{"id":42,"date":"2021-12-15T09:53:03","date_gmt":"2021-12-15T09:53:03","guid":{"rendered":"https:\/\/wp-prd.let.ethz.ch\/analysis19\/chapter\/summen-und-produkte\/"},"modified":"2021-12-15T09:53:03","modified_gmt":"2021-12-15T09:53:03","slug":"summen-und-produkte","status":"publish","type":"chapter","link":"https:\/\/wp-prd.let.ethz.ch\/analysis19\/chapter\/summen-und-produkte\/","title":{"raw":"Summen und Produkte","rendered":"Summen und Produkte"},"content":{"raw":"\n<style>.cmr-5{font-size:50%;}\n.cmr-7{font-size:70%;}\n.cmmi-5{font-size:50%;font-style: italic;}\n.cmmi-7{font-size:70%;font-style: italic;}\n.cmmi-10{font-style: italic;}\n.cmsy-5{font-size:50%;}\n.cmsy-7{font-size:70%;}\n.cmbx-10{ font-weight: bold;}\n.cmbsy-10{font-weight: bold;}\n.cmbsy-10{font-weight: bold;}\n.cmbsy-10{font-weight: bold;}\n.cmbsy-7{font-size:70%;font-weight: bold;}\n.cmbsy-7{font-weight: bold;}\n.cmbsy-7{font-weight: bold;}\n.cmbsy-5{font-size:50%;font-weight: bold;}\n.cmbsy-5{font-weight: bold;}\n.cmbsy-5{font-weight: bold;}\n.cmex-7{font-size:70%;}\n.cmex-7x-x-71{font-size:49%;}\n.msam-7{font-size:70%;}\n.msam-5{font-size:50%;}\n.msbm-7{font-size:70%;}\n.msbm-5{font-size:50%;}\n.cmr-17{font-size:170%;}\n.cmr-12{font-size:120%;}\n.cmti-10{ font-style: italic;}\np{margin-top:0;margin-bottom:0}\np.indent{text-indent:0;}\np + p{margin-top:1em;}\np + div, p + pre {margin-top:1em;}\ndiv + p, pre + p {margin-top:1em;}\n@media print {div.crosslinks {visibility:hidden;}}\na img { border-top: 0; 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\n}\ndiv.proof p:first-of-type {\n\tmargin: 0px;\n}\ndiv.qed {\n\tmargin-top: -25px;\n\tmargin-bottom: -7px;\n\ttext-align: right;\n}\ntable.equation+div.qed {\n\tmargin-top: -65px;\n}\n\n\/* The following is making also math-formulas inside the headers of Lemmas, etc., white. *\/\ndiv.melemma h4 span {\n    color: white;\n}\ndiv.metheorem h4 span {\n    color: white;\n}\n\n\/* The following are used to avoid fullstop, period, colon, semicolon, and endquote (broader) to move by itself to the next line after a formula.\n   The math-environment before needs to be wrapped in span.maperiod and the fullstop etc. in a span.period --- together they achieve what we want.  *\/\nspan.maperiod {\n       margin-right: 5px;\n}\nspan.period {\n       display: inline-block;\n       width: 0px;\n       margin-left: -5px;\n       margin-right: 4.9px;\n\t   text-indent: 0px;\n}\nspan.maendquote {\n       margin-right: 8px;\n}\nspan.endquote {\n       display: inline-block;\n       width: 0px;\n       margin-left: -8px;\n       margin-right: 7.9px;\n}\n\n\n\/* The following is removing an extra space left of the equation side in aligned equations *\/\nspan.mjx-mtd {\n    padding-left: 0em !important;\n}\n\n\/* The following fixes the weird problem that math appears smaller if it was rendered while the details tag was closed. *\/\ndetails span.mjx-chtml, details span.MathJax_CHTML {\n font-size: 100% !important;\n}\n\n\/* trying to fix line breaks in verbatim, new lines are missing *\/\npre.verbatim {\n\twhite-space: pre-wrap;\n\tfont-size: small;\n}\n<\/style><h3 id=\"z759f256f5507\" class=\"sectionHead\"><span class=\"titlemark\">3.1 <\/span> <a id=\"x1-770001\"><\/a>Summen und Produkte<\/h3> <p class=\"noindent\">Sei <math display=\"inline\"><mi>n<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2115<\/mi><\/math> und seien <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>\u2102<\/mi><\/math> oder <math display=\"inline\"><msub><mrow><mi>a<\/mi><\/mrow><mrow><mn>1<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-punc\">,<\/mo> <mi class=\"MathClass-op\">\u2026<\/mi><mo> <\/mo> <mo class=\"MathClass-punc\">,<\/mo> <msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi> <\/mrow> <\/msub> <\/math> Elemente eines Vektorraums&nbsp;<math display=\"inline\"><mi>V<\/mi> <\/math> (wie zum Beispiel <math display=\"inline\"><msup><mrow><mi>\u211d<\/mi><\/mrow><mrow><mi>d<\/mi><\/mrow><\/msup><\/math> f\u00fcr ein <math display=\"inline\"><mi>d<\/mi> <mo class=\"MathClass-rel\">\u2265<\/mo> <mn>1<\/mn><\/math>). Wir wollen hier f\u00fcr eine nat\u00fcrliche Zahl&nbsp;<math display=\"inline\"><mi>n<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2115<\/mi><\/math> die <span class=\"ecbx-1095\">Summe <\/span>von <math display=\"inline\"><msub><mrow><mi>a<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><\/math> bis <span class=\"maperiod\"><math display=\"inline\"><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi> <\/mrow> <\/msub> <\/math><\/span><span class=\"period\">,<\/span> also <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><munderover accent=\"false\" accentunder=\"false\"><mrow><mo> \u2211<\/mo> <\/mrow><mrow><mi>j<\/mi><mo class=\"MathClass-rel\">=<\/mo><mn>1<\/mn><\/mrow><mrow><mi>n<\/mi><\/mrow><\/munderover><msub><mrow><mi>a<\/mi><\/mrow><mrow> <mi>j<\/mi><\/mrow><\/msub> <mo class=\"MathClass-rel\">=<\/mo> <msub><mrow><mi>a<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub> <mo class=\"MathClass-bin\">+<\/mo> <mo>\u2026<\/mo> <mo class=\"MathClass-bin\">+<\/mo> <msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><mo class=\"MathClass-punc\">,<\/mo><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">besprechen und formal korrekt definieren. <\/p><p class=\"indent\">Vom formalen Standpunkt her gesehen ist <math display=\"inline\"><mi>j<\/mi><mo class=\"MathClass-rel\">\u21a6<\/mo><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>j<\/mi><\/mrow><\/msub> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>V<\/mi> <\/math> eine Funktion, (die oft auch durch eine konkrete Formel gegeben sein wird und) deren Definitionsbereich die Menge <\/p><table id=\"zd14d9022f276\" class=\"equation-star\"><tr><td> <math class=\"equation\" display=\"block\"> <mrow><mo fence=\"true\" form=\"prefix\"> {<\/mo><mrow><mi>j<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2115<\/mi><mo class=\"MathClass-rel\">\u2223<\/mo><mn>1<\/mn> <mo class=\"MathClass-rel\">\u2264<\/mo> <mi>j<\/mi> <mo class=\"MathClass-rel\">\u2264<\/mo> <mi>n<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">}<\/mo><\/mrow> <\/math><\/td><\/tr><\/table> <p class=\"indent\">enthalten muss. Wir k\u00f6nnen <math display=\"inline\"><msubsup><mrow><mi class=\"MathClass-op\"> \u2211<\/mi><mo> <\/mo> <\/mrow><mrow><mi>i<\/mi><mo class=\"MathClass-rel\">=<\/mo><mn>1<\/mn><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msubsup><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>j<\/mi><\/mrow><\/msub><\/math> rekursiv definieren durch <\/p><table id=\"z4d3f6041c5d4\" class=\"equation-star\"><tr><td> <math class=\"equation\" display=\"block\"> <munderover accent=\"false\" accentunder=\"false\"><mrow><mo>\u2211<\/mo> <\/mrow><mrow><mi>j<\/mi><mo class=\"MathClass-rel\">=<\/mo><mn>1<\/mn><\/mrow><mrow><mn>1<\/mn><\/mrow><\/munderover><msub><mrow><mi>a<\/mi><\/mrow><mrow> <mi>j<\/mi><\/mrow><\/msub> <mo class=\"MathClass-rel\">=<\/mo> <msub><mrow><mi>a<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><mstyle class=\"text\"><mtext>&nbsp;und&nbsp;<\/mtext><\/mstyle><munderover accent=\"false\" accentunder=\"false\"><mrow><mo> \u2211<\/mo> <\/mrow><mrow><mi>j<\/mi><mo class=\"MathClass-rel\">=<\/mo><mn>1<\/mn><\/mrow><mrow><mi>k<\/mi><mo class=\"MathClass-bin\">+<\/mo><mn>1<\/mn><\/mrow><\/munderover><msub><mrow><mi>a<\/mi><\/mrow><mrow> <mi>j<\/mi><\/mrow><\/msub> <mo class=\"MathClass-rel\">=<\/mo> <mstyle><mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow \/><mo fence=\"true\" form=\"postfix\" \/><\/mrow><\/mstyle><munderover accent=\"false\" accentunder=\"false\"><mrow><mo>\u2211<\/mo> <\/mrow><mrow><mi>j<\/mi><mo class=\"MathClass-rel\">=<\/mo><mn>1<\/mn><\/mrow><mrow><mi>k<\/mi><\/mrow><\/munderover><msub><mrow><mi>a<\/mi><\/mrow><mrow> <mi>j<\/mi><\/mrow><\/msub><mstyle><mrow><mo fence=\"true\" form=\"prefix\"> )<\/mo><mrow \/><mo fence=\"true\" form=\"postfix\" \/><\/mrow><\/mstyle> <mo class=\"MathClass-bin\">+<\/mo> <msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>k<\/mi><mo class=\"MathClass-bin\">+<\/mo><mn>1<\/mn><\/mrow><\/msub> <\/math><\/td><\/tr><\/table> <p class=\"indent\">f\u00fcr <span class=\"maperiod\"><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> <mo class=\"MathClass-bin\">\u2212<\/mo> <mn>1<\/mn><\/mrow><mo fence=\"true\" form=\"postfix\">}<\/mo><\/mrow><\/math><\/span><span class=\"period\">.