50ca220a\" class=\"subsectionHead\"><span class=\"titlemark\">9.6.3 <\/span> <a id=\"x1-2840003\"><\/a>Asymptotik der harmonischen Reihe<\/h4> <p class=\"noindent\">F\u00fcr den Beweis der Stirling-Formel wurde eine Kombination von Taylor-Approximation und Integration verwendet, um das asymptotische Verhalten der Reihe <math display=\"inline\"><mi class=\"qopname\">log<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>n<\/mi><mo class=\"MathClass-punc\">!<\/mo><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">=<\/mo><msubsup><mrow><mi class=\"qopname\"> \u2211<\/mi><mo>  <\/mo> <\/mrow><mrow><mi>k<\/mi><mo class=\"MathClass-rel\">=<\/mo><mn>1<\/mn><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msubsup><mi class=\"qopname\"> log<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>k<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math> zu bestimmen. Diese Methode kann auch allgemeiner zur Untersuchung des Divergenzverhaltens von Reihen eingesetzt werden. Zum Beispiel gilt die \u201easymptotische Entwicklung\u201c <\/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>k<\/mi><mo class=\"MathClass-rel\">=<\/mo><mn>1<\/mn><\/mrow><mrow><mi>n<\/mi><\/mrow><\/munderover><mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mi>k<\/mi><\/mrow><\/mfrac> <mo class=\"MathClass-rel\">=<\/mo><mi class=\"qopname\"> log<\/mi><mo>  <\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>n<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-bin\">+<\/mo> <mi>\u03b3<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mn>2<\/mn><mi>n<\/mi><\/mrow><\/mfrac> <mo class=\"MathClass-bin\">+<\/mo> <mi>O<\/mi><mo class=\"MathClass-open\">(<\/mo><mfrac><mrow> <mn>1<\/mn><\/mrow> <mrow><msup><mrow><mi>n<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><\/mrow><\/mfrac><mo class=\"MathClass-close\">)<\/mo><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"><mstyle class=\"label\" id=\"x1-284001r21\" \/><mstyle class=\"maketag\"><mtext>(9.21)<\/mtext><\/mstyle><mspace class=\"nbsp\" width=\"0.33em\" \/> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">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> wobei                                                                                                                                                                           <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mi>\u03b3<\/mi> <mo class=\"MathClass-rel\">=<\/mo><msubsup><mrow><mo> \u222b  <\/mo><\/mrow><mrow><mn>1<\/mn><\/mrow><mrow><mi>\u221e<\/mi><\/mrow><\/msubsup><mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow> <mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mo class=\"MathClass-open\">\u230a<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">\u230b<\/mo><\/mrow><\/mfrac><mo class=\"MathClass-bin\">\u2212<\/mo><mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mi>x<\/mi><\/mrow><\/mfrac> <\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mn>0<\/mn><mo class=\"MathClass-punc\">.<\/mo><mn>5<\/mn><mn>7<\/mn><mn>7<\/mn><mn>2<\/mn><mn>1<\/mn><mn>5<\/mn><mn>6<\/mn><mn>6<\/mn><mn>4<\/mn><mn>9<\/mn><mn>0<\/mn><mn>1<\/mn><mn>5<\/mn><mn>3<\/mn><mn>2<\/mn><mo>\u2026<\/mo><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">die <span class=\"ecbx-1095\">Euler-Mascheroni-Konstante <\/span>genannt wird. <\/p> <div class=\"me meexample\"> <p class=\"indent\"><\/p><h4 id=\"zbc75f545a61c\"> <a id=\"x1-284002r63\"><\/a> <span class=\"ecbx-1095\">\u00dc<\/span><span class=\"ecbx-1095\">bung 9.63.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Wir wollen die Asymptotik<\/span> (<a href=\"..\/..\/chapter\/drei-asymptotische-formeln#x1-284001r21\">9.21<\/a>) <span class=\"ecti-1095\">in dieser <\/span><span class=\"ecti-1095\">\u00dc<\/span><span class=\"ecti-1095\">bung beweisen.<\/span> <\/p><dl class=\"enumerate\"><dt class=\"enumerate\"> <span class=\"ecti-1095\">(a)<\/span><\/dt><dd class=\"enumerate\"><span class=\"ecti-1095\">Beweisen Sie, dass das Integral in der Definition von<\/span> <math display=\"inline\"><mi>\u03b3<\/mi><\/math> <span class=\"ecti-1095\">konvergiert und dass die Gleichung<\/span> <math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><munderover accent=\"false\" accentunder=\"false\"><mrow><mo>\u2211<\/mo> <\/mrow><mrow><mi>k<\/mi><mo class=\"MathClass-rel\">=<\/mo><mn>1<\/mn><\/mrow><mrow><mi>n<\/mi><\/mrow><\/munderover><mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mi>k<\/mi><\/mrow><\/mfrac> <mo class=\"MathClass-rel\">=<\/mo><mi class=\"qopname\"> log<\/mi><mo>  <\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>n<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-bin\">+<\/mo> <mi>\u03b3<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mi>n<\/mi><\/mrow><\/mfrac> <mo class=\"MathClass-bin\">\u2212<\/mo><msubsup><mrow><mo>\u222b  <\/mo><\/mrow><mrow><mi>n<\/mi><\/mrow><mrow><mi>\u221e<\/mi><\/mrow><\/msubsup><mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow> <mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mo class=\"MathClass-open\">\u230a<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">\u230b<\/mo><\/mrow><\/mfrac><mo class=\"MathClass-bin\">\u2212<\/mo><mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mi>x<\/mi><\/mrow><\/mfrac> <\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\"><span class=\"ecti-1095\">f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r alle <\/span><math display=\"inline\"><mi>n<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2115<\/mi><\/math> <span class=\"ecti-1095\">gilt.<\/span> <\/p><\/dd><dt class=\"enumerate\"> <span class=\"ecti-1095\">(b)<\/span><\/dt><dd class=\"enumerate\"><span class=\"ecti-1095\">Verwenden Sie Taylor-Approximation, um<\/span> <div class=\"custom-itemize\"><div class=\"item-head\"> <span class=\"tcrm-1095\">\u2022<\/span><\/div><div class=\"item-content\"><math display=\"inline\"><mi class=\"qopname\">log<\/mi><mo>  <\/mo><mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>m<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mn>1<\/mn><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-rel\">=<\/mo><mi class=\"qopname\"> log<\/mi><mo>  <\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>m<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-bin\">+<\/mo><mfrac><mrow> <mn>1<\/mn><\/mrow> <mrow><mi>m<\/mi><\/mrow><\/mfrac> <mo class=\"MathClass-bin\">\u2212<\/mo><mfrac><mrow> <mn>1<\/mn><\/mrow> <mrow><mn>2<\/mn><msup><mrow><mi>m<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><\/mrow><\/mfrac> <mo class=\"MathClass-bin\">+<\/mo> <mi>O<\/mi><mo class=\"MathClass-open\">(<\/mo><mfrac><mrow> <mn>1<\/mn><\/mrow> <mrow><msup><mrow><mi>m<\/mi><\/mrow><mrow><mn>3<\/mn><\/mrow><\/msup><\/mrow><\/mfrac><mo class=\"MathClass-close\">)<\/mo><\/math> <\/div><div class=\"item-head\"> <span class=\"tcrm-1095\">\u2022<\/span><\/div><div class=\"item-content\"><math display=\"inline\"><mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mi>m<\/mi><\/mrow><\/mfrac> <mo class=\"MathClass-bin\">\u2212<\/mo><msubsup><mrow><mi class=\"MathClass-op\">\u222b  <\/mi><mo> <\/mo><\/mrow><mrow><mi>m<\/mi><\/mrow><mrow><mi>m<\/mi><mo class=\"MathClass-bin\">+<\/mo><mn>1<\/mn><\/mrow><\/msubsup><mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mi>t<\/mi><\/mrow><\/mfrac> <mspace class=\"thinspace\" width=\"0.17em\" \/> <mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>t<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mn>2<\/mn><msup><mrow><mi>m<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><\/mrow><\/mfrac> <mo class=\"MathClass-bin\">+<\/mo> <mi>O<\/mi><mo class=\"MathClass-open\">(<\/mo><mfrac><mrow> <mn>1<\/mn><\/mrow> <mrow><msup><mrow><mi>m<\/mi><\/mrow><mrow><mn>3<\/mn><\/mrow><\/msup><\/mrow><\/mfrac><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">=<\/mo><mfrac><mrow> <mn>1<\/mn><\/mrow> <mrow><mn>2<\/mn><\/mrow><\/mfrac><msubsup><mrow><mi class=\"MathClass-op\"> \u222b  <\/mi><mo> <\/mo><\/mrow><mrow><mi>m<\/mi><\/mrow><mrow><mi>m<\/mi><mo class=\"MathClass-bin\">+<\/mo><mn>1<\/mn><\/mrow><\/msubsup><msup><mrow><mi>t<\/mi><\/mrow><mrow><mo class=\"MathClass-bin\">\u2212<\/mo><mn>2<\/mn><\/mrow><\/msup><mspace class=\"thinspace\" width=\"0.17em\" \/> <mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>t<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mi>O<\/mi><mo class=\"MathClass-open\">(<\/mo><mfrac><mrow> <mn>1<\/mn><\/mrow> <mrow><msup><mrow><mi>m<\/mi><\/mrow><mrow><mn>3<\/mn><\/mrow><\/msup><\/mrow><\/mfrac><mo class=\"MathClass-close\">)<\/mo><\/math><\/div><\/div> <p class=\"noindent\"><span class=\"ecti-1095\">f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r <\/span><math display=\"inline\"><mi>m<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2115<\/mi><\/math> <span class=\"ecti-1095\">zu<\/span> <span class=\"ecti-1095\">beweisen.<\/span> <\/p><\/dd><dt class=\"enumerate\"> <span class=\"ecti-1095\">(c)<\/span><\/dt><dd class=\"enumerate\"><span class=\"ecti-1095\">Verwenden Sie das Integralkriterium (Satz <\/span><a href=\"..\/..\/chapter\/das-uneigentliche-integral#x1-273007r35\"><span class=\"ecti-1095\">9.35<\/span><\/a><span class=\"ecti-1095\">) um zu zeigen, dass<\/span> <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>m<\/mi><mo class=\"MathClass-rel\">=<\/mo><mi>n<\/mi><\/mrow><mrow><mi>\u221e<\/mi><\/mrow><\/munderover> <mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><msup><mrow><mi>m<\/mi><\/mrow><mrow><mn>3<\/mn><\/mrow><\/msup><\/mrow><\/mfrac> <mo class=\"MathClass-rel\">=<\/mo> <mi>O<\/mi><mo class=\"MathClass-open\">(<\/mo><mfrac><mrow> <mn>1<\/mn><\/mrow> <mrow><msup><mrow><mi>n<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><\/mrow><\/mfrac><mo class=\"MathClass-close\">)<\/mo><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\"><span class=\"ecti-1095\">f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r <\/span><span class=\"maperiod\"><math display=\"inline\"><mi>n<\/mi> <mo class=\"MathClass-rel\">\u2192<\/mo> <mi>\u221e<\/mi><\/math><\/span><span class=\"period\">.<\/span> <\/p><\/dd><dt class=\"enumerate\"> <span class=\"ecti-1095\">(d)<\/span><\/dt><dd class=\"enumerate\"><span class=\"ecti-1095\">Verbinden Sie die obigen Aussagen zu einem Beweis von<\/span> (<a href=\"..\/..\/chapter\/drei-asymptotische-formeln#x1-284001r21\">9.21<\/a>)<span class=\"ecti-1095\">.<\/span> <\/dd><dt class=\"enumerate\"> <span class=\"ecti-1095\">(e)<\/span><\/dt><dd class=\"enumerate\"><span class=\"ecti-1095\">Wiederholen Sie Beispiel <\/span><a href=\"..\/..\/chapter\/reihen#x1-187005r5\"><span class=\"ecti-1095\">7.5<\/span><\/a> <span class=\"ecti-1095\">und approximieren Sie mit der Formel<\/span> (<a href=\"..\/..\/chapter\/drei-asymptotische-formeln#x1-284001r21\">9.21<\/a>) <span class=\"ecti-1095\">die notwendige Anzahl Bausteine<\/span> <math display=\"inline\"><mi>n<\/mi><\/math> <span class=\"ecti-1095\">f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r einen<\/span> <span class=\"ecti-1095\">Sprungturm, der <\/span><math display=\"inline\"><mn>1<\/mn><mn>0<\/mn><\/math><span class=\"ecti-1095\">m<\/span> <span class=\"ecti-1095\">in den See hinreicht. Vergleichen Sie dies dann zum exakten Resultat<\/span> <span class=\"maperiod\"><math display=\"inline\"><mi>n<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mn>1<\/mn><mn>2<\/mn><mn>3<\/mn><mn>6<\/mn><mn>7<\/mn><\/math><\/span><span class=\"period\">.<\/span><\/dd><\/dl> <\/div> <a id=\"x1-284008r281\"><\/a> \n","rendered":"\n<style scoped=\"scoped\">.cmr-5{font-size:50%;}\n.cmr-7{font-size:70%;}\n.cmmi-5{font-size:50%;font-style: italic;}\n.cmmi-7{font-size:70%;font-style: italic;}\n.cmmi-10{font-style: italic;}\n.cmsy-5{font-size:50%;}\n.cmsy-7{font-size:70%;}\n.cmbx-10{ font-weight: bold;}\n.cmbsy-10{font-weight: bold;}\n.cmbsy-10{font-weight: bold;}\n.cmbsy-10{font-weight: bold;}\n.cmbsy-7{font-size:70%;font-weight: bold;}\n.cmbsy-7{font-weight: bold;}\n.cmbsy-7{font-weight: bold;}\n.cmbsy-5{font-size:50%;font-weight: bold;}\n.cmbsy-5{font-weight: bold;}\n.cmbsy-5{font-weight: 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border-bottom-color: rgb(255,255,255); }\n#content table.equation-star, #content table.equation-star tbody tr td { border: 0px none rgb(255,255,255); }\nmtd.align-odd{margin-left:2em; text-align:right;}\nmtd.align-even{margin-right:2em; text-align:left;}\n.boxed{border: 1px solid black; padding-left:2px; padding-right:2px;}\n.rotatebox{display: inline-block;}\n.item-head{float:left;width:2em;clear:left;}\n.item-content{margin-left:2em;}\n .foreignobject {line-height:100%; font-size:120%; font-family:STIXgeneral,Times,Symbol,cmr10,CMSY10,CMEX10;padding:0; margin:0; text-align:center; }\nmath {vertical-align:baseline; line-height:100%; font-size:100%; font-family:STIXGeneral,Times,Symbol, cmr10,cmsy10,cmex10,cmmi10; font-style: normal; margin:0; padding:0; }\n\n.entry-title{display: none}\n\ndiv.newtheorem { margin-bottom: 2em; margin-top: 2em; border: 1px solid #333; background: #c7e4da; border-color: #4eb79e;}\ndiv.newtheorem h3 { background: #4eb79e; color: white; padding: 0px 15px 0px 15px; margin-top: 12px}\ndiv.newtheorem p { padding: 15px 15px 15px 15px; }\n\ndiv.newtheorem p span.head .ecbx-1095{font-weight: bold}\ndiv.newtheorem p .ecti-1095{font-style: italic}\ndiv.newtheorem div.custom-itemize{font-style: italic}\ndiv.quote{font-style: italic}\ndiv.newtheorem dl, dl.enumerate {display: grid; grid-template-columns: 5% auto; align-items: start; margin-top: 1em}\ndiv.newtheorem dl dd, dl.enumerate dd {margin-bottom: 0.5em}\ndiv.newtheorem dl dt, dl.enumerate dt {font-weight: normal; margin-top: 0px; text-align: right; margin-right: 15%}\ndiv.newtheorem dl dd {font-style: italic}\ndiv.newtheorem dl dt {font-style: italic}\ndiv.proof p span.ecti-1095 {font-style: italic}\ndiv.figure p img { margin-left: auto; margin-right: auto; display: block; }\ndiv.mefigcentered, div.figure { text-align: center }\n\ndl:after {content:\"\";display:table;clear:both;}\ndd {padding:.5em 0;}\ndl {width:100%;}\ndt, dd {display:inline-block; 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=\"ze5d76fd3a8ac\" class=\"sectionHead\"><span class=\"titlemark\">9.6 <\/span> <a id=\"x1-2810006\"><\/a>Drei asymptotische Formeln<\/h3> <a id=\"x1-281001r280\"><\/a> <h4 id=\"z003f5dd8e139\" class=\"subsectionHead\"><span class=\"titlemark\">9.6.1 <\/span> <a id=\"x1-2820001\"><\/a>Das Wallissche Produkt<\/h4> <p class=\"noindent\">Die Formel <\/p><math display=\"block\"><mtable class=\"align\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mi>\u03c0<\/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><mn>1<\/mn><\/mrow> <mrow><mi>n<\/mi><\/mrow><\/mfrac> <mfrac><mrow><msup><mrow><mo class=\"MathClass-open\">(<\/mo><msup><mrow><mn>2<\/mn><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msup><mi>n<\/mi><mo class=\"MathClass-punc\">!<\/mo><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mn>4<\/mn><\/mrow><\/msup><\/mrow> <mrow><msup><mrow><mo class=\"MathClass-open\">(<\/mo><mo class=\"MathClass-open\">(<\/mo><mn>2<\/mn><mi>n<\/mi><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-punc\">!<\/mo><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><\/mrow><\/mfrac><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"><mstyle class=\"label\" id=\"x1-282001r17\" \/><mstyle class=\"maketag\"><mtext>(9.17)<\/mtext><\/mstyle><mspace class=\"nbsp\" width=\"0.33em\" \/> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">nach John Wallis (1616-1703) wird das <span class=\"ecbx-1095\">Wallissche Produkt <\/span>genannt. <\/p><p class=\"indent\">Um Gleichung (<a href=\"..\/..\/chapter\/drei-asymptotische-formeln#x1-282001r17\">9.17<\/a>) zu beweisen, wollen wir zuerst f\u00fcr jedes <math display=\"inline\"><mi>k<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <msub><mrow><mi>\u2115<\/mi><\/mrow><mrow><mn>0<\/mn> <\/mrow> <\/msub> <\/math> das Integral <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><msub><mrow><mi>I<\/mi><\/mrow><mrow><mi>k<\/mi><\/mrow><\/msub> <mo class=\"MathClass-rel\">=<\/mo><msubsup><mrow><mo> \u222b  <\/mo><\/mrow><mrow><mn>0<\/mn><\/mrow><mrow><mi>\u03c0<\/mi><\/mrow><\/msubsup><msup><mrow><mi class=\"qopname\"> sin<\/mi><mo>  <\/mo><\/mrow><mrow><mi>k<\/mi><\/mrow><\/msup> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">mit vollst\u00e4ndiger Induktion berechnen. <\/p><p class=\"indent\">F\u00fcr <math display=\"inline\"><mi>k<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mn>0<\/mn><\/math> gilt <math display=\"inline\"><msub><mrow><mi>I<\/mi><\/mrow><mrow><mn>0<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-rel\">=<\/mo> <mi>\u03c0<\/mi><\/math> und f\u00fcr <math display=\"inline\"><mi>k<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mn>1<\/mn><\/math> gilt <span class=\"maperiod\"><math display=\"inline\"><msub><mrow><mi>I<\/mi><\/mrow><mrow><mn>1<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-rel\">=<\/mo><msubsup><mrow><mi class=\"MathClass-op\"> \u222b  <\/mi><mo> <\/mo><\/mrow><mrow><mn>0<\/mn><\/mrow><mrow><mi>\u03c0<\/mi><\/mrow><\/msubsup><mi class=\"qopname\"> sin<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mn>2<\/mn><\/math><\/span><span class=\"period\">.<\/span> Mit partieller Integration                                                                                                                                                                           erhalten wir f\u00fcr alle <math display=\"inline\"><mi>k<\/mi> <mo class=\"MathClass-rel\">\u2265<\/mo> <mn>2<\/mn><\/math> <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><msub><mrow><mi>I<\/mi><\/mrow><mrow><mi>k<\/mi><\/mrow><\/msub><\/mtd> <mtd class=\"align-even\"> <mo class=\"MathClass-rel\">=<\/mo><msubsup><mrow><mo> \u222b  <\/mo><\/mrow><mrow><mn>0<\/mn><\/mrow><mrow><mi>\u03c0<\/mi><\/mrow><\/msubsup><msup><mrow><mi class=\"qopname\"> sin<\/mi><mo>  <\/mo><\/mrow><mrow><mi>k<\/mi><\/mrow><\/msup> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi> <mo class=\"MathClass-rel\">=<\/mo><msubsup><mrow><mo> \u222b  <\/mo><\/mrow><mrow><mn>0<\/mn><\/mrow><mrow><mi>\u03c0<\/mi><\/mrow><\/msubsup><msup><mrow><mi class=\"qopname\"> sin<\/mi><mo>  <\/mo><\/mrow><mrow><mi>k<\/mi><mo class=\"MathClass-bin\">\u2212<\/mo><mn>1<\/mn><\/mrow><\/msup> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><mi class=\"qopname\">sin<\/mi><mo>  <\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\" \/> <mtd class=\"align-label\"> <mspace width=\"2em\" \/><\/mtd><\/mtr><mtr><mtd class=\"align-odd\" columnalign=\"right\" \/> <mtd class=\"align-even\"> <mo class=\"MathClass-rel\">=<\/mo><msubsup><mrow> <mrow><mo fence=\"true\" form=\"prefix\"> [<\/mo><mrow><msup><mrow><mi class=\"qopname\">sin<\/mi><mo>  <\/mo><\/mrow><mrow><mi>k<\/mi><mo class=\"MathClass-bin\">\u2212<\/mo><mn>1<\/mn><\/mrow><\/msup> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><mstyle><mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow \/><mo fence=\"true\" form=\"postfix\" \/><\/mrow><\/mstyle> <mo class=\"MathClass-bin\">\u2212<\/mo><mi class=\"qopname\"> cos<\/mi><mo>  <\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><mstyle><mrow><mo fence=\"true\" form=\"prefix\"> )<\/mo><mrow \/><mo fence=\"true\" form=\"postfix\" \/><\/mrow><\/mstyle><\/mrow><mo fence=\"true\" form=\"postfix\">]<\/mo><\/mrow><\/mrow><mrow> <mn>0<\/mn><\/mrow><mrow><mi>\u03c0<\/mi><\/mrow><\/msubsup> <mo class=\"MathClass-bin\">+<\/mo><msubsup><mrow><mo> \u222b  <\/mo><\/mrow><mrow><mn>0<\/mn><\/mrow><mrow><mi>\u03c0<\/mi><\/mrow><\/msubsup> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>k<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <mn>1<\/mn><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><msup><mrow><mi class=\"qopname\">sin<\/mi><mo>  <\/mo><\/mrow><mrow><mi>k<\/mi><mo class=\"MathClass-bin\">\u2212<\/mo><mn>2<\/mn><\/mrow><\/msup> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><msup><mrow><mi class=\"qopname\">cos<\/mi><mo>  <\/mo><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\" \/> <mtd class=\"align-label\"> <mspace width=\"2em\" \/><\/mtd><\/mtr><mtr><mtd class=\"align-odd\" columnalign=\"right\" \/> <mtd class=\"align-even\"> <mo class=\"MathClass-rel\">=<\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>k<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <mn>1<\/mn><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><msubsup><mrow><mo>\u222b  <\/mo><\/mrow><mrow><mn>0<\/mn><\/mrow><mrow><mi>\u03c0<\/mi><\/mrow><\/msubsup><msup><mrow><mi class=\"qopname\"> sin<\/mi><mo>  <\/mo><\/mrow><mrow><mi>k<\/mi><mo class=\"MathClass-bin\">\u2212<\/mo><mn>2<\/mn><\/mrow><\/msup> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mn>1<\/mn> <mo class=\"MathClass-bin\">\u2212<\/mo><msup><mrow><mi class=\"qopname\"> sin<\/mi><mo>  <\/mo><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\" \/> <mtd