<\/span> Diese Definition entspricht einem einfachen rekursiven Algorithmus, um die Summe <math display=\"inline\"><msubsup><mrow><mi class=\"MathClass-op\">\u2211<\/mi><mo> <\/mo> <\/mrow><mrow><mi>i<\/mi><mo class=\"MathClass-rel\">=<\/mo><mn>1<\/mn><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msubsup><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>j<\/mi><\/mrow><\/msub><\/math> zu berechnen. Allgemeiner ist die Summe <math display=\"inline\"><msubsup><mrow><mi class=\"MathClass-op\"> \u2211<\/mi><mo> <\/mo> <\/mrow><mrow><mi>i<\/mi><mo class=\"MathClass-rel\">=<\/mo><mi>m<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msubsup><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>j<\/mi><\/mrow><\/msub><\/math> f\u00fcr ganze Zahlen <math display=\"inline\"><mi>m<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>n<\/mi><\/math> ebenso rekursiv durch <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><munderover accent=\"false\" accentunder=\"false\"><mrow><mo> \u2211<\/mo> <\/mrow><mrow><mi>j<\/mi><mo class=\"MathClass-rel\">=<\/mo><mi>m<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/munderover><msub><mrow><mi>a<\/mi><\/mrow><mrow> <mi>j<\/mi><\/mrow><\/msub> <mo class=\"MathClass-rel\">=<\/mo> <mrow class=\"cases\"> <mrow><mo fence=\"true\" form=\"prefix\"> {<\/mo><mrow> <mtable align=\"axis\" class=\"array\" columnlines=\"none\" equalcolumns=\"false\" equalrows=\"false\"> <mtr><mtd class=\"array\" columnalign=\"left\"><mn>0<\/mn> <mspace class=\"quad\" width=\"1em\" \/><\/mtd><mtd class=\"array\" columnalign=\"left\"><mstyle class=\"text\"><mtext>falls&nbsp;<\/mtext><\/mstyle><mi>m<\/mi> <mo class=\"MathClass-rel\">&gt;<\/mo> <mi>n<\/mi><mo class=\"MathClass-punc\">,<\/mo> <\/mtd> <\/mtr> <mtr><mtd class=\"array\" columnalign=\"left\"><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>m<\/mi><\/mrow><\/msub> <mspace class=\"quad\" width=\"1em\" \/><\/mtd><mtd class=\"array\" columnalign=\"left\"><mstyle class=\"text\"><mtext>falls&nbsp;<\/mtext><\/mstyle><mi>m<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mi>n<\/mi><mstyle class=\"text\"><mtext>&nbsp;und<\/mtext><\/mstyle><\/mtd> <\/mtr> <mtr><mtd class=\"array\" columnalign=\"left\"><mstyle><mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow \/><mo fence=\"true\" form=\"postfix\" \/><\/mrow><\/mstyle><munderover accent=\"false\" accentunder=\"false\"><mrow><mo> \u2211<\/mo> <\/mrow><mrow><mi>j<\/mi><mo class=\"MathClass-rel\">=<\/mo><mi>m<\/mi><\/mrow><mrow><mi>n<\/mi><mo class=\"MathClass-bin\">\u2212<\/mo><mn>1<\/mn><\/mrow><\/munderover><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>j<\/mi><\/mrow><\/msub><mstyle><mrow><mo fence=\"true\" form=\"prefix\"> )<\/mo><mrow \/><mo fence=\"true\" form=\"postfix\" \/><\/mrow><\/mstyle> <mo class=\"MathClass-bin\">+<\/mo> <msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><mspace class=\"quad\" width=\"1em\" \/><\/mtd><mtd class=\"array\" columnalign=\"left\"><mstyle class=\"text\"><mtext>falls&nbsp;<\/mtext><\/mstyle><mi>m<\/mi> <mo class=\"MathClass-rel\">&lt;<\/mo> <mi>n<\/mi> <\/mtd><\/mtr> <\/mtable> <\/mrow><mo fence=\"true\" form=\"postfix\" \/><\/mrow><\/mrow><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">definiert. Wir werden&nbsp;<math display=\"inline\"><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>j<\/mi><\/mrow><\/msub><\/math> als die <span class=\"ecbx-1095\">Summanden <\/span>und&nbsp;<math display=\"inline\"><mi>j<\/mi><\/math> als den <span class=\"ecbx-1095\">Index <\/span>der Summe <math display=\"inline\"><msubsup><mrow><mi class=\"MathClass-op\"> \u2211<\/mi><mo> <\/mo> <\/mrow><mrow><mi>j<\/mi><mo class=\"MathClass-rel\">=<\/mo><mn>1<\/mn><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msubsup><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>j<\/mi><\/mrow><\/msub><\/math> bezeichnen. <\/p><p class=\"indent\">Falls nun <math display=\"inline\"><mi>m<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>n<\/mi><\/math> ganze Zahlen und <math display=\"inline\"><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>m<\/mi> <\/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>\u2102<\/mi><\/math> sind, dann k\u00f6nnen wir auch das <span class=\"ecbx-1095\">Produkt <\/span><math display=\"inline\"><msubsup><mrow><mi class=\"MathClass-op\">\u220f<\/mi><mo> <\/mo> <\/mrow><mrow><mi>j<\/mi><mo class=\"MathClass-rel\">=<\/mo><mi>m<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msubsup><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>j<\/mi><\/mrow><\/msub><\/math> von <math display=\"inline\"><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>m<\/mi> <\/mrow> <\/msub> <\/math> bis <math display=\"inline\"><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi> <\/mrow> <\/msub> <\/math> rekursiv durch <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><munderover accent=\"false\" accentunder=\"false\"><mrow><mo> \u220f<\/mo> <\/mrow><mrow><mi>j<\/mi><mo class=\"MathClass-rel\">=<\/mo><mi>m<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/munderover><msub><mrow><mi>a<\/mi><\/mrow><mrow> <mi>j<\/mi><\/mrow><\/msub> <mo class=\"MathClass-rel\">=<\/mo> <mrow class=\"cases\"> <mrow><mo fence=\"true\" form=\"prefix\"> {<\/mo><mrow> <mtable align=\"axis\" class=\"array\" columnlines=\"none\" equalcolumns=\"false\" equalrows=\"false\"> <mtr><mtd class=\"array\" columnalign=\"left\"><mn>1<\/mn> <mspace class=\"quad\" width=\"1em\" \/><\/mtd><mtd class=\"array\" columnalign=\"left\"><mstyle class=\"text\"><mtext>falls&nbsp;<\/mtext><\/mstyle><mi>m<\/mi> <mo class=\"MathClass-rel\">&gt;<\/mo> <mi>n<\/mi><mo class=\"MathClass-punc\">,<\/mo> <\/mtd> <\/mtr> <mtr><mtd class=\"array\" columnalign=\"left\"><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>m<\/mi><\/mrow><\/msub> <mspace class=\"quad\" width=\"1em\" \/><\/mtd><mtd class=\"array\" columnalign=\"left\"><mstyle class=\"text\"><mtext>falls&nbsp;<\/mtext><\/mstyle><mi>m<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mi>n<\/mi><mstyle class=\"text\"><mtext>&nbsp;und<\/mtext><\/mstyle><\/mtd> <\/mtr> <mtr><mtd class=\"array\" columnalign=\"left\"><mstyle><mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow \/><mo fence=\"true\" form=\"postfix\" \/><\/mrow><\/mstyle><munderover accent=\"false\" accentunder=\"false\"><mrow><mo> \u220f<\/mo> <\/mrow><mrow><mi>j<\/mi><mo class=\"MathClass-rel\">=<\/mo><mi>m<\/mi><\/mrow><mrow><mi>n<\/mi><mo class=\"MathClass-bin\">\u2212<\/mo><mn>1<\/mn><\/mrow><\/munderover><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>j<\/mi><\/mrow><\/msub><mstyle><mrow><mo fence=\"true\" form=\"prefix\"> )<\/mo><mrow \/><mo fence=\"true\" form=\"postfix\" \/><\/mrow><\/mstyle> <mo class=\"MathClass-bin\">\u22c5<\/mo> <msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><mspace class=\"quad\" width=\"1em\" \/><\/mtd><mtd class=\"array\" columnalign=\"left\"><mstyle class=\"text\"><mtext>falls&nbsp;<\/mtext><\/mstyle><mi>m<\/mi> <mo class=\"MathClass-rel\">&lt;<\/mo> <mi>n<\/mi> <\/mtd><\/mtr> <\/mtable> <\/mrow><mo fence=\"true\" form=\"postfix\" \/><\/mrow><\/mrow><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">definieren. Wir werden&nbsp;<math display=\"inline\"><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>j<\/mi><\/mrow><\/msub><\/math> als die <span class=\"ecbx-1095\">Faktoren <\/span>und&nbsp;<math display=\"inline\"><mi>j<\/mi><\/math> als den <span class=\"ecbx-1095\">Index <\/span>des Produkts <math display=\"inline\"><msubsup><mrow><mi class=\"MathClass-op\"> \u220f<\/mi><mo> <\/mo> <\/mrow><mrow><mi>j<\/mi><mo class=\"MathClass-rel\">=<\/mo><mi>m<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msubsup><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>j<\/mi><\/mrow><\/msub><\/math> bezeichnen. <\/p><p class=\"indent\">Der Index <math display=\"inline\"><mi>j<\/mi><\/math> in der Summe <math display=\"inline\"><msubsup><mrow><mi class=\"MathClass-op\"> \u2211<\/mi><mo> <\/mo> <\/mrow><mrow><mi>j<\/mi><mo class=\"MathClass-rel\">=<\/mo><mn>1<\/mn><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msubsup><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>j<\/mi><\/mrow><\/msub><\/math> oder dem Produkt <math display=\"inline\"><msubsup><mrow><mi class=\"MathClass-op\"> \u220f<\/mi><mo> <\/mo> <\/mrow><mrow><mi>j<\/mi><mo class=\"MathClass-rel\">=<\/mo><mn>1<\/mn><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msubsup><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>j<\/mi><\/mrow><\/msub><\/math> hat ausserhalb der Summe oder dem Produkt keinerlei Bedeutung; er ist sozusagen eine interne Variable f\u00fcr das rekursive Teilprogramm und wird von einem Programm, welches das Teilprogramm aufruft, nicht gesehen. Insbesondere gilt                                                                                                                                                                           <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><munderover accent=\"false\" accentunder=\"false\"><mrow><mo> \u2211<\/mo> <\/mrow><mrow><mi>j<\/mi><mo class=\"MathClass-rel\">=<\/mo><mi>m<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/munderover><msub><mrow><mi>a<\/mi><\/mrow><mrow> <mi>j<\/mi><\/mrow><\/msub> <mo class=\"MathClass-rel\">=<\/mo><munderover accent=\"false\" accentunder=\"false\"><mrow><mo> \u2211<\/mo> <\/mrow><mrow><mi>k<\/mi><mo class=\"MathClass-rel\">=<\/mo><mi>m<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/munderover><msub><mrow><mi>a<\/mi><\/mrow><mrow> <mi>k<\/mi><\/mrow><\/msub> <mo class=\"MathClass-rel\">=<\/mo><munderover accent=\"false\" accentunder=\"false\"><mrow><mo> \u2211<\/mo> <\/mrow><mrow><mi>\u2113<\/mi><mo class=\"MathClass-rel\">=<\/mo><mi>m<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/munderover><msub><mrow><mi>a<\/mi><\/mrow><mrow> <mi>\u2113<\/mi><\/mrow><\/msub><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">und analog f\u00fcr das Produkt. Manchmal werden wir auch eine <span class=\"ecbx-1095\">Indexverschiebung <\/span>anwenden, wie zum Beispiel in <\/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>j<\/mi><mo class=\"MathClass-rel\">=<\/mo><mi>m<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/munderover><msub><mrow><mi>a<\/mi><\/mrow><mrow> <mi>j<\/mi><\/mrow><\/msub> <mo class=\"MathClass-rel\">=<\/mo><munderover accent=\"false\" accentunder=\"false\"><mrow><mo> \u2211<\/mo> <\/mrow><mrow><mi>k<\/mi><mo class=\"MathClass-rel\">=<\/mo><mi>m<\/mi><mo class=\"MathClass-bin\">\u2212<\/mo><mn>1<\/mn><\/mrow><mrow><mi>n<\/mi><mo class=\"MathClass-bin\">\u2212<\/mo><mn>1<\/mn><\/mrow><\/munderover><msub><mrow><mi>a<\/mi><\/mrow><mrow> <mi>k<\/mi><mo class=\"MathClass-bin\">+<\/mo><mn>1<\/mn><\/mrow><\/msub> <mo class=\"MathClass-rel\">=<\/mo><munderover accent=\"false\" accentunder=\"false\"><mrow><mo> \u2211<\/mo> <\/mrow><mrow><mi>\u2113<\/mi><mo class=\"MathClass-rel\">=<\/mo><mi>m<\/mi><mo class=\"MathClass-bin\">+<\/mo><mn>1<\/mn><\/mrow><mrow><mi>n<\/mi><mo class=\"MathClass-bin\">+<\/mo><mn>1<\/mn><\/mrow><\/munderover><msub><mrow><mi>a<\/mi><\/mrow><mrow> <mi>\u2113<\/mi><mo class=\"MathClass-bin\">\u2212<\/mo><mn>1<\/mn><\/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=\"x1-77001r1\" \/><mstyle class=\"maketag\"><mtext>(3.1)<\/mtext><\/mstyle><mspace class=\"nbsp\" width=\"0.33em\" \/> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">Dies l\u00e4sst sich direkt mittels vollst\u00e4ndiger Induktion beweisen (siehe die folgende \u00dcbung), doch wollen wir bemerken, dass es leicht ist, sich diese Formeln zu merken. Statt diese auswendig zu lernen, \u00fcberpr\u00fcfen Sie einfach bei Auftreten von Indexverschiebungen dieser Form bei beiden Summen, ob jeweils diesselben Ausdr\u00fccke f\u00fcr den ersten und den letzten Summanden auftreten. <\/p> <div class=\"me meexample\"> <p class=\"indent\"><\/p><h4 id=\"z591d29b41c63\"> <a id=\"x1-77002r1\"><\/a> <span class=\"ecbx-1095\">\u00dc<\/span><span class=\"ecbx-1095\">bung 3.1 <\/span>(Indexverschiebung)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Beweisen Sie  Gleichung<\/span>  (<a href=\"..