class=\"align-label\"> <mspace width=\"2em\" \/><\/mtd><\/mtr><mtr><mtd class=\"align-odd\" columnalign=\"right\" \/> <mtd class=\"align-even\"> <mo class=\"MathClass-rel\">=<\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>k<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <mn>1<\/mn><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><msub><mrow><mi>I<\/mi><\/mrow><mrow><mi>k<\/mi><mo class=\"MathClass-bin\">\u2212<\/mo><mn>2<\/mn><\/mrow><\/msub> <mo class=\"MathClass-bin\">\u2212<\/mo> <msub><mrow><mi>I<\/mi><\/mrow><mrow><mi>k<\/mi><\/mrow><\/msub><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-punc\">,<\/mo><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\" \/> <mtd class=\"align-label\"> <mspace width=\"2em\" \/><\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">woraus sich die Rekursionsformel <\/p><table id=\"z030660109524\" class=\"equation\"><tr><td> <a id=\"x1-282002r18\"><\/a> <math class=\"equation\" display=\"block\"> <msub><mrow><mi>I<\/mi><\/mrow><mrow><mi>k<\/mi><\/mrow><\/msub> <mo class=\"MathClass-rel\">=<\/mo> <mfrac><mrow><mi>k<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <mn>1<\/mn><\/mrow> <mrow><mi>k<\/mi><\/mrow><\/mfrac> <msub><mrow><mi>I<\/mi><\/mrow><mrow><mi>k<\/mi><mo class=\"MathClass-bin\">\u2212<\/mo><mn>2<\/mn><\/mrow><\/msub> <\/math><\/td><td class=\"eq-no\">(9.18)<\/td><\/tr><\/table> <p class=\"indent\">f\u00fcr jedes ganze <math display=\"inline\"><mi>k<\/mi> <mo class=\"MathClass-rel\">\u2265<\/mo> <mn>2<\/mn><\/math> ergibt. <\/p><p class=\"indent\">Somit ist f\u00fcr <math display=\"inline\"><mi>n<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2115<\/mi><\/math> <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><msub><mrow><mi>I<\/mi><\/mrow><mrow><mn>2<\/mn><mi>n<\/mi><\/mrow><\/msub> <mo class=\"MathClass-rel\">=<\/mo> <mfrac><mrow><mn>2<\/mn><mi>n<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <mn>1<\/mn><\/mrow> <mrow><mn>2<\/mn><mi>n<\/mi><\/mrow><\/mfrac> <msub><mrow><mi>I<\/mi><\/mrow><mrow><mn>2<\/mn><mi>n<\/mi><mo class=\"MathClass-bin\">\u2212<\/mo><mn>2<\/mn><\/mrow><\/msub><\/mtd> <mtd class=\"align-even\"> <mo class=\"MathClass-rel\">=<\/mo> <mfrac><mrow><mn>2<\/mn><mi>n<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <mn>1<\/mn><\/mrow> <mrow><mn>2<\/mn><mi>n<\/mi><\/mrow><\/mfrac> <mo class=\"MathClass-bin\">\u22c5<\/mo><mfrac><mrow><mn>2<\/mn><mi>n<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <mn>3<\/mn><\/mrow> <mrow><mn>2<\/mn><mi>n<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <mn>2<\/mn><\/mrow><\/mfrac><mo class=\"MathClass-rel\">\u22ef<\/mo><mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mn>2<\/mn><\/mrow><\/mfrac><msub><mrow><mi>I<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><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> <mfrac><mrow><mo class=\"MathClass-open\">(<\/mo><mn>2<\/mn><mi>n<\/mi><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-open\">(<\/mo><mn>2<\/mn><mi>n<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <mn>1<\/mn><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-open\">(<\/mo><mn>2<\/mn><mi>n<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <mn>2<\/mn><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-rel\">\u22ef<\/mo><mn>1<\/mn><\/mrow> <mrow><mstyle><mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow \/><mo fence=\"true\" form=\"postfix\" \/><\/mrow><\/mstyle><mo class=\"MathClass-open\">(<\/mo><mn>2<\/mn><mi>n<\/mi><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-open\">(<\/mo><mn>2<\/mn><mi>n<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <mn>2<\/mn><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-rel\">\u22ef<\/mo><mn>2<\/mn><msup><mrow><mstyle><mrow><mo fence=\"true\" form=\"prefix\"> )<\/mo><mrow \/><mo fence=\"true\" form=\"postfix\" \/><\/mrow><\/mstyle><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><\/mrow><\/mfrac> <msub><mrow><mi>I<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub> <mo class=\"MathClass-rel\">=<\/mo> <mfrac><mrow><mo class=\"MathClass-open\">(<\/mo><mn>2<\/mn><mi>n<\/mi><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-punc\">!<\/mo><\/mrow> <mrow><msup><mrow><mo class=\"MathClass-open\">(<\/mo><msup><mrow><mn>2<\/mn><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msup><mi>n<\/mi><mo class=\"MathClass-punc\">!<\/mo><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><\/mrow><\/mfrac> <mo class=\"MathClass-bin\">\u22c5<\/mo> <mi>\u03c0<\/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 den Bruch mit&nbsp;<math display=\"inline\"><mn>2<\/mn><mi>n<\/mi><mo class=\"MathClass-open\">(<\/mo><mn>2<\/mn><mi>n<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <mn>2<\/mn><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-rel\">\u22ef<\/mo><mn>2<\/mn><\/math> erweitert haben um im Z\u00e4hler&nbsp;<math display=\"inline\"><mo class=\"MathClass-open\">(<\/mo><mn>2<\/mn><mi>n<\/mi><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-punc\">!<\/mo><\/math> zu erhalten. Auf dieselbe Weise ergibt sich <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><msub><mrow><mi>I<\/mi><\/mrow><mrow><mn>2<\/mn><mi>n<\/mi><mo class=\"MathClass-bin\">+<\/mo><mn>1<\/mn><\/mrow><\/msub> <mo class=\"MathClass-rel\">=<\/mo> <mfrac><mrow><mn>2<\/mn><mi>n<\/mi><\/mrow> <mrow><mn>2<\/mn><mi>n<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mn>1<\/mn><\/mrow><\/mfrac> <mo class=\"MathClass-bin\">\u22c5<\/mo><mfrac><mrow><mn>2<\/mn><mi>n<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <mn>2<\/mn><\/mrow> <mrow><mn>2<\/mn><mi>n<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <mn>1<\/mn><\/mrow><\/mfrac><mo class=\"MathClass-rel\">\u22ef<\/mo><mfrac><mrow><mn>2<\/mn><\/mrow> <mrow><mn>3<\/mn><\/mrow><\/mfrac><msub><mrow><mi>I<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub> <mo class=\"MathClass-rel\">=<\/mo> <mfrac><mrow><mstyle><mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow \/><mo fence=\"true\" form=\"postfix\" \/><\/mrow><\/mstyle><mo class=\"MathClass-open\">(<\/mo><mn>2<\/mn><mi>n<\/mi><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-open\">(<\/mo><mn>2<\/mn><mi>n<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <mn>2<\/mn><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-rel\">\u22ef<\/mo><mn>2<\/mn><msup><mrow><mstyle><mrow><mo fence=\"true\" form=\"prefix\"> )<\/mo><mrow \/><mo fence=\"true\" form=\"postfix\" \/><\/mrow><\/mstyle><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><\/mrow> <mrow><mo class=\"MathClass-open\">(<\/mo><mn>2<\/mn><mi>n<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mn>1<\/mn><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-open\">(<\/mo><mn>2<\/mn><mi>n<\/mi><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-rel\">\u22ef<\/mo><mn>1<\/mn><\/mrow><\/mfrac> <mo class=\"MathClass-bin\">\u22c5<\/mo> <mn>2<\/mn> <mo class=\"MathClass-rel\">=<\/mo> <mfrac><mrow><msup><mrow><mo class=\"MathClass-open\">(<\/mo><msup><mrow><mn>2<\/mn><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msup><mi>n<\/mi><mo class=\"MathClass-punc\">!<\/mo><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><\/mrow> <mrow><mo class=\"MathClass-open\">(<\/mo><mn>2<\/mn><mi>n<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mn>1<\/mn><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-punc\">!<\/mo><\/mrow><\/mfrac> <mo class=\"MathClass-bin\">\u22c5<\/mo> <mn>2<\/mn><mo class=\"MathClass-punc\">.<\/mo><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">Die Folge&nbsp;<math display=\"inline\"><msub><mrow><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>I<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><\/math> erf\u00fcllt noch eine weitere Eigenschaft: Denn f\u00fcr <math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> [<\/mo><mrow><mn>0<\/mn><mo class=\"MathClass-punc\">,<\/mo><mi>\u03c0<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">]<\/mo><\/mrow><\/math> ist <math display=\"inline\"><mi class=\"qopname\"> sin<\/mi><mo>  <\/mo> <mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">\u2208<\/mo> <mo class=\"MathClass-open\">[<\/mo><mn>0<\/mn><mo class=\"MathClass-punc\">,<\/mo><mn>1<\/mn><mo class=\"MathClass-close\">]<\/mo><\/math> und somit <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><msup><mrow><mi class=\"qopname\"> sin<\/mi><mo>  <\/mo><\/mrow><mrow><mn>2<\/mn><mi>n<\/mi><mo class=\"MathClass-bin\">+<\/mo><mn>2<\/mn><\/mrow><\/msup><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">\u2264<\/mo><msup><mrow><mi class=\"qopname\"> sin<\/mi><mo>  <\/mo><\/mrow><mrow><mn>2<\/mn><mi>n<\/mi><mo class=\"MathClass-bin\">+<\/mo><mn>1<\/mn><\/mrow><\/msup><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">\u2264<\/mo><msup><mrow><mi class=\"qopname\"> sin<\/mi><mo>  <\/mo><\/mrow><mrow><mn>2<\/mn><mi>n<\/mi><\/mrow><\/msup><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">f\u00fcr alle <math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mo class=\"MathClass-open\">[<\/mo><mn>0<\/mn><mo class=\"MathClass-punc\">,<\/mo><mi>\u03c0<\/mi><mo class=\"MathClass-close\">]<\/mo><\/math> and <span class=\"maperiod\"><math display=\"inline\"><mi>n<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2115<\/mi><\/math><\/span><span class=\"period\">.<\/span> Durch Integration folgt die Monotonieungleichung <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><msub><mrow><mi>I<\/mi><\/mrow><mrow><mn>2<\/mn><mi>n<\/mi><mo class=\"MathClass-bin\">+<\/mo><mn>2<\/mn><\/mrow><\/msub> <mo class=\"MathClass-rel\">=<\/mo><msubsup><mrow><mo> \u222b  <\/mo><\/mrow><mrow><mn>0<\/mn><\/mrow><mrow><mi>\u03c0<\/mi><\/mrow><\/msubsup><msup><mrow><mi class=\"qopname\"> sin<\/mi><mo>  <\/mo><\/mrow><mrow><mn>2<\/mn><mi>n<\/mi><mo class=\"MathClass-bin\">+<\/mo><mn>2<\/mn><\/mrow><\/msup><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2264<\/mo> <msub><mrow><mi>I<\/mi><\/mrow><mrow> <mn>2<\/mn><mi>n<\/mi><mo class=\"MathClass-bin\">+<\/mo><mn>1<\/mn><\/mrow><\/msub> <mo class=\"MathClass-rel\">=<\/mo><msubsup><mrow><mo> \u222b  <\/mo><\/mrow><mrow><mn>0<\/mn><\/mrow><mrow><mi>\u03c0<\/mi><\/mrow><\/msubsup><msup><mrow><mi class=\"qopname\"> sin<\/mi><mo>  <\/mo><\/mrow><mrow><mn>2<\/mn><mi>n<\/mi><mo class=\"MathClass-bin\">+<\/mo><mn>1<\/mn><\/mrow><\/msup><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2264<\/mo> <msub><mrow><mi>I<\/mi><\/mrow><mrow> <mn>2<\/mn><mi>n<\/mi><\/mrow><\/msub> <mo class=\"MathClass-rel\">=<\/mo><msubsup><mrow><mo> \u222b  <\/mo><\/mrow><mrow><mn>0<\/mn><\/mrow><mrow><mi>\u03c0<\/mi><\/mrow><\/msubsup><msup><mrow><mi class=\"qopname\"> sin<\/mi><mo>  <\/mo><\/mrow><mrow><mn>2<\/mn><mi>n<\/mi><\/mrow><\/msup><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">und mittels (<a href=\"..\/..\/chapter\/drei-asymptotische-formeln#x1-282002r18\">9.18<\/a>) auch <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><msub><mrow><mi>I<\/mi><\/mrow><mrow><mn>2<\/mn><mi>n<\/mi><mo class=\"MathClass-bin\">+<\/mo><mn>2<\/mn><\/mrow><\/msub> <mo class=\"MathClass-rel\">=<\/mo> <mfrac><mrow><mn>2<\/mn><mi>n<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mn>1<\/mn><\/mrow> <mrow><mn>2<\/mn><mi>n<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mn>2<\/mn><\/mrow><\/mfrac><munder class=\"msub\"><mrow><munder accentunder=\"false\"><mrow> <mfrac><mrow><mo class=\"MathClass-open\">(<\/mo><mn>2<\/mn><mi>n<\/mi><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-punc\">!<\/mo><\/mrow> <mrow><msup><mrow><mo class=\"MathClass-open\">(<\/mo><msup><mrow><mn>2<\/mn><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msup><mi>n<\/mi><mo class=\"MathClass-punc\">!<\/mo><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><\/mrow><\/mfrac><mi>\u03c0<\/mi><\/mrow><mo>\ufe38<\/mo><\/munder> <\/mrow><mrow><msub><mrow><mi>I<\/mi><\/mrow><mrow><mn>2<\/mn><mi>n<\/mi><\/mrow><\/msub><\/mrow><\/munder> <mo class=\"MathClass-rel\">\u2264<\/mo> <mfrac><mrow><msup><mrow><mo class=\"MathClass-open\">(<\/mo><msup><mrow><mn>2<\/mn><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msup><mi>n<\/mi><mo class=\"MathClass-punc\">!<\/mo><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><\/mrow> <mrow><mo class=\"MathClass-open\">(<\/mo><mn>2<\/mn><mi>n<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mn>1<\/mn><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-punc\">!<\/mo><\/mrow><\/mfrac><mn>2<\/mn> <mo class=\"MathClass-rel\">\u2264<\/mo> <msub><mrow><mi>I<\/mi><\/mrow><mrow><mn>2<\/mn><mi>n<\/mi><\/mrow><\/msub> <mo class=\"MathClass-rel\">=<\/mo> <mfrac><mrow><mo class=\"MathClass-open\">(<\/mo><mn>2<\/mn><mi>n<\/mi><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-punc\">!<\/mo><\/mrow> <mrow><msup><mrow><mo class=\"MathClass-open\">(<\/mo><msup><mrow><mn>2<\/mn><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msup><mi>n<\/mi><mo class=\"MathClass-punc\">!<\/mo><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><\/mrow><\/mfrac><mi>\u03c0<\/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\">Wir multiplizieren diese Ungleichungskette mit <math display=\"inline\"><mfrac><mrow><msup><mrow><mo class=\"MathClass-open\">(<\/mo><msup><mrow><mn>2<\/mn><\/mrow><mrow><mi>n<\/mi> <\/mrow> <\/msup> <mi>n<\/mi><mo class=\"MathClass-punc\">!<\/mo><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mn>2<\/mn> <\/mrow> <\/msup> <\/mrow> <mrow><mo class=\"MathClass-open\">(<\/mo><mn>2<\/mn><mi>n<\/mi><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-punc\">!<\/mo><\/mrow><\/mfrac> <mfrac><mrow><mn>2<\/mn><mi>n<\/mi><mo class=\"MathClass-bin\">+<\/mo><mn>1<\/mn><\/mrow> <mrow><mn>2<\/mn><mi>n<\/mi><\/mrow><\/mfrac> <\/math> und erhalten                                                                                                                                                                           <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"> <mfrac><mrow><msup><mrow><mo class=\"MathClass-open\">(<\/mo><mn>2<\/mn><mi>n<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mn>1<\/mn><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><\/mrow> <mrow><mn>2<\/mn><mi>n<\/mi><mo class=\"MathClass-open\">(<\/mo><mn>2<\/mn><mi>n<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mn>2<\/mn><mo class=\"MathClass-close\">)<\/mo><\/mrow><\/mfrac><mi>\u03c0<\/mi> <mo class=\"MathClass-rel\">\u2264<\/mo> <mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mi>n<\/mi><\/mrow><\/mfrac> <mfrac><mrow><msup><mrow><mo class=\"MathClass-open\">(<\/mo><msup><mrow><mn>2<\/mn><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msup><mi>n<\/mi><mo class=\"MathClass-punc\">!<\/mo><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mn>4<\/mn><\/mrow><\/msup><\/mrow> <mrow><msup><mrow><mo class=\"MathClass-open\">(<\/mo><mo class=\"MathClass-open\">(<\/mo><mn>2<\/mn><mi>n<\/mi><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-punc\">!<\/mo><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><\/mrow><\/mfrac> <mo class=\"MathClass-rel\">\u2264<\/mo> <mfrac><mrow><mn>2<\/mn><mi>n<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mn>1<\/mn><\/mrow> <mrow><mn>2<\/mn><mi>n<\/mi><\/mrow><\/mfrac> <mi>\u03c0<\/mi><mo class=\"MathClass-punc\">.<\/mo><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">Dies beweist die Formel (<a href=\"..\/..\/chapter\/drei-asymptotische-formeln#x1-282001r17\">9.17<\/a>) nach Wallis unter Anwendung des Sandwich-Lemmas (Lemma&nbsp;<a href=\"..\/..\/chapter\/reelle-folgen#x1-157004r2\">6.2<\/a>). <a id=\"x1-282003r282\"><\/a> <\/p> <h4 id=\"zf481290825ce\" class=\"subsectionHead\"><span class=\"titlemark\">9.6.2 <\/span> <a id=\"x1-2830002\"><\/a>Stirling-Formel<\/h4> <p class=\"noindent\">Wir verwenden nun das Wallissche Produkt und Taylor-Approximationen, um das asymptotische Verhalten von <math display=\"inline\"><mi>n<\/mi><mo class=\"MathClass-punc\">!<\/mo><\/math> zu bestimmen. Genauer formuliert wollen wir die <span class=\"ecbx-1095\">Stirling-Formel<\/span> <\/p><math display=\"block\"><mtable class=\"align\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><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><mi>n<\/mi><mo class=\"MathClass-punc\">!<\/mo><\/mrow> <mrow><msqrt><mrow><mn>2<\/mn><mi>\u03c0<\/mi><mi>n<\/mi><\/mrow><\/msqrt><msup><mrow> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mfrac><mrow><mi>n<\/mi><\/mrow> <mrow><mi class=\"qopname\"> e<\/mi><mo>  <\/mo> <\/mrow><\/mfrac> <\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <\/mrow><mrow><mi>n<\/mi><\/mrow><\/msup><\/mrow><\/mfrac> <mo class=\"MathClass-rel\">=<\/mo> <mn>1<\/mn><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"><mstyle class=\"label\" id=\"x1-283001r19\" \/><mstyle class=\"maketag\"><mtext>(9.19)<\/mtext><\/mstyle><mspace class=\"nbsp\" width=\"0.33em\" \/> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">beweisen. Diese ist nach James Stirling (1692-1770) benannt und wird auch als <math display=\"inline\"><mi>n<\/mi><mo class=\"MathClass-punc\">!<\/mo> <mo class=\"MathClass-rel\">\u2248<\/mo> <msqrt> <mrow><mn>2<\/mn><mi>\u03c0<\/mi><mi>n<\/mi><\/mrow><\/msqrt><msup><mrow> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow> <mfrac> <mrow> <mi>n<\/mi><\/mrow> <mrow><mi class=\"qopname\"> e<\/mi><mo>  <\/mo> <\/mrow><\/mfrac> <\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <\/mrow><mrow><mi>n<\/mi><\/mrow><\/msup><\/math> geschrieben. <\/p><p class=\"indent\">Der Grundgedanke f\u00fcr den Beweis von (<a href=\"..\/..\/chapter\/drei-asymptotische-formeln#x1-283001r19\">9.19<\/a>) ist die Gleichung                                                                                                                                                                           <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mi class=\"qopname\"> log<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>n<\/mi><mo class=\"MathClass-punc\">!<\/mo><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">=<\/mo><mi class=\"qopname\"> log<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mn>1<\/mn><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">+<\/mo><mi class=\"qopname\"> log<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mn>2<\/mn><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">+<\/mo> <mi class=\"qopname\">\u2026<\/mi><mo>  <\/mo> <mo class=\"MathClass-bin\">+<\/mo><mi class=\"qopname\"> log<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/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 <math display=\"inline\"><mi>n<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2115<\/mi><\/math> genauer zu betrachten, wobei wir die Summe aber durch das Integral <math display=\"inline\"><msubsup><mrow><mi class=\"MathClass-op\">\u222b  <\/mi><mo> <\/mo><\/mrow><mrow><mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mn>2<\/mn><\/mrow><\/mfrac> <\/mrow><mrow><mi>n<\/mi><mo class=\"MathClass-bin\">+<\/mo><mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mn>2<\/mn><\/mrow><\/mfrac> <\/mrow><\/msubsup><mi class=\"qopname\"> log<\/mi><mo>  <\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi><\/math> ersetzen wollen, da dieses einfacher zu berechnen ist. <\/p><p class=\"indent\">F\u00fcr <math display=\"inline\"><mi>k<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2115<\/mi><\/math> und <math display=\"inline\"><mi>t<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> [<\/mo><mrow><mi>k<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo><mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mn>2<\/mn><\/mrow><\/mfrac><mo class=\"MathClass-punc\">,<\/mo><mi>k<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mn>2<\/mn><\/mrow><\/mfrac> <\/mrow><mo fence=\"true\" form=\"postfix\">]<\/mo><\/mrow><\/math> gilt nach Beispiel <a href=\"..\/..\/chapter\/numerische-integration#x1-280002r59\">9.59<\/a> <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mi class=\"qopname\"> log<\/mi><mo>  <\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>t<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-rel\">=<\/mo><mi class=\"qopname\"> log<\/mi><mo>  <\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>k<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-bin\">+<\/mo><mfrac><mrow> <mn>1<\/mn><\/mrow> <mrow><mi>k<\/mi><\/mrow><\/mfrac> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>t<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>k<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-bin\">+<\/mo> <mi>O<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mfrac><mrow> <mn>1<\/mn><\/mrow> <mrow><msup><mrow><mi>k<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><\/mrow><\/mfrac> <\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-punc\">.