\/..\/chapter\/summen-und-produkte#x1-77001r1\">3.1<\/a>)  <span class=\"ecti-1095\">und  eine  analoge  Formel  f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r  das  Produkt  mittels<\/span> <span class=\"ecti-1095\">vollst<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">ndiger Induktion.<\/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\">Bei         dem         Induktionsbeweis         bez<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">glich         der         Variable<\/span> <math display=\"inline\"><mi>n<\/mi> <mo class=\"MathClass-rel\">\u2265<\/mo> <mi>m<\/mi><\/math> <span class=\"ecti-1095\">werden                                            Sie                                            feststellen,<\/span> <span class=\"ecti-1095\">dass man genau die Eigenschaften in obiger Merkregel verwendet: F<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r den Induktionsanfang<\/span> <math display=\"inline\"><mi>n<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mi>m<\/mi><\/math> <span class=\"ecti-1095\">m<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">ssen die   ersten   Summanden   <\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">bereinstimmen.   F<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r   den   Induktionsschritt   von<\/span> <math display=\"inline\"><mi>n<\/mi><\/math> <span class=\"ecti-1095\">nach<\/span> <math display=\"inline\"><mi>n<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mn>1<\/mn><\/math> <span class=\"ecti-1095\">m<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">ssen die zus<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">tzlichen (also die letzten) Summanden <\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">bereinstimmen.<\/span><\/p><\/details>  <\/div> <p class=\"indent\">Der einfachste Fall einer Funktion <math display=\"inline\"><mi>j<\/mi><mo class=\"MathClass-rel\">\u21a6<\/mo><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>j<\/mi><\/mrow><\/msub><\/math> ist der Fall der konstanten Funktion <math display=\"inline\"><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>j<\/mi><\/mrow><\/msub> <mo class=\"MathClass-rel\">=<\/mo> <mi>z<\/mi><\/math> f\u00fcr ein <math display=\"inline\"><mi>z<\/mi><\/math> und f\u00fcr alle <span class=\"maperiod\"><math display=\"inline\"><mi>j<\/mi><\/math><\/span><span class=\"period\">.<\/span> In diesem Fall ergibt sich die Summe zu&nbsp;<math display=\"inline\"><msubsup><mrow><mi class=\"MathClass-op\"> \u2211<\/mi><mo> <\/mo> <\/mrow><mrow><mi>j<\/mi><mo class=\"MathClass-rel\">=<\/mo><mn>1<\/mn><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msubsup><mi>z<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mi>n<\/mi><mi>z<\/mi><\/math> f\u00fcr alle <math display=\"inline\"><mi>n<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2115<\/mi><\/math> und <math display=\"inline\"><mi>z<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2102<\/mi><\/math> (oder <math display=\"inline\"><mi>z<\/mi><\/math> in einem Vektorraum). Im Falle des Produkts erhalten wir aber die Definition der <span class=\"ecbx-1095\">Potenzfunktion<\/span> f\u00fcr <math display=\"inline\"><mi>z<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2102<\/mi><\/math> und <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\"><msup><mrow><mi>z<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msup> <mo class=\"MathClass-rel\">=<\/mo><munderover accent=\"false\" accentunder=\"false\"><mrow><mo> \u220f<\/mo> <\/mrow><mrow><mi>j<\/mi><mo class=\"MathClass-rel\">=<\/mo><mn>1<\/mn><\/mrow><mrow><mi>n<\/mi><\/mrow><\/munderover><mi>z<\/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\">die somit rekursiv durch <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><msup><mrow><mi>z<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msup> <mo class=\"MathClass-rel\">=<\/mo> <mi>z<\/mi><mstyle class=\"text\"><mtext>&nbsp;und&nbsp;<\/mtext><\/mstyle><msup><mrow><mi>z<\/mi><\/mrow><mrow><mi>n<\/mi><mo class=\"MathClass-bin\">+<\/mo><mn>1<\/mn><\/mrow><\/msup> <mo class=\"MathClass-rel\">=<\/mo> <msup><mrow><mi>z<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msup><mi>z<\/mi><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">f\u00fcr alle <math display=\"inline\"><mi>n<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2115<\/mi><\/math> definiert ist. Wir nennen&nbsp;<math display=\"inline\"><mi>z<\/mi><\/math> die <span class=\"ecbx-1095\">Basis <\/span>und&nbsp;<math display=\"inline\"><mi>n<\/mi><\/math> den <span class=\"ecbx-1095\">Exponenten<\/span>. Wir erweitern diese Definition durch <\/p><math display=\"block\"><mtable class=\"align\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><msup><mrow><mi>z<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msup> <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=\"x1-77003r2\" \/><mstyle class=\"maketag\"><mtext>(3.2)<\/mtext><\/mstyle><mspace class=\"nbsp\" width=\"0.33em\" \/> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">f\u00fcr alle <math display=\"inline\"><mi>z<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2102<\/mi><\/math> (insbesondere<button class=\"hover-trigger\" style=\"vertical-align: super;font: smaller\">\u2020<\/button><span class=\"hover-text\"><span class=\"marginpar\">\u2020 Da <math display=\"inline\"><mi>z<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2102<\/mi><mo class=\"MathClass-rel\">\u21a6<\/mo> <msup><mrow><mi>z<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msup><\/math> die konstante Funktion mit Wert <math display=\"inline\"><mn>1<\/mn><\/math> darstellt, w\u00e4re es sehr eigenartig (und im n\u00e4chsten Abschnitt bei der Diskussion von Polynomen extrem st\u00f6rend), wenn wir diese Funktion f\u00fcr <math display=\"inline\"><mi>z<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mn>0<\/mn><\/math> undefiniert lassen oder mit einem anderen Wert versehen. Trotzdem ist der Ausdruck <math display=\"inline\"><msup><mrow><mn>0<\/mn><\/mrow><mrow><mn>0<\/mn> <\/mrow> <\/msup> <\/math> undefiniert, wenn dieser losgel\u00f6st von der Diskussion der Potenzfunktion <math display=\"inline\"><mi>z<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2102<\/mi><mo class=\"MathClass-rel\">\u21a6<\/mo> <msup><mrow><mi>z<\/mi><\/mrow><mrow><mi>n<\/mi> <\/mrow> <\/msup> <\/math> f\u00fcr <math display=\"inline\"><mi>n<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mn>0<\/mn><\/math> auftritt.<\/span><\/span> f\u00fcr <math display=\"inline\"><mi>z<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mn>0<\/mn><\/math>) und <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><msup><mrow><mi>z<\/mi><\/mrow><mrow><mo class=\"MathClass-bin\">\u2212<\/mo><mi>n<\/mi><\/mrow><\/msup> <mo class=\"MathClass-rel\">=<\/mo> <msup><mrow><mo class=\"MathClass-open\">(<\/mo><msup><mrow><mi>z<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msup><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mo class=\"MathClass-bin\">\u2212<\/mo><mn>1<\/mn><\/mrow><\/msup><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">f\u00fcr alle <math display=\"inline\"><mi>z<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <msup><mrow><mi>\u2102<\/mi><\/mrow><mrow><mo class=\"MathClass-bin\">\u00d7<\/mo><\/mrow><\/msup> <mo class=\"MathClass-punc\">:<\/mo><mo class=\"MathClass-rel\">=<\/mo> <mi>\u2102<\/mi> <mo class=\"MathClass-bin\">\u2216<\/mo><mrow><mo fence=\"true\" form=\"prefix\"> {<\/mo><mrow><mn>0<\/mn><\/mrow><mo fence=\"true\" form=\"postfix\">}<\/mo><\/mrow><\/math> und <span class=\"maperiod\"><math display=\"inline\"><mi>n<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2115<\/mi><\/math><\/span><span class=\"period\">.<\/span> Die n\u00e4chste \u00dcbung zeigt, dass die so definierte Potenzfunktion die \u00fcblichen Rechenregeln erf\u00fcllt. <\/p> <div class=\"me melemma\"> <p class=\"indent\"><\/p><h4 id=\"zd9cc817b329e\"> <a id=\"x1-77004r2\"><\/a> <span class=\"ecbx-1095\">Wichtige <\/span><span class=\"ecbx-1095\">\u00dc<\/span><span class=\"ecbx-1095\">bung 3.2.