<\/mo><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">Wir erhalten <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><msubsup><mrow><mo> \u222b  <\/mo><\/mrow><mrow><mi>k<\/mi><mo class=\"MathClass-bin\">\u2212<\/mo><mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mn>2<\/mn><\/mrow><\/mfrac> <\/mrow><mrow><mi>k<\/mi><mo class=\"MathClass-bin\">+<\/mo><mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mn>2<\/mn><\/mrow><\/mfrac> <\/mrow><\/msubsup><mi class=\"qopname\"> log<\/mi><mo>  <\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>t<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>t<\/mi> <mo class=\"MathClass-rel\">=<\/mo><mi class=\"qopname\"> log<\/mi><mo>  <\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>k<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-bin\">+<\/mo> <mi>O<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mfrac><mrow> <mn>1<\/mn><\/mrow><mrow><msup><mrow><mi>k<\/mi><\/mrow><mrow><mn>2<\/mn> <\/mrow><\/msup><\/mrow><\/mfrac> <\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">f\u00fcr alle <span class=\"maperiod\"><math display=\"inline\"><mi>k<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2115<\/mi><\/math><\/span><span class=\"period\">,<\/span> da sich der lineare Term bei der Integration wegl\u00f6scht.                                                                                                                                                                           <\/p><p class=\"indent\">F\u00fcr ein festes&nbsp;<math display=\"inline\"><mi>n<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2115<\/mi><\/math> und die Summe \u00fcber <math display=\"inline\"><mi>k<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> {<\/mo><mrow><mn>1<\/mn><mo class=\"MathClass-punc\">,<\/mo><mi class=\"MathClass-op\">\u2026<\/mi><mo> <\/mo><mo class=\"MathClass-punc\">,<\/mo><mi>n<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">}<\/mo><\/mrow><\/math> folgt daraus <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mi class=\"qopname\"> log<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>n<\/mi><mo class=\"MathClass-punc\">!<\/mo><mo class=\"MathClass-close\">)<\/mo><\/mtd> <mtd class=\"align-even\"> <mo class=\"MathClass-rel\">=<\/mo><munderover accent=\"false\" accentunder=\"false\"><mrow><mo> \u2211<\/mo> <\/mrow><mrow><mi>k<\/mi><mo class=\"MathClass-rel\">=<\/mo><mn>1<\/mn><\/mrow><mrow><mi>n<\/mi><\/mrow><\/munderover><mi class=\"qopname\"> log<\/mi><mo>  <\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>k<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-rel\">=<\/mo><munderover accent=\"false\" accentunder=\"false\"><mrow><mo> \u2211<\/mo> <\/mrow><mrow><mi>k<\/mi><mo class=\"MathClass-rel\">=<\/mo><mn>1<\/mn><\/mrow><mrow><mi>n<\/mi><\/mrow><\/munderover><msubsup><mrow><mo> \u222b  <\/mo><\/mrow><mrow><mi>k<\/mi><mo class=\"MathClass-bin\">\u2212<\/mo><mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mn>2<\/mn><\/mrow><\/mfrac> <\/mrow><mrow><mi>k<\/mi><mo class=\"MathClass-bin\">+<\/mo><mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mn>2<\/mn><\/mrow><\/mfrac> <\/mrow><\/msubsup><mi class=\"qopname\"> log<\/mi><mo>  <\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>t<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>t<\/mi> <mo class=\"MathClass-bin\">+<\/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><mi>O<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mfrac><mrow> <mn>1<\/mn><\/mrow> <mrow><msup><mrow><mi>k<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><\/mrow><\/mfrac> <\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-rel\">=<\/mo><msubsup><mrow><mo> \u222b  <\/mo><\/mrow><mrow><mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mn>2<\/mn><\/mrow><\/mfrac> <\/mrow><mrow><mi>n<\/mi><mo class=\"MathClass-bin\">+<\/mo><mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mn>2<\/mn><\/mrow><\/mfrac> <\/mrow><\/msubsup><mi class=\"qopname\"> log<\/mi><mo>  <\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>t<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>t<\/mi> <mo class=\"MathClass-bin\">+<\/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><mi>O<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mfrac><mrow> <mn>1<\/mn><\/mrow> <mrow><msup><mrow><mi>k<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><\/mrow><\/mfrac> <\/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\">=<\/mo> <mo class=\"MathClass-open\">(<\/mo><mi>n<\/mi> <mo class=\"MathClass-bin\">+<\/mo><mfrac><mrow> <mn>1<\/mn><\/mrow> <mrow><mn>2<\/mn><\/mrow><\/mfrac><mo class=\"MathClass-close\">)<\/mo><mi class=\"qopname\">log<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>n<\/mi> <mo class=\"MathClass-bin\">+<\/mo><mfrac><mrow> <mn>1<\/mn><\/mrow> <mrow><mn>2<\/mn><\/mrow><\/mfrac><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">\u2212<\/mo> <mo class=\"MathClass-open\">(<\/mo><mi>n<\/mi> <mo class=\"MathClass-bin\">+<\/mo><mfrac><mrow> <mn>1<\/mn><\/mrow> <mrow><mn>2<\/mn><\/mrow><\/mfrac><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>c<\/mi> <mo class=\"MathClass-bin\">+<\/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><mi>O<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mfrac><mrow> <mn>1<\/mn><\/mrow> <mrow><msup><mrow><mi>k<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><\/mrow><\/mfrac> <\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-punc\">,<\/mo><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\" \/> <mtd class=\"align-label\"> <mspace width=\"2em\" \/><\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">wobei sich die Konstante <math display=\"inline\"><mi>c<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u211d<\/mi><\/math> durch Auswerten der Stammfunktion bei der unteren Grenze <math display=\"inline\"><mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mn>2<\/mn><\/mrow><\/mfrac><\/math> ergibt. Wiederum nach Taylorapproximation gilt <span class=\"maperiod\"><math display=\"inline\"><mi class=\"qopname\"> log<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>n<\/mi> <mo class=\"MathClass-bin\">+<\/mo><mfrac><mrow> <mn>1<\/mn><\/mrow> <mrow><mn>2<\/mn><\/mrow><\/mfrac><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">=<\/mo><mi class=\"qopname\"> log<\/mi><mo>  <\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>n<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-bin\">+<\/mo><mfrac><mrow> <mn>1<\/mn><\/mrow> <mrow><mi>n<\/mi><\/mrow><\/mfrac> <mo class=\"MathClass-bin\">\u22c5<\/mo><mfrac><mrow> <mn>1<\/mn><\/mrow> <mrow><mn>2<\/mn><\/mrow><\/mfrac> <mo class=\"MathClass-bin\">+<\/mo> <mi>O<\/mi><mo class=\"MathClass-open\">(<\/mo><mfrac><mrow> <mn>1<\/mn><\/mrow> <mrow><msup><mrow><mi>n<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><\/mrow><\/mfrac><mo class=\"MathClass-close\">)<\/mo><\/math><\/span><span class=\"period\">.<\/span> Wir multiplizieren diese Approximation mit <math display=\"inline\"><mo class=\"MathClass-open\">(<\/mo><mi>n<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mn>2<\/mn><\/mrow><\/mfrac><mo class=\"MathClass-close\">)<\/mo><\/math> und erhalten (mit&nbsp;<math display=\"inline\"><mo class=\"MathClass-open\">(<\/mo><mi>n<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mn>2<\/mn><\/mrow><\/mfrac><mo class=\"MathClass-close\">)<\/mo><mi>O<\/mi><mo class=\"MathClass-open\">(<\/mo> <mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><msup><mrow><mi>n<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><\/mrow><\/mfrac><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <mi>O<\/mi><mo class=\"MathClass-open\">(<\/mo><mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mi>n<\/mi><\/mrow><\/mfrac><mo class=\"MathClass-close\">)<\/mo><\/math>) somit <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mi class=\"qopname\"> log<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>n<\/mi><mo class=\"MathClass-punc\">!<\/mo><mo class=\"MathClass-close\">)<\/mo><\/mtd> <mtd class=\"align-even\"> <mo class=\"MathClass-rel\">=<\/mo> <mo class=\"MathClass-open\">(<\/mo><mi>n<\/mi> <mo class=\"MathClass-bin\">+<\/mo><mfrac><mrow> <mn>1<\/mn><\/mrow> <mrow><mn>2<\/mn><\/mrow><\/mfrac><mo class=\"MathClass-close\">)<\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi class=\"qopname\">log<\/mi><mo>  <\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>n<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-bin\">+<\/mo><mfrac><mrow> <mn>1<\/mn><\/mrow> <mrow><mn>2<\/mn><mi>n<\/mi><\/mrow><\/mfrac> <mo class=\"MathClass-bin\">+<\/mo> <mi>O<\/mi><mo class=\"MathClass-open\">(<\/mo><mfrac><mrow> <mn>1<\/mn><\/mrow> <mrow><msup><mrow><mi>n<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><\/mrow><\/mfrac><mo class=\"MathClass-close\">)<\/mo><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-bin\">\u2212<\/mo> <mo class=\"MathClass-open\">(<\/mo><mi>n<\/mi> <mo class=\"MathClass-bin\">+<\/mo><mfrac><mrow> <mn>1<\/mn><\/mrow> <mrow><mn>2<\/mn><\/mrow><\/mfrac><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>c<\/mi> <mo class=\"MathClass-bin\">+<\/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><mi>O<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mfrac><mrow> <mn>1<\/mn><\/mrow> <mrow><msup><mrow><mi>k<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><\/mrow><\/mfrac> <\/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\">=<\/mo> <mo class=\"MathClass-open\">(<\/mo><mi>n<\/mi> <mo class=\"MathClass-bin\">+<\/mo><mfrac><mrow> <mn>1<\/mn><\/mrow> <mrow><mn>2<\/mn><\/mrow><\/mfrac><mo class=\"MathClass-close\">)<\/mo><mi class=\"qopname\">log<\/mi><mo>  <\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>n<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-bin\">+<\/mo><mfrac><mrow> <mn>1<\/mn><\/mrow> <mrow><mn>2<\/mn><\/mrow><\/mfrac> <mo class=\"MathClass-bin\">+<\/mo> <mi>O<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mfrac><mrow> <mn>1<\/mn><\/mrow> <mrow><mi>n<\/mi><\/mrow><\/mfrac> <\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>n<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo><mfrac><mrow> <mn>1<\/mn><\/mrow> <mrow><mn>2<\/mn><\/mrow><\/mfrac> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>c<\/mi> <mo class=\"MathClass-bin\">+<\/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><mi>O<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mfrac><mrow> <mn>1<\/mn><\/mrow> <mrow><msup><mrow><mi>k<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><\/mrow><\/mfrac> <\/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\">=<\/mo> <mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mn>2<\/mn><\/mrow><\/mfrac><mi class=\"qopname\">log<\/mi><mo>  <\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>n<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-bin\">+<\/mo> <mi>n<\/mi><mi class=\"qopname\">log<\/mi><mo>  <\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>n<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>n<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\" \/> <mtd class=\"align-label\"> <mspace width=\"2em\" \/><\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">f\u00fcr eine konvergente 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> in <math display=\"inline\"><mi>\u211d<\/mi><\/math> (wobei wir Proposition&nbsp;<a href=\"..\/..\/chapter\/absolute-konvergenz#x1-192001r28\">7.28<\/a> verwendet haben um die absolut konvergenten Fehlerterme&nbsp;<math display=\"inline\"><msub><mrow><mi>\ud835\udf00<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub> <mo class=\"MathClass-rel\">=<\/mo> <mi>O<\/mi><mo class=\"MathClass-open\">(<\/mo> <mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><msup><mrow><mi>n<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><\/mrow><\/mfrac><mo class=\"MathClass-close\">)<\/mo><\/math> aufzusummieren). Wenden wir die Exponentialabbildung an, so erhalten wir f\u00fcr alle <math display=\"inline\"><mi>n<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2115<\/mi><\/math> <\/p><math display=\"block\"><mtable class=\"align\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mi>n<\/mi><mo class=\"MathClass-punc\">!<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <msqrt><mrow><mi>n<\/mi><\/mrow><\/msqrt><msup><mrow><mi>n<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msup><msup><mrow><mi class=\"qopname\"> e<\/mi><mo>  <\/mo><\/mrow><mrow><mo class=\"MathClass-bin\">\u2212<\/mo><mi>n<\/mi><\/mrow><\/msup><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\"><mstyle class=\"label\" id=\"x1-283002r20\" \/><mstyle class=\"maketag\"><mtext>(9.20)<\/mtext><\/mstyle><mspace class=\"nbsp\" width=\"0.33em\" \/> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">wobei <math display=\"inline\"><msub><mrow><mi>b<\/mi><\/mrow><mrow><mi>n<\/mi> <\/mrow> <\/msub> <mo class=\"MathClass-rel\">=<\/mo><mi class=\"qopname\"> exp<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><\/math> ist. Da <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> konvergent ist, ist auch <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> konvergent und der Grenzwert <math display=\"inline\"><mi>B<\/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><mi class=\"qopname\">exp<\/mi><mo>  <\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-rel\">=<\/mo><mi class=\"qopname\"> exp<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><munder class=\"msub\"><mrow><mi class=\"qopname\">lim<\/mi><mo>  <\/mo><\/mrow><mrow><mi>n<\/mi><mo class=\"MathClass-rel\">\u2192<\/mo><mi>\u221e<\/mi><\/mrow><\/munder><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><\/math> ist positiv. <\/p><p class=\"indent\">Um <math display=\"inline\"><mi>B<\/mi><\/math> zu berechnen, verwenden wir (<a href=\"..\/..\/chapter\/drei-asymptotische-formeln#x1-283002r20\">9.20<\/a>) gemeinsam mit dem Wallisschen Produkt (<a href=\"..\/..\/chapter\/drei-asymptotische-formeln#x1-282001r17\">9.17<\/a>). Es gilt <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mi>\u03c0<\/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><mn>1<\/mn><\/mrow> <mrow><mi>n<\/mi><\/mrow><\/mfrac> <mfrac><mrow><msup><mrow><mo class=\"MathClass-open\">(<\/mo><msup><mrow><mn>2<\/mn><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msup><mi>n<\/mi><mo class=\"MathClass-punc\">!<\/mo><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mn>4<\/mn><\/mrow><\/msup><\/mrow> <mrow><msup><mrow><mo class=\"MathClass-open\">(<\/mo><mo class=\"MathClass-open\">(<\/mo><mn>2<\/mn><mi>n<\/mi><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-punc\">!<\/mo><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><\/mrow><\/mfrac> <mo class=\"MathClass-rel\">=<\/mo><munder class=\"msub\"><mrow><mi class=\"qopname\"> lim<\/mi><mo>  <\/mo><\/mrow><mrow><mi>n<\/mi><mo class=\"MathClass-rel\">\u2192<\/mo><mi>\u221e<\/mi><\/mrow><\/munder><mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mi>n<\/mi><\/mrow><\/mfrac> <mfrac><mrow><msup><mrow><mrow><mo class=\"MathClass-open\" fence=\"true\" mathsize=\"1.19em\">(<\/mo><mrow><msup><mrow><mn>2<\/mn><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msup><msqrt><mrow><mi>n<\/mi><\/mrow><\/msqrt><msup><mrow><mi>n<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msup><msup><mrow><mi class=\"qopname\"> e<\/mi><mo>  <\/mo><\/mrow><mrow><mo class=\"MathClass-bin\">\u2212<\/mo><mi>n<\/mi><\/mrow><\/msup><msub><mrow><mi>b<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><\/mrow><mo class=\"MathClass-close\" fence=\"true\" mathsize=\"1.19em\">)<\/mo><\/mrow><\/mrow><mrow><mn>4<\/mn><\/mrow><\/msup><\/mrow> <mrow><msup><mrow><mrow><mo class=\"MathClass-open\" fence=\"true\" mathsize=\"1.19em\">(<\/mo><mrow><msqrt><mrow><mn>2<\/mn><mi>n<\/mi><\/mrow><\/msqrt><msup><mrow><mo class=\"MathClass-open\">(<\/mo><mn>2<\/mn><mi>n<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mn>2<\/mn><mi>n<\/mi><\/mrow><\/msup><msup><mrow><mi>e<\/mi><\/mrow><mrow><mo class=\"MathClass-bin\">\u2212<\/mo><mn>2<\/mn><mi>n<\/mi><\/mrow><\/msup><msub><mrow><mi>b<\/mi><\/mrow><mrow><mn>2<\/mn><mi>n<\/mi><\/mrow><\/msub><\/mrow><mo class=\"MathClass-close\" fence=\"true\" mathsize=\"1.19em\">)<\/mo><\/mrow><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><\/mrow><\/mfrac> <mo class=\"MathClass-rel\">=<\/mo><munder class=\"msub\"><mrow><mi class=\"qopname\"> lim<\/mi><mo>  <\/mo><\/mrow><mrow><mi>n<\/mi><mo class=\"MathClass-rel\">\u2192<\/mo><mi>\u221e<\/mi><\/mrow><\/munder> <mfrac><mrow><msubsup><mrow><mi>b<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><mrow><mn>4<\/mn><\/mrow><\/msubsup><\/mrow> <mrow><mn>2<\/mn><msubsup><mrow><mi>b<\/mi><\/mrow><mrow><mn>2<\/mn><mi>n<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msubsup><\/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\">da die Potenzen von <span class=\"maperiod\"><math display=\"inline\"><mn>2<\/mn><\/math><\/span><span class=\"period\">,<\/span> von&nbsp;<math display=\"inline\"><mi>n<\/mi><\/math> und von&nbsp;<math display=\"inline\"><mi>e<\/mi><\/math> sich jeweils direkt k\u00fcrzen, und die Wurzelterme gemeinsam mit dem&nbsp;<math display=\"inline\"> <mfrac> <mrow> <mn>1<\/mn><\/mrow> <mrow><mi>n<\/mi><\/mrow><\/mfrac><\/math> vor dem Bruch sich zu einem&nbsp;<math display=\"inline\"><mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mn>2<\/mn><\/mrow><\/mfrac><\/math> vereinfachen. Daraus folgt <\/p><table id=\"z45543ab021a2\" class=\"equation-star\"><tr><td> <math class=\"equation\" display=\"block\"> <mi>\u03c0<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mfrac><mrow><msup><mrow><mi>B<\/mi><\/mrow><mrow><mn>4<\/mn><\/mrow><\/msup><\/mrow> <mrow><mn>2<\/mn><msup><mrow><mi>B<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><\/mrow><\/mfrac> <mo class=\"MathClass-rel\">=<\/mo><mfrac><mrow> <mn>1<\/mn><\/mrow> <mrow><mn>2<\/mn><\/mrow><\/mfrac><msup><mrow><mi>B<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup> <\/math><\/td><\/tr><\/table> <p class=\"indent\">und somit muss <math display=\"inline\"><mi>B<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <msqrt><mrow><mn>2<\/mn><mi>\u03c0<\/mi><\/mrow><\/msqrt><\/math> gelten. Aus Gleichung (<a href=\"..\/..\/chapter\/drei-asymptotische-formeln#x1-283002r20\">9.20<\/a>) erhalten wir <\/p><table id=\"z6dbf5813031d\" class=\"equation-star\"><tr><td> <math class=\"equation\" display=\"block\"> <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><mi>n<\/mi><mo class=\"MathClass-punc\">!<\/mo><\/mrow> <mrow><msqrt><mrow><mi>n<\/mi><\/mrow><\/msqrt><msup><mrow> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mfrac><mrow><mi>n<\/mi><\/mrow> <mrow><mi class=\"qopname\"> e<\/mi><mo>  <\/mo> <\/mrow><\/mfrac> <\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <\/mrow><mrow><mi>n<\/mi><\/mrow><\/msup><\/mrow><\/mfrac> <mo class=\"MathClass-rel\">=<\/mo><munder class=\"msub\"><mrow><mi class=\"qopname\"> lim<\/mi><mo>  <\/mo><\/mrow><mrow><mi>n<\/mi><mo class=\"MathClass-rel\">\u2192<\/mo><mi>\u221e<\/mi><\/mrow><\/munder><msub><mrow><mi>b<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub> <mo class=\"MathClass-rel\">=<\/mo> <msqrt><mrow><mn>2<\/mn><mi>\u03c0<\/mi><\/mrow><\/msqrt><mo class=\"MathClass-punc\">,<\/mo> <\/math><\/td><\/tr><\/table> <p class=\"indent\">was zur Stirling-Formel (<a href=\"..\/..\/chapter\/drei-asymptotische-formeln#x1-283001r19\">9.19<\/a>) \u00e4quivalent ist. <\/p><p class=\"indent\">Auf Grund der Bedeutung von der Fakult\u00e4tsfunktion, zum Beispiel in kombinatorischen \u00dcberlegungen, ist es nicht verwunderlich, dass die Sterling-Formel vielfache Anwendungen besitzt. In den folgenden \u00dcbungen wollen wir einige wenige solcher Anwendungen pr\u00e4sentieren. <\/p> <div class=\"me meexample\"> <div class=\"wp-nocaption \"><\/div><h4 id=\"z3d03330a7bbf\"> <a id=\"x1-283003r60\"><\/a> <span class=\"ecbx-1095\">\u00dc<\/span><span class=\"ecbx-1095\">bung 9.60 <\/span>(Besuche bei der Irrfahrt auf <math display=\"inline\"><mi>\u2124<\/mi><\/math>)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Wir betrachten eine (diskrete) zuf<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">llige Bewegung auf den ganzen Zahlen ausgehend von<\/span> <span class=\"ecti-1095\">Null. Also beginnen wir zur Zeit <\/span><math display=\"inline\"><mi>t<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mn>0<\/mn><\/math> <span class=\"ecti-1095\">im Ursprung, h<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">pfen dann mit Wahrscheinlichkeit <\/span><math display=\"inline\"><mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mn>2<\/mn><\/mrow><\/mfrac><\/math> <span class=\"ecti-1095\">entweder nach links auf die Position <\/span><math display=\"inline\"> <mo class=\"MathClass-bin\">\u2212<\/mo> <mn>1<\/mn><\/math> <span class=\"ecti-1095\">oder nach rechts auf die Position <\/span><span class=\"maperiod\"><math display=\"inline\"><mn>1<\/mn><\/math><\/span><span class=\"period\">.