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Beweisen Sie <\/span><span class=\"maperiod\"><math display=\"inline\"><msup><mrow><mo class=\"MathClass-open\">(<\/mo><mi>z<\/mi><mi>w<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mi>m<\/mi><\/mrow><\/msup> <mo class=\"MathClass-rel\">=<\/mo> <msup><mrow><mi>z<\/mi><\/mrow><mrow><mi>m<\/mi><\/mrow><\/msup><msup><mrow><mi>w<\/mi><\/mrow><mrow><mi>m<\/mi><\/mrow><\/msup><\/math><\/span><span class=\"period\">,<\/span> <math display=\"inline\"><msup><mrow><mi>z<\/mi><\/mrow><mrow><mi>m<\/mi><mo class=\"MathClass-bin\">+<\/mo><mi>n<\/mi> <\/mrow> <\/msup> <mo class=\"MathClass-rel\">=<\/mo> <msup><mrow><mi>z<\/mi><\/mrow><mrow><mi>m<\/mi> <\/mrow> <\/msup> <msup><mrow><mi>z<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msup><\/math> <span class=\"ecti-1095\">und<\/span> <math display=\"inline\"><msup><mrow><mo class=\"MathClass-open\">(<\/mo><msup><mrow><mi>z<\/mi><\/mrow><mrow><mi>m<\/mi> <\/mrow> <\/msup> <mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mi>n<\/mi> <\/mrow> <\/msup> <mo class=\"MathClass-rel\">=<\/mo> <msup><mrow><mi>z<\/mi><\/mrow><mrow><mi>m<\/mi><mi>n<\/mi><\/mrow><\/msup><\/math> <span class=\"ecti-1095\">zuerst f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r<\/span> <span class=\"ecti-1095\">alle <\/span><math display=\"inline\"><mi>z<\/mi><mo class=\"MathClass-punc\">,<\/mo> <mi>w<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2102<\/mi><\/math> <span class=\"ecti-1095\">und<\/span> <math display=\"inline\"><mi>m<\/mi><mo class=\"MathClass-punc\">,<\/mo> <mi>n<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <msub><mrow><mi>\u2115<\/mi><\/mrow><mrow><mn>0<\/mn> <\/mrow> <\/msub> <\/math> <span class=\"ecti-1095\">mit vollst<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">ndiger<\/span> <span class=\"ecti-1095\">Induktion und dann f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r alle <\/span><math display=\"inline\"><mi>z<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>w<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <msup><mrow><mi>\u2102<\/mi><\/mrow><mrow><mo class=\"MathClass-bin\">\u00d7<\/mo><\/mrow><\/msup><\/math> <span class=\"ecti-1095\">und <\/span><math display=\"inline\"><mi>m<\/mi><mo class=\"MathClass-punc\">,<\/mo> <mi>n<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2124<\/mi><\/math><span class=\"ecti-1095\">. <\/span><\/p><details><summary style=\"color:#FF7F00\"><span class=\"ecti-1095\">Teill<\/span><span class=\"ecti-1095\">\u00f6<\/span><span class=\"ecti-1095\">sung.<\/span><\/summary><p class=\"indent\" style=\"margin-top: 0\"> <span class=\"ecti-1095\">Wir beweisen <\/span><math display=\"inline\"><msup><mrow><mo class=\"MathClass-open\">(<\/mo><mi>z<\/mi><mi>w<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mi>m<\/mi><\/mrow><\/msup> <mo class=\"MathClass-rel\">=<\/mo> <msup><mrow><mi>z<\/mi><\/mrow><mrow><mi>m<\/mi><\/mrow><\/msup><msup><mrow><mi>w<\/mi><\/mrow><mrow><mi>m<\/mi><\/mrow><\/msup><\/math> <span class=\"ecti-1095\">f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r alle <\/span><math display=\"inline\"><mi>z<\/mi><mo class=\"MathClass-punc\">,<\/mo> <mi>w<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2102<\/mi><\/math> <span class=\"ecti-1095\">und<\/span> <math display=\"inline\"><mi>m<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <msub><mrow><mi>\u2115<\/mi><\/mrow><mrow><mn>0<\/mn> <\/mrow> <\/msub> <\/math> <span class=\"ecti-1095\">mittels<\/span> <span class=\"ecti-1095\">Induktion nach <\/span><span class=\"maperiod\"><math display=\"inline\"><mi>m<\/mi><\/math><\/span><span class=\"period\">.<\/span> <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\">=<\/mo> <mn>0<\/mn><\/math> <span class=\"ecti-1095\">ergibt<\/span> <span class=\"ecti-1095\">sich <\/span><math display=\"inline\"><mn>1<\/mn> <mo class=\"MathClass-rel\">=<\/mo> <mn>1<\/mn> <mo class=\"MathClass-bin\">\u22c5<\/mo> <mn>1<\/mn><\/math> <span class=\"ecti-1095\">nach Definition. Angenommen die Aussage gilt bereits f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r ein<\/span> <math display=\"inline\"><mi>m<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <msub><mrow><mi>\u2115<\/mi><\/mrow><mrow><mn>0<\/mn> <\/mrow> <\/msub> <\/math><span class=\"ecti-1095\">, dann<\/span> <span class=\"ecti-1095\">gilt ebenso<\/span> <\/p> <table id=\"z7fa83074ab6a\" class=\"equation-star\"><tr><td> <math class=\"equation\" display=\"block\"> <msup><mrow><mo class=\"MathClass-open\">(<\/mo><mi>z<\/mi><mi>w<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mi>m<\/mi><mo class=\"MathClass-bin\">+<\/mo><mn>1<\/mn><\/mrow><\/msup> <mo class=\"MathClass-rel\">=<\/mo> <msup><mrow><mo class=\"MathClass-open\">(<\/mo><mi>z<\/mi><mi>w<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mi>m<\/mi><\/mrow><\/msup><mo class=\"MathClass-open\">(<\/mo><mi>z<\/mi><mi>w<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <msup><mrow><mi>z<\/mi><\/mrow><mrow><mi>m<\/mi><\/mrow><\/msup><msup><mrow><mi>w<\/mi><\/mrow><mrow><mi>m<\/mi><\/mrow><\/msup><mi>z<\/mi><mi>w<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <msup><mrow><mi>z<\/mi><\/mrow><mrow><mi>m<\/mi><mo class=\"MathClass-bin\">+<\/mo><mn>1<\/mn><\/mrow><\/msup><msup><mrow><mi>w<\/mi><\/mrow><mrow><mi>m<\/mi><mo class=\"MathClass-bin\">+<\/mo><mn>1<\/mn><\/mrow><\/msup><mo class=\"MathClass-punc\">.<\/mo> <\/math><\/td><\/tr><\/table> <p class=\"indent\"><span class=\"ecti-1095\">Dies beweist den Induktionsschritt und damit die gew<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">nschte Aussage. <\/span><\/p><\/details>  <\/div> <div class=\"me meexample\"> <p class=\"indent\"><\/p><h4 id=\"zc9b857a8d07e\"> <span class=\"ecti-1095\">Bemerkung.<\/span><\/h4> <p class=\"indent\">Formal gesehen  sollten  wir  auch  alle  weiteren  Rechenregeln  in  diesem  Abschnitt  mit vollst\u00e4ndiger Induktion beweisen. Da uns diese Beweise aber sehr wenig lehren, werden wir darauf verzichten. <\/p> <\/div> <a id=\"x1-77005r75\"><\/a> <h4 id=\"z8b67de9a107d\" class=\"subsectionHead\"><span class=\"titlemark\">3.1.1 <\/span> <a id=\"x1-780001\"><\/a>Rechenregeln f\u00fcr die Summe<\/h4> <p class=\"noindent\">Die Summe erf\u00fcllt f\u00fcr gegebene ganze Zahlen <math display=\"inline\"><mi>m<\/mi><mo class=\"MathClass-punc\">,<\/mo> <mi>n<\/mi><\/math> mit <math display=\"inline\"><mi>m<\/mi> <mo class=\"MathClass-rel\">\u2264<\/mo> <mi>n<\/mi><\/math> die Gleichungen <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><munderover accent=\"false\" accentunder=\"false\"><mrow><mo> \u2211<\/mo> <\/mrow><mrow><mi>k<\/mi><mo class=\"MathClass-rel\">=<\/mo><mi>m<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/munderover><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>a<\/mi><\/mrow><mrow> <mi>k<\/mi><\/mrow><\/msub> <mo class=\"MathClass-bin\">+<\/mo> <msub><mrow><mi>b<\/mi><\/mrow><mrow><mi>k<\/mi><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">=<\/mo><munderover accent=\"false\" accentunder=\"false\"><mrow><mo> \u2211<\/mo> <\/mrow><mrow><mi>k<\/mi><mo class=\"MathClass-rel\">=<\/mo><mi>m<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/munderover><msub><mrow><mi>a<\/mi><\/mrow><mrow> <mi>k<\/mi><\/mrow><\/msub> <mo class=\"MathClass-bin\">+<\/mo><munderover accent=\"false\" accentunder=\"false\"><mrow><mo> \u2211<\/mo> <\/mrow><mrow><mi>k<\/mi><mo class=\"MathClass-rel\">=<\/mo><mi>m<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/munderover><msub><mrow><mi>b<\/mi><\/mrow><mrow> <mi>k<\/mi><\/mrow><\/msub><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">und <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><munderover accent=\"false\" accentunder=\"false\"><mrow><mo> \u2211<\/mo> <\/mrow><mrow><mi>k<\/mi><mo class=\"MathClass-rel\">=<\/mo><mi>m<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/munderover><mo class=\"MathClass-open\">(<\/mo><mi>c<\/mi><msub><mrow><mi>a<\/mi><\/mrow><mrow> <mi>k<\/mi><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <mi>c<\/mi><munderover accent=\"false\" accentunder=\"false\"><mrow><mo>\u2211<\/mo> <\/mrow><mrow><mi>k<\/mi><mo class=\"MathClass-rel\">=<\/mo><mi>m<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/munderover><msub><mrow><mi>a<\/mi><\/mrow><mrow> <mi>k<\/mi><\/mrow><\/msub><mo class=\"MathClass-punc\">,<\/mo><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">wobei <span class=\"maperiod\"><math display=\"inline\"><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>m<\/mi> <\/mrow> <\/msub> <mo class=\"MathClass-punc\">,<\/mo> <mi class=\"MathClass-op\">\u2026<\/mi><mo> <\/mo> <mo class=\"MathClass-punc\">,<\/mo><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><\/math><\/span><span class=\"period\">,<\/span> <math display=\"inline\"><msub><mrow><mi>b<\/mi><\/mrow><mrow><mi>m<\/mi> <\/mrow> <\/msub> <mo class=\"MathClass-punc\">,<\/mo> <mi class=\"MathClass-op\">\u2026<\/mi><mo> <\/mo> <mo class=\"MathClass-punc\">,<\/mo> <msub><mrow><mi>b<\/mi><\/mrow><mrow><mi>n<\/mi> <\/mrow> <\/msub> <\/math> in einem reellen (respektive komplexen) Vektorraum&nbsp;<math display=\"inline\"><mi>V<\/mi> <\/math> liegen und <math display=\"inline\"><mi>c<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u211d<\/mi><\/math> (respektive <math display=\"inline\"><mi>c<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2102<\/mi><\/math>) ein Skalar ist. (Die erste Eigenschaft ist eine Mischung aus Assoziativgesetz und Kommutativgesetz f\u00fcr die Addition, und die zweite Eigenschaft ist eine Verallgemeinerung des Distributivgesetzes.) <\/p><p class=\"indent\">Diese beiden Eigenschaften (die Summe wird auf die Summe und das skalare Vielfache auf das skalare Vielfache