<\/span> <span class=\"ecti-1095\">Bezeichnet im Allgemeinen <\/span><math display=\"inline\"><msub><mrow><mi>x<\/mi><\/mrow><mrow><mi>t<\/mi><\/mrow><\/msub> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2124<\/mi><\/math> <span class=\"ecti-1095\">die Position zur Zeit <\/span><span class=\"maperiod\"><math display=\"inline\"><mi>t<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2115<\/mi><\/math><\/span><span class=\"period\">,<\/span> <span class=\"ecti-1095\">so bewegen wir uns auf die Zeit <\/span><math display=\"inline\"><mi>t<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mn>1<\/mn><\/math> <span class=\"ecti-1095\">hin wieder entweder links oder rechts mit gleicher Wahrscheinlichkeit von <\/span><span class=\"maperiod\"><math display=\"inline\"><mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mn>2<\/mn><\/mrow><\/mfrac><\/math><\/span><span class=\"period\">.<\/span> <\/p><p class=\"indent\"><span class=\"ecti-1095\">Berechnen     Sie     die     Wahrscheinlichkeit,     dass     man     sich     zur     Zeit<\/span> <math display=\"inline\"><mi>t<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2115<\/mi><\/math> <span class=\"ecti-1095\">bei Null befindet und approximieren Sie diese mit der Stirling-Formel.<\/span> <\/p> <\/div> <div class=\"me meexample\"> <div class=\"wp-nocaption \"><\/div><h4 id=\"zc173ee8030df\"> <a id=\"x1-283004r61\"><\/a> <span class=\"ecbx-1095\">\u00dc<\/span><span class=\"ecbx-1095\">bung 9.61.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Eine M<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">nze wird <\/span><math display=\"inline\"><mi>n<\/mi><\/math><span class=\"ecti-1095\">-mal<\/span> <span class=\"ecti-1095\">geworfen. Sei <\/span><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> <span class=\"ecti-1095\">Zeigen Sie, dass die Wahrscheinlichkeit, dass die M<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">nze h<\/span><span class=\"ecti-1095\">\u00f6<\/span><span class=\"ecti-1095\">chstens <\/span><math display=\"inline\"><mo class=\"MathClass-open\">(<\/mo><mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mn>2<\/mn><\/mrow><\/mfrac> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>\ud835\udf00<\/mi><mo class=\"MathClass-close\">)<\/mo><mi>n<\/mi><\/math><span class=\"ecti-1095\">-mal<\/span> <span class=\"ecti-1095\">auf Kopf f<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">llt, 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\">gegen Null geht.<\/span> <\/p><p class=\"indent\"><span class=\"ecti-1095\">Hinweis: Sie d<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">rfen dabei annehmen, dass <\/span><math display=\"inline\"><msup><mrow><mn>2<\/mn><\/mrow><mrow><mo class=\"MathClass-bin\">\u2212<\/mo><mi>n<\/mi><\/mrow><\/msup><mfenced close=\")\" open=\"(\" separators><mfrac linethickness=\"0.0pt\"><mrow><mi>n<\/mi><\/mrow> <mrow><mi>k<\/mi><\/mrow><\/mfrac><\/mfenced> <\/math> <span class=\"ecti-1095\">die Wahrscheinlichkeit ist, dass bei <\/span><math display=\"inline\"><mi>n<\/mi><\/math><span class=\"ecti-1095\">-maligem<\/span> <span class=\"ecti-1095\">Wurf die M<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">nze <\/span><math display=\"inline\"><mi>k<\/mi><\/math><span class=\"ecti-1095\">-mal<\/span> <span class=\"ecti-1095\">auf Kopf f<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">llt. Zeigen Sie auch, dass die Funktion <\/span><math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mo class=\"MathClass-open\">(<\/mo><mn>0<\/mn><mo class=\"MathClass-punc\">,<\/mo><mn>1<\/mn><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-rel\">\u21a6<\/mo><mi>x<\/mi><mi class=\"qopname\">log<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">+<\/mo> <mo class=\"MathClass-open\">(<\/mo><mn>1<\/mn> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><mi class=\"qopname\">log<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mn>1<\/mn> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math> <span class=\"ecti-1095\">ein striktes lokales Maximum bei <\/span><math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mn>2<\/mn><\/mrow><\/mfrac><\/math> <span class=\"ecti-1095\">hat.<\/span> <\/p> <\/div> <div class=\"me meexample\"> <div class=\"wp-nocaption \"><\/div><h4 id=\"z8b5c72abd994\"> <a id=\"x1-283005r62\"><\/a> <span class=\"ecbx-1095\">\u00dc<\/span><span class=\"ecbx-1095\">bung 9.62 <\/span>(Ein Ausblick auf Algorithmik)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">In dieser <\/span><span class=\"ecti-1095\">\u00dc<\/span><span class=\"ecti-1095\">bung wollen wir die worst case Geschwindigkeit eines Sortieralgorithmus absch<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">tzen.<\/span> <span class=\"ecti-1095\">Sei also gegeben ein Algorithmus, der zu einem Datensatz <\/span><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> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2124<\/mi><\/math> <span class=\"ecti-1095\">diesen umordnet, das heisst, eine Abbildung <\/span><math display=\"inline\"><mi>j<\/mi> <mo class=\"MathClass-punc\">:<\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> {<\/mo><mrow><mn>1<\/mn><mo class=\"MathClass-punc\">,<\/mo><mi class=\"MathClass-op\">\u2026<\/mi><mo> <\/mo><mo class=\"MathClass-punc\">,<\/mo><mi>n<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">}<\/mo><\/mrow> <mo class=\"MathClass-rel\">\u2192<\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> {<\/mo><mrow><mn>1<\/mn><mo class=\"MathClass-punc\">,<\/mo><mi class=\"MathClass-op\">\u2026<\/mi><mo> <\/mo><mo class=\"MathClass-punc\">,<\/mo><mi>n<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">}<\/mo><\/mrow><\/math> <span class=\"ecti-1095\">findet, so dass <\/span><span class=\"maperiod\"><math display=\"inline\"><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>j<\/mi><mo class=\"MathClass-open\">(<\/mo><mn>1<\/mn><mo class=\"MathClass-close\">)<\/mo><\/mrow><\/msub> <mo class=\"MathClass-rel\">\u2264<\/mo><mi class=\"MathClass-op\">\u2026<\/mi><mo> <\/mo> <mo class=\"MathClass-rel\">\u2264<\/mo> <msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>j<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>n<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow><\/msub><\/math><\/span><span class=\"period\">.<\/span> <span class=\"ecti-1095\">Wir interessieren uns nun f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r die Zahl <\/span><math display=\"inline\"><mi>W<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>n<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math> <span class=\"ecti-1095\">der Anzahl Vergleiche zweier Zahlen des Datensatzes, die der Algorithmus schlimmstenfalls<\/span> <span class=\"ecti-1095\">durchf<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">hren muss, um den Datensatz zu sortieren. Zeigen Sie, dass eine von <\/span><math display=\"inline\"><mi>n<\/mi><\/math> <span class=\"ecti-1095\">unabh<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">ngige Konstante <\/span><math display=\"inline\"><mi>C<\/mi> <mo class=\"MathClass-rel\">&gt;<\/mo> <mn>0<\/mn><\/math> <span class=\"ecti-1095\">existiert, so dass <\/span><span class=\"maperiod\"><math display=\"inline\"><mi>W<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>n<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">\u2265<\/mo> <mi>C<\/mi><mi>n<\/mi><mi class=\"qopname\">log<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>n<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math><\/span><span class=\"period\">.<\/span> <\/p><div class=\"wp-nocaption \"><\/div><details><summary style=\"color:#FF7F00\"><span class=\"ecti-1095\">Hinweis.<\/span><\/summary><p class=\"indent\" style=\"margin-top: 0\"><span class=\"ecti-1095\">Z<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">hlen         Sie         die         Anzahl         Datens<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">tze,         die         mit<\/span> <math display=\"inline\"><mi>A<\/mi><\/math> <span class=\"ecti-1095\">Umsortierungen aus einem fixen Datensatz konstruiert werden k<\/span><span class=\"ecti-1095\">\u00f6<\/span><span class=\"ecti-1095\">nnen.<\/span><\/p><\/details>  <\/div> <a id=\"x1-283006r283\"><\/a> <h4 id=\"z717d50ca220a\" class=\"subsectionHead\"><span class=\"titlemark\">9.6.3 <\/span> <a id=\"x1-2840003\"><\/a>Asymptotik der harmonischen Reihe<\/h4> <p class=\"noindent\">F\u00fcr den Beweis der Stirling-Formel wurde eine Kombination von Taylor-Approximation und Integration verwendet, um das asymptotische Verhalten der Reihe <math display=\"inline\"><mi class=\"qopname\">log<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>n<\/mi><mo class=\"MathClass-punc\">!<\/mo><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">=<\/mo><msubsup><mrow><mi class=\"qopname\"> \u2211<\/mi><mo>  <\/mo> <\/mrow><mrow><mi>k<\/mi><mo class=\"MathClass-rel\">=<\/mo><mn>1<\/mn><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msubsup><mi class=\"qopname\"> log<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>k<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math> zu bestimmen. Diese Methode kann auch allgemeiner zur Untersuchung des Divergenzverhaltens von Reihen eingesetzt werden. Zum Beispiel gilt die \u201easymptotische Entwicklung\u201c <\/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>k<\/mi><mo class=\"MathClass-rel\">=<\/mo><mn>1<\/mn><\/mrow><mrow><mi>n<\/mi><\/mrow><\/munderover><mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mi>k<\/mi><\/mrow><\/mfrac> <mo class=\"MathClass-rel\">=<\/mo><mi class=\"qopname\"> log<\/mi><mo>  <\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>n<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-bin\">+<\/mo> <mi>\u03b3<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mn>2<\/mn><mi>n<\/mi><\/mrow><\/mfrac> <mo class=\"MathClass-bin\">+<\/mo> <mi>O<\/mi><mo class=\"MathClass-open\">(<\/mo><mfrac><mrow> <mn>1<\/mn><\/mrow> <mrow><msup><mrow><mi>n<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><\/mrow><\/mfrac><mo class=\"MathClass-close\">)<\/mo><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"><mstyle class=\"label\" id=\"x1-284001r21\" \/><mstyle class=\"maketag\"><mtext>(9.21)<\/mtext><\/mstyle><mspace class=\"nbsp\" width=\"0.33em\" \/> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">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> wobei                                                                                                                                                                           <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mi>\u03b3<\/mi> <mo class=\"MathClass-rel\">=<\/mo><msubsup><mrow><mo> \u222b  <\/mo><\/mrow><mrow><mn>1<\/mn><\/mrow><mrow><mi>\u221e<\/mi><\/mrow><\/msubsup><mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow> <mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mo class=\"MathClass-open\">\u230a<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">\u230b<\/mo><\/mrow><\/mfrac><mo class=\"MathClass-bin\">\u2212<\/mo><mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mi>x<\/mi><\/mrow><\/mfrac> <\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mn>0<\/mn><mo class=\"MathClass-punc\">.