abgebildet) werden auch als <span class=\"ecbx-1095\">Linearit<\/span><span class=\"ecbx-1095\">\u00e4<\/span><span class=\"ecbx-1095\">t der<\/span> <span class=\"ecbx-1095\">Abbildung<\/span>&nbsp;<math display=\"inline\"><mi class=\"MathClass-op\"> \u2211<\/mi><mo> <\/mo> <\/math> bezeichnet, wobei&nbsp;<math display=\"inline\"><mi class=\"MathClass-op\"> \u2211<\/mi><mo> <\/mo> <\/math> auf dem Vektorraum <math display=\"inline\"><msup><mrow><mi>V<\/mi> <\/mrow><mrow><mo class=\"MathClass-open\">{<\/mo><mi>m<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi class=\"MathClass-op\">\u2026<\/mi><mo> <\/mo><mo class=\"MathClass-punc\">,<\/mo><mi>n<\/mi><mo class=\"MathClass-close\">}<\/mo><\/mrow><\/msup><\/math> der Funktionen von&nbsp;<math display=\"inline\"><mo class=\"MathClass-open\">{<\/mo><mi>m<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi class=\"MathClass-op\">\u2026<\/mi><mo> <\/mo><mo class=\"MathClass-punc\">,<\/mo><mi>n<\/mi><mo class=\"MathClass-close\">}<\/mo><\/math> nach <math display=\"inline\"><mi>V<\/mi> <\/math> definiert ist, den Vektorraum <math display=\"inline\"><mi>V<\/mi> <\/math> als Zielbereich besitzt, und&nbsp;<math display=\"inline\"><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>m<\/mi><\/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-close\">)<\/mo> <mo class=\"MathClass-rel\">\u2208<\/mo> <msup><mrow><mi>V<\/mi> <\/mrow><mrow><mo class=\"MathClass-open\">{<\/mo><mi>m<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi class=\"MathClass-op\">\u2026<\/mi><mo> <\/mo><mo class=\"MathClass-punc\">,<\/mo><mi>n<\/mi><mo class=\"MathClass-close\">}<\/mo><\/mrow><\/msup><\/math> auf&nbsp;<math display=\"inline\"><msubsup><mrow><mi class=\"MathClass-op\"> \u2211<\/mi><mo> <\/mo> <\/mrow><mrow><mi>k<\/mi><mo class=\"MathClass-rel\">=<\/mo><mi>m<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msubsup><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>k<\/mi><\/mrow><\/msub><\/math> abbildet. Wie der Name sagt, wird Linearit\u00e4t ausf\u00fchrlicher in der Linearen Algebra besprochen. Es handelt sich dabei aber auch um eine wichtige Eigenschaft f\u00fcr die Analysis, welche also h\u00e4ufig auftreten. <\/p><p class=\"indent\">Des Weiteren gilt die Formel f\u00fcr die <span class=\"ecbx-1095\">Teleskopsumme<\/span> <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><munderover accent=\"false\" accentunder=\"false\"><mrow><mo>\u2211<\/mo> <\/mrow><mrow><mi>k<\/mi><mo class=\"MathClass-rel\">=<\/mo><mi>m<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/munderover><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>a<\/mi><\/mrow><mrow> <mi>k<\/mi><mo class=\"MathClass-bin\">+<\/mo><mn>1<\/mn><\/mrow><\/msub> <mo class=\"MathClass-bin\">\u2212<\/mo> <msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>k<\/mi><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><\/mtd> <mtd class=\"align-even\"> <mo class=\"MathClass-rel\">=<\/mo> <mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>m<\/mi><mo class=\"MathClass-bin\">+<\/mo><mn>1<\/mn><\/mrow><\/msub> <mo class=\"MathClass-bin\">\u2212<\/mo> <msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>m<\/mi><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">+<\/mo> <mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>m<\/mi><mo class=\"MathClass-bin\">+<\/mo><mn>2<\/mn><\/mrow><\/msub> <mo class=\"MathClass-bin\">\u2212<\/mo> <msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>m<\/mi><mo class=\"MathClass-bin\">+<\/mo><mn>1<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">+<\/mo> <mi class=\"MathClass-op\">\u2026<\/mi><mo> <\/mo> <mo class=\"MathClass-bin\">+<\/mo> <mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub> <mo class=\"MathClass-bin\">\u2212<\/mo> <msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi><mo class=\"MathClass-bin\">\u2212<\/mo><mn>1<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">+<\/mo> <mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi><mo class=\"MathClass-bin\">+<\/mo><mn>1<\/mn><\/mrow><\/msub> <mo class=\"MathClass-bin\">\u2212<\/mo> <msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\" \/> <mtd class=\"align-label\"> <mspace width=\"2em\" \/><\/mtd><\/mtr><mtr><mtd class=\"align-odd\" columnalign=\"right\" \/> <mtd class=\"align-even\"> <mo class=\"MathClass-rel\">=<\/mo> <msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi><mo class=\"MathClass-bin\">+<\/mo><mn>1<\/mn><\/mrow><\/msub> <mo class=\"MathClass-bin\">\u2212<\/mo> <msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>m<\/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\">wobei <math display=\"inline\"><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>m<\/mi> <\/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><mo class=\"MathClass-bin\">+<\/mo><mn>1<\/mn><\/mrow><\/msub><\/math> in einem reellen oder einem komplexen Vektorraum liegen. Formaler argumentiert gilt                                                                                                                                                                           <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><munderover accent=\"false\" accentunder=\"false\"><mrow><mo> \u2211<\/mo> <\/mrow><mrow><mi>k<\/mi><mo class=\"MathClass-rel\">=<\/mo><mi>m<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/munderover><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>a<\/mi><\/mrow><mrow> <mi>k<\/mi><mo class=\"MathClass-bin\">+<\/mo><mn>1<\/mn><\/mrow><\/msub> <mo class=\"MathClass-bin\">\u2212<\/mo> <msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>k<\/mi><\/mrow><\/msub><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><mi>m<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/munderover><msub><mrow><mi>a<\/mi><\/mrow><mrow> <mi>k<\/mi><mo class=\"MathClass-bin\">+<\/mo><mn>1<\/mn><\/mrow><\/msub> <mo class=\"MathClass-bin\">\u2212<\/mo><munderover accent=\"false\" accentunder=\"false\"><mrow><mo>\u2211<\/mo> <\/mrow><mrow><mi>k<\/mi><mo class=\"MathClass-rel\">=<\/mo><mi>m<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/munderover><msub><mrow><mi>a<\/mi><\/mrow><mrow> <mi>k<\/mi><\/mrow><\/msub> <mo class=\"MathClass-rel\">=<\/mo><munderover accent=\"false\" accentunder=\"false\"><mrow><mo> \u2211<\/mo> <\/mrow><mrow><mi>j<\/mi><mo class=\"MathClass-rel\">=<\/mo><mi>m<\/mi><mo class=\"MathClass-bin\">+<\/mo><mn>1<\/mn><\/mrow><mrow><mi>n<\/mi><mo class=\"MathClass-bin\">+<\/mo><mn>1<\/mn><\/mrow><\/munderover><msub><mrow><mi>a<\/mi><\/mrow><mrow> <mi>j<\/mi><\/mrow><\/msub> <mo class=\"MathClass-bin\">\u2212<\/mo><munderover accent=\"false\" accentunder=\"false\"><mrow><mo>\u2211<\/mo> <\/mrow><mrow><mi>k<\/mi><mo class=\"MathClass-rel\">=<\/mo><mi>m<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/munderover><msub><mrow><mi>a<\/mi><\/mrow><mrow> <mi>k<\/mi><\/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> <mstyle><mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow \/><mo fence=\"true\" form=\"postfix\" \/><\/mrow><\/mstyle><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi><mo class=\"MathClass-bin\">+<\/mo><mn>1<\/mn><\/mrow><\/msub> <mo class=\"MathClass-bin\">+<\/mo><munderover accent=\"false\" accentunder=\"false\"><mrow><mo> \u2211<\/mo> <\/mrow><mrow><mi>j<\/mi><mo class=\"MathClass-rel\">=<\/mo><mi>m<\/mi><mo class=\"MathClass-bin\">+<\/mo><mn>1<\/mn><\/mrow><mrow><mi>n<\/mi><\/mrow><\/munderover><msub><mrow><mi>a<\/mi><\/mrow><mrow> <mi>j<\/mi><\/mrow><\/msub><mstyle><mrow><mo fence=\"true\" form=\"prefix\"> )<\/mo><mrow \/><mo fence=\"true\" form=\"postfix\" \/><\/mrow><\/mstyle> <mo class=\"MathClass-bin\">\u2212<\/mo><mstyle><mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow \/><mo fence=\"true\" form=\"postfix\" \/><\/mrow><\/mstyle><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>m<\/mi><\/mrow><\/msub> <mo class=\"MathClass-bin\">+<\/mo><munderover accent=\"false\" accentunder=\"false\"><mrow><mo> \u2211<\/mo> <\/mrow><mrow><mi>k<\/mi><mo class=\"MathClass-rel\">=<\/mo><mi>m<\/mi><mo class=\"MathClass-bin\">+<\/mo><mn>1<\/mn><\/mrow><mrow><mi>n<\/mi><\/mrow><\/munderover><msub><mrow><mi>a<\/mi><\/mrow><mrow> <mi>k<\/mi><\/mrow><\/msub><mstyle><mrow><mo fence=\"true\" form=\"prefix\"> )<\/mo><mrow \/><mo fence=\"true\" form=\"postfix\" \/><\/mrow><\/mstyle> <mo class=\"MathClass-rel\">=<\/mo> <msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi><mo class=\"MathClass-bin\">+<\/mo><mn>1<\/mn><\/mrow><\/msub> <mo class=\"MathClass-bin\">\u2212<\/mo> <msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>m<\/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\">wie bereits behauptet. Die Formel f\u00fcr die Teleskopsumme l\u00e4sst sich zur Abel-Summationsformel verallgemeinern, welche \u00fcberraschend viele Anwendungen in der Analysis und Zahlentheorie findet. <\/p> <div class=\"me meexample\"> <p class=\"indent\"><\/p><h4 id=\"zfd32429ac61d\"> <a id=\"x1-78001r3\"><\/a> <span class=\"ecbx-1095\">\u00dc<\/span><span class=\"ecbx-1095\">bung 3.3 <\/span>(Abel-Summation)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Seien <\/span><span class=\"maperiod\"><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-punc\">,<\/mo><msub><mrow><mi>b<\/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>b<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2102<\/mi><\/math><\/span><span class=\"period\">.<\/span> <span class=\"ecti-1095\">Wir setzen <\/span><math display=\"inline\"><msub><mrow><mi>A<\/mi><\/mrow><mrow><mi>k<\/mi><\/mrow><\/msub> <mo class=\"MathClass-rel\">=<\/mo><msubsup><mrow><mi class=\"MathClass-op\"> \u2211<\/mi><mo> <\/mo> <\/mrow><mrow><mi>j<\/mi><mo class=\"MathClass-rel\">=<\/mo><mn>1<\/mn><\/mrow><mrow><mi>k<\/mi><\/mrow><\/msubsup><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>j<\/mi><\/mrow><\/msub><\/math> <span class=\"ecti-1095\">f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r <\/span><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> <span class=\"ecti-1095\">mit <\/span><span class=\"maperiod\"><math display=\"inline\"><mi>k<\/mi> <mo class=\"MathClass-rel\">\u2264<\/mo> <mi>n<\/mi><\/math><\/span><span class=\"period\">.<\/span> <span class=\"ecti-1095\">Zeigen Sie die Abel-Summationsformel<\/span> <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><munderover accent=\"false\" accentunder=\"false\"><mrow><mo>\u2211<\/mo> <\/mrow><mrow><mi>k<\/mi><mo class=\"MathClass-rel\">=<\/mo><mn>1<\/mn><\/mrow><mrow><mi>n<\/mi><\/mrow><\/munderover><msub><mrow><mi>a<\/mi><\/mrow><mrow> <mi>k<\/mi><\/mrow><\/msub><msub><mrow><mi>b<\/mi><\/mrow><mrow><mi>k<\/mi><\/mrow><\/msub> <mo class=\"MathClass-rel\">=<\/mo> <msub><mrow><mi>A<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><msub><mrow><mi>b<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub> <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><mo class=\"MathClass-bin\">\u2212<\/mo><mn>1<\/mn><\/mrow><\/munderover><msub><mrow><mi>A<\/mi><\/mrow><mrow> <mi>k<\/mi><\/mrow><\/msub> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><msub><mrow><mi>b<\/mi><\/mrow><mrow><mi>k<\/mi><\/mrow><\/msub> <mo class=\"MathClass-bin\">\u2212<\/mo> <msub><mrow><mi>b<\/mi><\/mrow><mrow><mi>k<\/mi><mo class=\"MathClass-bin\">+<\/mo><mn>1<\/mn><\/mrow><\/msub><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-punc\">.<\/mo><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\"><span class=\"ecti-1095\">Verwenden Sie dazu die Gleichung <\/span><math display=\"inline\"><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>k<\/mi><\/mrow><\/msub> <mo class=\"MathClass-rel\">=<\/mo> <msub><mrow><mi>A<\/mi><\/mrow><mrow><mi>k<\/mi><\/mrow><\/msub> <mo class=\"MathClass-bin\">\u2212<\/mo> <msub><mrow><mi>A<\/mi><\/mrow><mrow><mi>k<\/mi><mo class=\"MathClass-bin\">\u2212<\/mo><mn>1<\/mn><\/mrow><\/msub><\/math> <span class=\"ecti-1095\">f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r alle <\/span><math display=\"inline\"><mi>k<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2115<\/mi><\/math> <span class=\"ecti-1095\">mit <\/span><span class=\"maperiod\"><math display=\"inline\"><mi>k<\/mi> <mo class=\"MathClass-rel\">\u2264<\/mo> <mi>n<\/mi><\/math><\/span><span class=\"period\">.<\/span> <span class=\"ecti-1095\">Wenden Sie des Weiteren die Abel-Summation auf die Summe<\/span> <math display=\"inline\"><msubsup><mrow><mi class=\"MathClass-op\">\u2211<\/mi><mo> <\/mo> <\/mrow><mrow><mi>k<\/mi><mo class=\"MathClass-rel\">=<\/mo><mn>1<\/mn><\/mrow><mrow><mn>2<\/mn><mi>n<\/mi><\/mrow><\/msubsup><mfrac><mrow><msup><mrow><mo class=\"MathClass-open\">(<\/mo><mo class=\"MathClass-bin\">\u2212<\/mo><mn>1<\/mn><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mi>k<\/mi><\/mrow><\/msup><\/mrow> <mrow><mi>k<\/mi><\/mrow><\/mfrac> <\/math> <span class=\"ecti-1095\">an.<\/span> <\/p><p class=\"indent\"><\/p><details><summary style=\"color:#FF7F00\"><span class=\"ecti-1095\">L<\/span><span class=\"ecti-1095\">\u00f6<\/span><span class=\"ecti-1095\">sung.<\/span><\/summary><p class=\"indent\" style=\"margin-top: 0\"> <span class=\"ecti-1095\">Wir bemerken zuerst, dass <\/span><math display=\"inline\"><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>k<\/mi><\/mrow><\/msub> <mo class=\"MathClass-rel\">=<\/mo> <msub><mrow><mi>A<\/mi><\/mrow><mrow><mi>k<\/mi><\/mrow><\/msub> <mo class=\"MathClass-bin\">\u2212<\/mo> <msub><mrow><mi>A<\/mi><\/mrow><mrow><mi>k<\/mi><mo class=\"MathClass-bin\">\u2212<\/mo><mn>1<\/mn><\/mrow><\/msub><\/math> <span class=\"ecti-1095\">f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r alle <\/span><math display=\"inline\"><mi>k<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2115<\/mi><\/math> <span class=\"ecti-1095\">mit<\/span> <math display=\"inline\"><mi>k<\/mi> <mo class=\"MathClass-rel\">\u2264<\/mo> <mi>n<\/mi><\/math> <span class=\"ecti-1095\">auf Grund von<\/span> <math display=\"inline\"><msub><mrow><mi>A<\/mi><\/mrow><mrow><mn>0<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-rel\">=<\/mo> <mn>0<\/mn><\/math> <span class=\"ecti-1095\">und der rekursiven<\/span> <span class=\"ecti-1095\">Definition der Summe <\/span><math display=\"inline\"><msub><mrow><mi>A<\/mi><\/mrow><mrow><mi>k<\/mi><\/mrow><\/msub> <mo class=\"MathClass-rel\">=<\/mo><msubsup><mrow><mi class=\"MathClass-op\"> \u2211<\/mi><mo> <\/mo> <\/mrow><mrow><mi>j<\/mi><mo class=\"MathClass-rel\">=<\/mo><mn>1<\/mn><\/mrow><mrow><mi>k<\/mi><\/mrow><\/msubsup><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>j<\/mi><\/mrow><\/msub><\/math> <span class=\"ecti-1095\">gilt. Nun gehen wir wie vorgeschlagen vor und erhalten<\/span> <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><munderover accent=\"false\" accentunder=\"false\"><mrow><mo>\u2211<\/mo> <\/mrow><mrow><mi>k<\/mi><mo class=\"MathClass-rel\">=<\/mo><mn>1<\/mn><\/mrow><mrow><mi>n<\/mi><\/mrow><\/munderover><msub><mrow><mi>a<\/mi><\/mrow><mrow> <mi>k<\/mi><\/mrow><\/msub><msub><mrow><mi>b<\/mi><\/mrow><mrow><mi>k<\/mi><\/mrow><\/msub><\/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><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>A<\/mi><\/mrow><mrow> <mi>k<\/mi><\/mrow><\/msub> <mo class=\"MathClass-bin\">\u2212<\/mo> <msub><mrow><mi>A<\/mi><\/mrow><mrow><mi>k<\/mi><mo class=\"MathClass-bin\">\u2212<\/mo><mn>1<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><msub><mrow><mi>b<\/mi><\/mrow><mrow><mi>k<\/mi><\/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><munderover accent=\"false\" accentunder=\"false\"><mrow><mo> \u2211<\/mo> <\/mrow><mrow><mi>k<\/mi><mo class=\"MathClass-rel\">=<\/mo><mn>1<\/mn><\/mrow><mrow><mi>n<\/mi><\/mrow><\/munderover><msub><mrow><mi>A<\/mi><\/mrow><mrow> <mi>k<\/mi><\/mrow><\/msub><msub><mrow><mi>b<\/mi><\/mrow><mrow><mi>k<\/mi><\/mrow><\/msub> <mo class=\"MathClass-bin\">\u2212<\/mo><munderover accent=\"false\" accentunder=\"false\"><mrow><mo>\u2211<\/mo> <\/mrow><mrow><mi>j<\/mi><mo class=\"MathClass-rel\">=<\/mo><mn>0<\/mn><\/mrow><mrow><mi>n<\/mi><mo class=\"MathClass-bin\">\u2212<\/mo><mn>1<\/mn><\/mrow><\/munderover><msub><mrow><mi>A<\/mi><\/mrow><mrow> <mi>j<\/mi><\/mrow><\/msub><msub><mrow><mi>b<\/mi><\/mrow><mrow><mi>j<\/mi><mo class=\"MathClass-bin\">+<\/mo><mn>1<\/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> <msub><mrow><mi>A<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><msub><mrow><mi>b<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub> <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><mo class=\"MathClass-bin\">\u2212<\/mo><mn>1<\/mn><\/mrow><\/munderover><msub><mrow><mi>A<\/mi><\/mrow><mrow> <mi>k<\/mi><\/mrow><\/msub><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>b<\/mi><\/mrow><mrow><mi>k<\/mi><\/mrow><\/msub> <mo class=\"MathClass-bin\">\u2212<\/mo> <msub><mrow><mi>b<\/mi><\/mrow><mrow><mi>k<\/mi><mo class=\"MathClass-bin\">+<\/mo><mn>1<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-punc\">,<\/mo><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\" \/> <mtd class=\"align-label\"> <mspace width=\"2em\" \/><\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\"><span class=\"ecti-1095\">wobei wir die Linearit<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">t der Summe, die Rekursionsformel der Summe und Indexverschiebung<\/span> <span class=\"ecti-1095\">verwendet haben.<\/span> <\/p><p class=\"indent\"><span class=\"ecti-1095\">In dem Spezialfall <\/span><math display=\"inline\"><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>k<\/mi><\/mrow><\/msub> <mo class=\"MathClass-rel\">=<\/mo> <msup><mrow><mo class=\"MathClass-open\">(<\/mo><mo class=\"MathClass-bin\">\u2212<\/mo><mn>1<\/mn><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mi>k<\/mi><\/mrow><\/msup><\/math> <span class=\"ecti-1095\">f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r <\/span><math display=\"inline\"><mn>1<\/mn> <mo class=\"MathClass-rel\">\u2264<\/mo> <mi>k<\/mi> <mo class=\"MathClass-rel\">\u2264<\/mo> <mn>2<\/mn><mi>n<\/mi><\/math> <span class=\"ecti-1095\">ergibt sich (mittels<\/span> <span class=\"ecti-1095\">vollst<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">ndiger Induktion), dass <\/span><math display=\"inline\"><msub><mrow><mi>A<\/mi><\/mrow><mrow><mi>k<\/mi><\/mrow><\/msub> <mo class=\"MathClass-rel\">=<\/mo> <mn>0<\/mn><\/math> <span class=\"ecti-1095\">f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r alle geraden <\/span><span class=\"maperiod\"><math display=\"inline\"><mi>k<\/mi> <mo class=\"MathClass-rel\">\u2264<\/mo> <mn>2<\/mn><mi>n<\/mi><\/math><\/span><span class=\"period\">,<\/span> <span class=\"ecti-1095\">insbesondere <\/span><math display=\"inline\"><msub><mrow><mi>A<\/mi><\/mrow><mrow><mn>2<\/mn><mi>n<\/mi><\/mrow><\/msub> <mo class=\"MathClass-rel\">=<\/mo> <mn>0<\/mn><\/math><span class=\"ecti-1095\">, und<\/span> <math display=\"inline\"><msub><mrow><mi>A<\/mi><\/mrow><mrow><mi>k<\/mi> <\/mrow> <\/msub> <mo class=\"MathClass-rel\">=<\/mo> <mo class=\"MathClass-bin\">\u2212<\/mo><mn>1<\/mn><\/math> <span class=\"ecti-1095\">f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r alle ungeraden<\/span> <math display=\"inline\"><mi>k<\/mi> <mo class=\"MathClass-rel\">\u2264<\/mo> <mn>2<\/mn><mi>n<\/mi><\/math><span