<\/mo><mn>5<\/mn><mn>7<\/mn><mn>7<\/mn><mn>2<\/mn><mn>1<\/mn><mn>5<\/mn><mn>6<\/mn><mn>6<\/mn><mn>4<\/mn><mn>9<\/mn><mn>0<\/mn><mn>1<\/mn><mn>5<\/mn><mn>3<\/mn><mn>2<\/mn><mo>\u2026<\/mo><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">die <span class=\"ecbx-1095\">Euler-Mascheroni-Konstante <\/span>genannt wird. <\/p> <div class=\"me meexample\"> <div class=\"wp-nocaption \"><\/div><h4 id=\"zbc75f545a61c\"> <a id=\"x1-284002r63\"><\/a> <span class=\"ecbx-1095\">\u00dc<\/span><span class=\"ecbx-1095\">bung 9.63.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Wir wollen die Asymptotik<\/span> (<a href=\"..\/..\/chapter\/drei-asymptotische-formeln#x1-284001r21\">9.21<\/a>) <span class=\"ecti-1095\">in dieser <\/span><span class=\"ecti-1095\">\u00dc<\/span><span class=\"ecti-1095\">bung beweisen.<\/span> <\/p><dl class=\"enumerate\"><dt class=\"enumerate\"> <span class=\"ecti-1095\">(a)<\/span><\/dt><dd class=\"enumerate\"><span class=\"ecti-1095\">Beweisen Sie, dass das Integral in der Definition von<\/span> <math display=\"inline\"><mi>\u03b3<\/mi><\/math> <span class=\"ecti-1095\">konvergiert und dass die Gleichung<\/span> <math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><munderover accent=\"false\" accentunder=\"false\"><mrow><mo>\u2211<\/mo> <\/mrow><mrow><mi>k<\/mi><mo class=\"MathClass-rel\">=<\/mo><mn>1<\/mn><\/mrow><mrow><mi>n<\/mi><\/mrow><\/munderover><mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mi>k<\/mi><\/mrow><\/mfrac> <mo class=\"MathClass-rel\">=<\/mo><mi class=\"qopname\"> log<\/mi><mo>  <\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>n<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-bin\">+<\/mo> <mi>\u03b3<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mi>n<\/mi><\/mrow><\/mfrac> <mo class=\"MathClass-bin\">\u2212<\/mo><msubsup><mrow><mo>\u222b  <\/mo><\/mrow><mrow><mi>n<\/mi><\/mrow><mrow><mi>\u221e<\/mi><\/mrow><\/msubsup><mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow> <mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mo class=\"MathClass-open\">\u230a<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">\u230b<\/mo><\/mrow><\/mfrac><mo class=\"MathClass-bin\">\u2212<\/mo><mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mi>x<\/mi><\/mrow><\/mfrac> <\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\"><span class=\"ecti-1095\">f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r alle <\/span><math display=\"inline\"><mi>n<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2115<\/mi><\/math> <span class=\"ecti-1095\">gilt.<\/span> <\/p><\/dd><dt class=\"enumerate\"> <span class=\"ecti-1095\">(b)<\/span><\/dt><dd class=\"enumerate\"><span class=\"ecti-1095\">Verwenden Sie Taylor-Approximation, um<\/span> <div class=\"custom-itemize\"><div class=\"item-head\"> <span class=\"tcrm-1095\">\u2022<\/span><\/div><div class=\"item-content\"><math display=\"inline\"><mi class=\"qopname\">log<\/mi><mo>  <\/mo><mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>m<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mn>1<\/mn><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-rel\">=<\/mo><mi class=\"qopname\"> log<\/mi><mo>  <\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>m<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-bin\">+<\/mo><mfrac><mrow> <mn>1<\/mn><\/mrow> <mrow><mi>m<\/mi><\/mrow><\/mfrac> <mo class=\"MathClass-bin\">\u2212<\/mo><mfrac><mrow> <mn>1<\/mn><\/mrow> <mrow><mn>2<\/mn><msup><mrow><mi>m<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><\/mrow><\/mfrac> <mo class=\"MathClass-bin\">+<\/mo> <mi>O<\/mi><mo class=\"MathClass-open\">(<\/mo><mfrac><mrow> <mn>1<\/mn><\/mrow> <mrow><msup><mrow><mi>m<\/mi><\/mrow><mrow><mn>3<\/mn><\/mrow><\/msup><\/mrow><\/mfrac><mo class=\"MathClass-close\">)<\/mo><\/math> <\/div><div class=\"item-head\"> <span class=\"tcrm-1095\">\u2022<\/span><\/div><div class=\"item-content\"><math display=\"inline\"><mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mi>m<\/mi><\/mrow><\/mfrac> <mo class=\"MathClass-bin\">\u2212<\/mo><msubsup><mrow><mi class=\"MathClass-op\">\u222b  <\/mi><mo> <\/mo><\/mrow><mrow><mi>m<\/mi><\/mrow><mrow><mi>m<\/mi><mo class=\"MathClass-bin\">+<\/mo><mn>1<\/mn><\/mrow><\/msubsup><mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mi>t<\/mi><\/mrow><\/mfrac> <mspace class=\"thinspace\" width=\"0.17em\" \/> <mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>t<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mn>2<\/mn><msup><mrow><mi>m<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><\/mrow><\/mfrac> <mo class=\"MathClass-bin\">+<\/mo> <mi>O<\/mi><mo class=\"MathClass-open\">(<\/mo><mfrac><mrow> <mn>1<\/mn><\/mrow> <mrow><msup><mrow><mi>m<\/mi><\/mrow><mrow><mn>3<\/mn><\/mrow><\/msup><\/mrow><\/mfrac><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">=<\/mo><mfrac><mrow> <mn>1<\/mn><\/mrow> <mrow><mn>2<\/mn><\/mrow><\/mfrac><msubsup><mrow><mi class=\"MathClass-op\"> \u222b  <\/mi><mo> <\/mo><\/mrow><mrow><mi>m<\/mi><\/mrow><mrow><mi>m<\/mi><mo class=\"MathClass-bin\">+<\/mo><mn>1<\/mn><\/mrow><\/msubsup><msup><mrow><mi>t<\/mi><\/mrow><mrow><mo class=\"MathClass-bin\">\u2212<\/mo><mn>2<\/mn><\/mrow><\/msup><mspace class=\"thinspace\" width=\"0.17em\" \/> <mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>t<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mi>O<\/mi><mo class=\"MathClass-open\">(<\/mo><mfrac><mrow> <mn>1<\/mn><\/mrow> <mrow><msup><mrow><mi>m<\/mi><\/mrow><mrow><mn>3<\/mn><\/mrow><\/msup><\/mrow><\/mfrac><mo class=\"MathClass-close\">)<\/mo><\/math><\/div><\/div> <p class=\"noindent\"><span class=\"ecti-1095\">f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r <\/span><math display=\"inline\"><mi>m<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2115<\/mi><\/math> <span class=\"ecti-1095\">zu<\/span> <span class=\"ecti-1095\">beweisen.<\/span> <\/p><\/dd><dt class=\"enumerate\"> <span class=\"ecti-1095\">(c)<\/span><\/dt><dd class=\"enumerate\"><span class=\"ecti-1095\">Verwenden Sie das Integralkriterium (Satz <\/span><a href=\"..\/..\/chapter\/das-uneigentliche-integral#x1-273007r35\"><span class=\"ecti-1095\">9.35<\/span><\/a><span class=\"ecti-1095\">) um zu zeigen, dass<\/span> <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>m<\/mi><mo class=\"MathClass-rel\">=<\/mo><mi>n<\/mi><\/mrow><mrow><mi>\u221e<\/mi><\/mrow><\/munderover> <mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><msup><mrow><mi>m<\/mi><\/mrow><mrow><mn>3<\/mn><\/mrow><\/msup><\/mrow><\/mfrac> <mo class=\"MathClass-rel\">=<\/mo> <mi>O<\/mi><mo class=\"MathClass-open\">(<\/mo><mfrac><mrow> <mn>1<\/mn><\/mrow> <mrow><msup><mrow><mi>n<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><\/mrow><\/mfrac><mo class=\"MathClass-close\">)<\/mo><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\"><span class=\"ecti-1095\">f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r <\/span><span class=\"maperiod\"><math display=\"inline\"><mi>n<\/mi> <mo class=\"MathClass-rel\">\u2192<\/mo> <mi>\u221e<\/mi><\/math><\/span><span class=\"period\">.<\/span> <\/p><\/dd><dt class=\"enumerate\"> <span class=\"ecti-1095\">(d)<\/span><\/dt><dd class=\"enumerate\"><span class=\"ecti-1095\">Verbinden Sie die obigen Aussagen zu einem Beweis von<\/span> (<a href=\"..\/..\/chapter\/drei-asymptotische-formeln#x1-284001r21\">9.21<\/a>)<span class=\"ecti-1095\">.<\/span> <\/dd><dt class=\"enumerate\"> <span class=\"ecti-1095\">(e)<\/span><\/dt><dd class=\"enumerate\"><span class=\"ecti-1095\">Wiederholen Sie Beispiel <\/span><a href=\"..\/..\/chapter\/reihen#x1-187005r5\"><span class=\"ecti-1095\">7.5<\/span><\/a> <span class=\"ecti-1095\">und approximieren Sie mit der Formel<\/span> (<a href=\"..\/..\/chapter\/drei-asymptotische-formeln#x1-284001r21\">9.21<\/a>) <span class=\"ecti-1095\">die notwendige Anzahl Bausteine<\/span> <math display=\"inline\"><mi>n<\/mi><\/math> <span class=\"ecti-1095\">f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r einen<\/span> <span class=\"ecti-1095\">Sprungturm, der <\/span><math display=\"inline\"><mn>1<\/mn><mn>0<\/mn><\/math><span class=\"ecti-1095\">m<\/span> <span class=\"ecti-1095\">in den See hinreicht. Vergleichen Sie dies dann zum exakten Resultat<\/span> <span class=\"maperiod\"><math display=\"inline\"><mi>n<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mn>1<\/mn><mn>2<\/mn><mn>3<\/mn><mn>6<\/mn><mn>7<\/mn><\/math><\/span><span class=\"period\">.<\/span><\/dd><\/dl> <\/div> <a id=\"x1-284008r281\"><\/a> \n","protected":false},"author":1089,"menu_order":6,"template":"","meta":{"pb_show_title":"","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-98","chapter","type-chapter","status-publish","hentry"],"part":92,"_links":{"self":[{"href":"https:\/\/wp-prd.let.ethz.ch\/analysis19\/wp-json\/pressbooks\/v2\/chapters\/98","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\/98\/revisions"}],"part":[{"href":"https:\/\/wp-prd.let.ethz.ch\/analysis19\/wp-json\/pressbooks\/v2\/parts\/92"}],"metadata":[{"href":"https:\/\/wp-prd.let.ethz.ch\/analysis19\/wp-json\/pressbooks\/v2\/chapters\/98\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/wp-prd.let.ethz.ch\/analysis19\/wp-json\/wp\/v2\/media?parent=98"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/wp-prd.let.ethz.ch\/analysis19\/wp-json\/pressbooks\/v2\/chapter-type?post=98"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/wp-prd.let.ethz.ch\/analysis19\/wp-json\/wp\/v2\/contributor?post=98"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/wp-prd.let.ethz.ch\/analysis19\/wp-json\/wp\/v2\/license?post=98"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}