class=\"ecti-1095\">. Da die ungeraden<\/span> <span class=\"ecti-1095\">ganzen Zahlen <\/span><math display=\"inline\"><mi>k<\/mi><\/math> <span class=\"ecti-1095\">mit <\/span><math display=\"inline\"><mn>1<\/mn> <mo class=\"MathClass-rel\">\u2264<\/mo> <mi>k<\/mi> <mo class=\"MathClass-rel\">\u2264<\/mo> <mn>2<\/mn><mi>n<\/mi><\/math> <span class=\"ecti-1095\">genau<\/span> <span class=\"ecti-1095\">die Form <\/span><math display=\"inline\"><mi>k<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mn>2<\/mn><mi>\u2113<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <mn>1<\/mn><\/math> <span class=\"ecti-1095\">f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r <\/span><math display=\"inline\"><mi>\u2113<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mn>1<\/mn><mo class=\"MathClass-punc\">,<\/mo> <mi class=\"MathClass-op\">\u2026<\/mi><mo> <\/mo><mo class=\"MathClass-punc\">,<\/mo><mi>n<\/mi><\/math> <span class=\"ecti-1095\">haben, erhalten wir daraus<\/span> <\/p><table id=\"z3a544a809f19\" class=\"equation-star\"><tr><td> <math class=\"equation\" display=\"block\"> <munderover accent=\"false\" accentunder=\"false\"><mrow><mo>\u2211<\/mo> <\/mrow><mrow><mi>k<\/mi><mo class=\"MathClass-rel\">=<\/mo><mn>1<\/mn><\/mrow><mrow><mn>2<\/mn><mi>n<\/mi><\/mrow><\/munderover><msub><mrow><mi>b<\/mi><\/mrow><mrow> <mi>k<\/mi><\/mrow><\/msub> <mo class=\"MathClass-rel\">=<\/mo> <mo class=\"MathClass-bin\">\u2212<\/mo><munderover accent=\"false\" accentunder=\"false\"><mrow><mo>\u2211<\/mo> <\/mrow><mrow><mi>\u2113<\/mi><mo class=\"MathClass-rel\">=<\/mo><mn>1<\/mn><\/mrow><mrow><mi>n<\/mi><\/mrow><\/munderover> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><msub><mrow><mi>b<\/mi><\/mrow><mrow> <mn>2<\/mn><mi>\u2113<\/mi><mo class=\"MathClass-bin\">\u2212<\/mo><mn>1<\/mn><\/mrow><\/msub> <mo class=\"MathClass-bin\">\u2212<\/mo> <msub><mrow><mi>b<\/mi><\/mrow><mrow><mn>2<\/mn><mi>\u2113<\/mi><\/mrow><\/msub><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-punc\">.<\/mo> <\/math><\/td><\/tr><\/table> <p class=\"indent\"><span class=\"ecti-1095\">Also entspricht die Abel-Summation in diesem Fall einer Zusammenfassung von jeweils zwei benachbarten<\/span> <span class=\"ecti-1095\">Summanden. Falls <\/span><math display=\"inline\"><msub><mrow><mi>b<\/mi><\/mrow><mrow><mi>k<\/mi><\/mrow><\/msub> <mo class=\"MathClass-rel\">=<\/mo> <mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mi>k<\/mi><\/mrow><\/mfrac><\/math> <span class=\"ecti-1095\">f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r <\/span><math display=\"inline\"><mn>1<\/mn> <mo class=\"MathClass-rel\">\u2264<\/mo> <mi>k<\/mi> <mo class=\"MathClass-rel\">\u2264<\/mo> <mn>2<\/mn><mi>n<\/mi><\/math><span class=\"ecti-1095\">, k<\/span><span class=\"ecti-1095\">\u00f6<\/span><span class=\"ecti-1095\">nnen<\/span> <span class=\"ecti-1095\">wir noch <\/span><math display=\"inline\"><msub><mrow><mi>b<\/mi><\/mrow><mrow><mn>2<\/mn><mi>\u2113<\/mi><mo class=\"MathClass-bin\">\u2212<\/mo><mn>1<\/mn><\/mrow><\/msub> <mo class=\"MathClass-bin\">\u2212<\/mo> <msub><mrow><mi>b<\/mi><\/mrow><mrow><mn>2<\/mn><mi>\u2113<\/mi><\/mrow><\/msub> <mo class=\"MathClass-rel\">=<\/mo> <mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mn>2<\/mn><mi>\u2113<\/mi><mo class=\"MathClass-open\">(<\/mo><mn>2<\/mn><mi>\u2113<\/mi><mo class=\"MathClass-bin\">\u2212<\/mo><mn>1<\/mn><mo class=\"MathClass-close\">)<\/mo><\/mrow><\/mfrac><\/math> <span class=\"ecti-1095\">in diese Formel einsetzen.<\/span><\/p><\/details>  <\/div> <p class=\"indent\">Anstelle der Dreiecksungleichung werden wir oft auch folgende verallgemeinerte Dreiecksungleichung f\u00fcr Summen verwenden. <\/p> <div class=\"me melemma\"> <p class=\"indent\"><\/p><h4 id=\"z095794f0b523\"> <a id=\"x1-78002r4\"><\/a> <span class=\"ecbx-1095\">Wichtige <\/span><span class=\"ecbx-1095\">\u00dc<\/span><span class=\"ecbx-1095\">bung 3.4 <\/span>(Verallgemeinerte Dreiecksungleichung)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Zeigen Sie, dass f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r alle Zahlen <\/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>\u2102<\/mi><\/math> <span class=\"ecti-1095\">die Ungleichung<\/span> <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mstyle><mrow><mo fence=\"true\" form=\"prefix\"> |<\/mo><mrow \/><mo fence=\"true\" form=\"postfix\" \/><\/mrow><\/mstyle><munderover accent=\"false\" accentunder=\"false\"><mrow><mo>\u2211<\/mo> <\/mrow><mrow><mi>i<\/mi><mo class=\"MathClass-rel\">=<\/mo><mn>1<\/mn><\/mrow><mrow><mi>n<\/mi><\/mrow><\/munderover><msub><mrow><mi>a<\/mi><\/mrow><mrow> <mi>i<\/mi><\/mrow><\/msub><mstyle><mrow><mo fence=\"true\" form=\"prefix\"> |<\/mo><mrow \/><mo fence=\"true\" form=\"postfix\" \/><\/mrow><\/mstyle> <mo class=\"MathClass-rel\">\u2264<\/mo><munderover accent=\"false\" accentunder=\"false\"><mrow><mo>\u2211<\/mo> <\/mrow><mrow><mi>i<\/mi><mo class=\"MathClass-rel\">=<\/mo><mn>1<\/mn><\/mrow><mrow><mi>n<\/mi><\/mrow><\/munderover><mo class=\"MathClass-rel\">|<\/mo><msub><mrow><mi>a<\/mi><\/mrow><mrow> <mi>i<\/mi><\/mrow><\/msub><mo class=\"MathClass-rel\">|<\/mo><mo class=\"MathClass-punc\">.<\/mo><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\"><span class=\"ecti-1095\">gilt.<\/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\">Wenden Sie Induktion nach <\/span><math display=\"inline\"><mi>n<\/mi><\/math> <span class=\"ecti-1095\">an und verwenden Sie die regul<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">re Dreiecksungleichung und die rekursive Definition der Summe f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r<\/span> <span class=\"ecti-1095\">den Induktionsschritt.<\/span><\/p><\/details>  <\/div> <p class=\"indent\">Manchmal wollen wir in einer Summe einen gewissen Summanden getrennt betrachten und dazu die Summe aufteilen. Dies kann dann zum Beispiel f\u00fcr&nbsp;<math display=\"inline\"><mn>1<\/mn> <mo class=\"MathClass-rel\">\u2264<\/mo> <mi>k<\/mi> <mo class=\"MathClass-rel\">\u2264<\/mo> <mi>n<\/mi><\/math> die Form <\/p> <table id=\"z7ec56ba905ce\" class=\"equation-star\"><tr><td> <math class=\"equation\" display=\"block\"> <munderover accent=\"false\" accentunder=\"false\"><mrow><mo>\u2211<\/mo> <\/mrow><mrow><mi>j<\/mi><mo class=\"MathClass-rel\">=<\/mo><mn>1<\/mn><\/mrow><mrow><mi>n<\/mi><\/mrow><\/munderover><msub><mrow><mi>a<\/mi><\/mrow><mrow> <mi>j<\/mi><\/mrow><\/msub> <mo class=\"MathClass-rel\">=<\/mo><munderover accent=\"false\" accentunder=\"false\"><mrow><mo> \u2211<\/mo> <\/mrow><mrow><mi>j<\/mi><mo class=\"MathClass-rel\">=<\/mo><mn>1<\/mn><\/mrow><mrow><mi>k<\/mi><\/mrow><\/munderover><msub><mrow><mi>a<\/mi><\/mrow><mrow> <mi>j<\/mi><\/mrow><\/msub> <mo class=\"MathClass-bin\">+<\/mo><munderover accent=\"false\" accentunder=\"false\"><mrow><mo> \u2211<\/mo> <\/mrow><mrow><mi>j<\/mi><mo class=\"MathClass-rel\">=<\/mo><mi>k<\/mi><mo class=\"MathClass-bin\">+<\/mo><mn>1<\/mn><\/mrow><mrow><mi>n<\/mi><\/mrow><\/munderover><msub><mrow><mi>a<\/mi><\/mrow><mrow> <mi>j<\/mi><\/mrow><\/msub> <mo class=\"MathClass-rel\">=<\/mo><munderover accent=\"false\" accentunder=\"false\"><mrow><mo> \u2211<\/mo> <\/mrow><mrow><mi>j<\/mi><mo class=\"MathClass-rel\">=<\/mo><mn>1<\/mn><\/mrow><mrow><mi>k<\/mi><mo class=\"MathClass-bin\">\u2212<\/mo><mn>1<\/mn><\/mrow><\/munderover><msub><mrow><mi>a<\/mi><\/mrow><mrow> <mi>j<\/mi><\/mrow><\/msub> <mo class=\"MathClass-bin\">+<\/mo> <msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>k<\/mi><\/mrow><\/msub> <mo class=\"MathClass-bin\">+<\/mo><munderover accent=\"false\" accentunder=\"false\"><mrow><mo> \u2211<\/mo> <\/mrow><mrow><mi>j<\/mi><mo class=\"MathClass-rel\">=<\/mo><mi>k<\/mi><mo class=\"MathClass-bin\">+<\/mo><mn>1<\/mn><\/mrow><mrow><mi>n<\/mi><\/mrow><\/munderover><msub><mrow><mi>a<\/mi><\/mrow><mrow> <mi>j<\/mi><\/mrow><\/msub> <\/math><\/td><\/tr><\/table> <p class=\"indent\">annehmen. In dem Spezialfall&nbsp;<math display=\"inline\"><mi>k<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mn>1<\/mn><\/math> sollte dies aber mit&nbsp;<math display=\"inline\"><msub><mrow><mi>a<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub> <mo class=\"MathClass-bin\">+<\/mo><msubsup><mrow><mi class=\"MathClass-op\"> \u2211<\/mi><mo> <\/mo> <\/mrow><mrow><mi>j<\/mi><mo class=\"MathClass-rel\">=<\/mo><mn>2<\/mn><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msubsup><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>j<\/mi><\/mrow><\/msub><\/math> und in dem Spezialfall&nbsp;<math display=\"inline\"><mi>k<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mi>n<\/mi><\/math> mit&nbsp;<math display=\"inline\"><msubsup><mrow><mi class=\"MathClass-op\"> \u2211<\/mi><mo> <\/mo> <\/mrow><mrow><mi>j<\/mi><mo class=\"MathClass-rel\">=<\/mo><mn>1<\/mn><\/mrow><mrow><mi>n<\/mi><mo class=\"MathClass-bin\">\u2212<\/mo><mn>1<\/mn><\/mrow><\/msubsup><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>j<\/mi><\/mrow><\/msub> <mo class=\"MathClass-bin\">+<\/mo> <msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><\/math> \u00fcbereinstimmen, was auf Grund unserer Definitionen&nbsp;<math display=\"inline\"><msubsup><mrow><mi class=\"MathClass-op\"> \u2211<\/mi><mo> <\/mo> <\/mrow><mrow><mi>j<\/mi><mo class=\"MathClass-rel\">=<\/mo><mn>1<\/mn><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msubsup><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>j<\/mi><\/mrow><\/msub> <mo class=\"MathClass-rel\">=<\/mo> <mn>0<\/mn><\/math> und&nbsp;<math display=\"inline\"><msubsup><mrow><mi class=\"MathClass-op\"> \u2211<\/mi><mo> <\/mo> <\/mrow><mrow><mi>j<\/mi><mo class=\"MathClass-rel\">=<\/mo><mi>n<\/mi><mo class=\"MathClass-bin\">+<\/mo><mn>1<\/mn><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msubsup><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>j<\/mi><\/mrow><\/msub> <mo class=\"MathClass-rel\">=<\/mo> <mn>0<\/mn><\/math> in der Tat gilt. (Der formale Beweis erfolgt wiederum mit Induktion nach <span class=\"maperiod\"><math display=\"inline\"><mi>n<\/mi> <mo class=\"MathClass-rel\">\u2265<\/mo> <mi>k<\/mi><\/math><\/span><span class=\"period\">.<\/span>) <a id=\"x1-78003r78\"><\/a> <\/p> <h4 id=\"z8611c8aa8059\" class=\"subsectionHead\"><span class=\"titlemark\">3.1.2 <\/span> <a id=\"x1-790002\"><\/a>Rechenregeln f\u00fcr das Produkt<\/h4> <p class=\"noindent\">F\u00fcr ganze Zahlen <math display=\"inline\"><mi>m<\/mi> <mo class=\"MathClass-rel\">\u2264<\/mo> <mi>n<\/mi><\/math> und <math display=\"inline\"><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>m<\/mi> <\/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-punc\">,<\/mo><msub><mrow><mi>b<\/mi><\/mrow><mrow><mi>m<\/mi><\/mrow><\/msub><mo class=\"MathClass-punc\">,<\/mo><mi class=\"MathClass-op\">\u2026<\/mi><mo> <\/mo><mo class=\"MathClass-punc\">,<\/mo><msub><mrow><mi>b<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2102<\/mi><\/math> gilt                                                                                                                                                                           <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><munderover accent=\"false\" accentunder=\"false\"><mrow><mo> \u220f<\/mo> <\/mrow><mrow><mi>k<\/mi><mo class=\"MathClass-rel\">=<\/mo><mi>m<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/munderover><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>a<\/mi><\/mrow><mrow> <mi>k<\/mi><\/mrow><\/msub><msub><mrow><mi>b<\/mi><\/mrow><mrow><mi>k<\/mi><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <mstyle><mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow \/><mo fence=\"true\" form=\"postfix\" \/><\/mrow><\/mstyle><munderover accent=\"false\" accentunder=\"false\"><mrow><mo>\u220f<\/mo> <\/mrow><mrow><mi>k<\/mi><mo class=\"MathClass-rel\">=<\/mo><mi>m<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/munderover><msub><mrow><mi>a<\/mi><\/mrow><mrow> <mi>k<\/mi><\/mrow><\/msub><mstyle><mrow><mo fence=\"true\" form=\"prefix\"> )<\/mo><mrow \/><mo fence=\"true\" form=\"postfix\" \/><\/mrow><\/mstyle><mstyle><mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow \/><mo fence=\"true\" form=\"postfix\" \/><\/mrow><\/mstyle><munderover accent=\"false\" accentunder=\"false\"><mrow><mo> \u220f<\/mo> <\/mrow><mrow><mi>k<\/mi><mo class=\"MathClass-rel\">=<\/mo><mi>m<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/munderover><msub><mrow><mi>b<\/mi><\/mrow><mrow> <mi>k<\/mi><\/mrow><\/msub><mstyle><mrow><mo fence=\"true\" form=\"prefix\"> )<\/mo><mrow \/><mo fence=\"true\" form=\"postfix\" \/><\/mrow><\/mstyle><mo class=\"MathClass-punc\">.<\/mo><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">Insbesondere ist f\u00fcr alle <math display=\"inline\"><mi>c<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2102<\/mi><\/math> <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><munderover accent=\"false\" accentunder=\"false\"><mrow><mo> \u220f<\/mo> <\/mrow><mrow><mi>k<\/mi><mo class=\"MathClass-rel\">=<\/mo><mi>m<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/munderover><mo class=\"MathClass-open\">(<\/mo><mi>c<\/mi><msub><mrow><mi>a<\/mi><\/mrow><mrow> <mi>k<\/mi><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <msup><mrow><mi>c<\/mi><\/mrow><mrow><mi>n<\/mi><mo class=\"MathClass-bin\">\u2212<\/mo><mi>m<\/mi><mo class=\"MathClass-bin\">+<\/mo><mn>1<\/mn><\/mrow><\/msup><mstyle><mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow \/><mo fence=\"true\" form=\"postfix\" \/><\/mrow><\/mstyle><munderover accent=\"false\" accentunder=\"false\"><mrow><mo> \u220f<\/mo> <\/mrow><mrow><mi>k<\/mi><mo class=\"MathClass-rel\">=<\/mo><mi>m<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/munderover><msub><mrow><mi>a<\/mi><\/mrow><mrow> <mi>k<\/mi><\/mrow><\/msub><mstyle><mrow><mo fence=\"true\" form=\"prefix\"> )<\/mo><mrow \/><mo fence=\"true\" form=\"postfix\" \/><\/mrow><\/mstyle><mo class=\"MathClass-punc\">.<\/mo><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">Des Weiteren gilt f\u00fcr alle <math display=\"inline\"><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>m<\/mi><\/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>\u2102<\/mi> <mo class=\"MathClass-bin\">\u2216<\/mo><mo class=\"MathClass-open\">{<\/mo><mn>0<\/mn><mo class=\"MathClass-close\">}<\/mo><\/math> die Formel f\u00fcr das <span class=\"ecbx-1095\">Teleskopprodukt<\/span> <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><munderover accent=\"false\" accentunder=\"false\"><mrow><mo>\u220f<\/mo> <\/mrow><mrow><mi>k<\/mi><mo class=\"MathClass-rel\">=<\/mo><mi>m<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/munderover><mfrac><mrow><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>k<\/mi><mo class=\"MathClass-bin\">+<\/mo><mn>1<\/mn><\/mrow><\/msub><\/mrow> <mrow><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>k<\/mi><\/mrow><\/msub><\/mrow><\/mfrac> <\/mtd> <mtd class=\"align-even\"> <mo class=\"MathClass-rel\">=<\/mo> <mstyle><mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow \/><mo fence=\"true\" form=\"postfix\" \/><\/mrow><\/mstyle><munderover accent=\"false\" accentunder=\"false\"><mrow><mo>\u220f<\/mo> <\/mrow><mrow><mi>k<\/mi><mo class=\"MathClass-rel\">=<\/mo><mi>m<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/munderover><msub><mrow><mi>a<\/mi><\/mrow><mrow> <mi>k<\/mi><mo class=\"MathClass-bin\">+<\/mo><mn>1<\/mn><\/mrow><\/msub><mstyle><mrow><mo fence=\"true\" form=\"prefix\"> )<\/mo><mrow \/><mo fence=\"true\" form=\"postfix\" \/><\/mrow><\/mstyle><mstyle><mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow \/><mo fence=\"true\" form=\"postfix\" \/><\/mrow><\/mstyle><munderover accent=\"false\" accentunder=\"false\"><mrow><mo> \u220f<\/mo> <\/mrow><mrow><mi>k<\/mi><mo class=\"MathClass-rel\">=<\/mo><mi>m<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/munderover> <mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>k<\/mi><\/mrow><\/msub><\/mrow><\/mfrac><mstyle><mrow><mo fence=\"true\" form=\"prefix\"> )<\/mo><mrow \/><mo fence=\"true\" form=\"postfix\" \/><\/mrow><\/mstyle> <mo class=\"MathClass-rel\">=<\/mo> <mstyle><mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow \/><mo fence=\"true\" form=\"postfix\" \/><\/mrow><\/mstyle><munderover accent=\"false\" accentunder=\"false\"><mrow><mo>\u220f<\/mo> <\/mrow><mrow><mi>k<\/mi><mo class=\"MathClass-rel\">=<\/mo><mi>m<\/mi><mo class=\"MathClass-bin\">+<\/mo><mn>1<\/mn><\/mrow><mrow><mi>n<\/mi><mo class=\"MathClass-bin\">+<\/mo><mn>1<\/mn><\/mrow><\/munderover><msub><mrow><mi>a<\/mi><\/mrow><mrow> <mi>k<\/mi><\/mrow><\/msub><mstyle><mrow><mo fence=\"true\" form=\"prefix\"> )<\/mo><mrow \/><mo fence=\"true\" form=\"postfix\" \/><\/mrow><\/mstyle><mstyle><mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow \/><mo fence=\"true\" form=\"postfix\" \/><\/mrow><\/mstyle><munderover accent=\"false\" accentunder=\"false\"><mrow><mo> \u220f<\/mo> <\/mrow><mrow><mi>k<\/mi><mo class=\"MathClass-rel\">=<\/mo><mi>m<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/munderover> <mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>k<\/mi><\/mrow><\/msub><\/mrow><\/mfrac><mstyle><mrow><mo fence=\"true\" form=\"prefix\"> )<\/mo><mrow \/><mo fence=\"true\" form=\"postfix\" \/><\/mrow><\/mstyle><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\" \/> <mtd class=\"align-label\"> <mspace width=\"2em\" \/><\/mtd><\/mtr><mtr><mtd class=\"align-odd\" columnalign=\"right\" \/> <mtd class=\"align-even\"> <mo class=\"MathClass-rel\">=<\/mo> <msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi><mo class=\"MathClass-bin\">+<\/mo><mn>1<\/mn><\/mrow><\/msub><mstyle><mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow \/><mo fence=\"true\" form=\"postfix\" \/><\/mrow><\/mstyle><munderover accent=\"false\" accentunder=\"false\"><mrow><mo> \u220f<\/mo> <\/mrow><mrow><mi>k<\/mi><mo class=\"MathClass-rel\">=<\/mo><mi>m<\/mi><mo class=\"MathClass-bin\">+<\/mo><mn>1<\/mn><\/mrow><mrow><mi>n<\/mi><\/mrow><\/munderover><msub><mrow><mi>a<\/mi><\/mrow><mrow> <mi>k<\/mi><\/mrow><\/msub><mstyle><mrow><mo fence=\"true\" form=\"prefix\"> )<\/mo><mrow \/><mo fence=\"true\" form=\"postfix\" \/><\/mrow><\/mstyle><mstyle><mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow \/><mo fence=\"true\" form=\"postfix\" \/><\/mrow><\/mstyle><munderover accent=\"false\" accentunder=\"false\"><mrow><mo> \u220f<\/mo> <\/mrow><mrow><mi>k<\/mi><mo class=\"MathClass-rel\">=<\/mo><mi>m<\/mi><mo class=\"MathClass-bin\">+<\/mo><mn>1<\/mn><\/mrow><mrow><mi>n<\/mi><\/mrow><\/munderover> <mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>k<\/mi><\/mrow><\/msub><\/mrow><\/mfrac><mstyle><mrow><mo fence=\"true\" form=\"prefix\"> )<\/mo><mrow \/><mo fence=\"true\" form=\"postfix\" \/><\/mrow><\/mstyle> <mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>m<\/mi><\/mrow><\/msub><\/mrow><\/mfrac> <mo class=\"MathClass-rel\">=<\/mo> <mfrac><mrow><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi><mo class=\"MathClass-bin\">+<\/mo><mn>1<\/mn><\/mrow><\/msub><\/mrow> <mrow><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>m<\/mi><\/mrow><\/msub><\/mrow><\/mfrac> <mo class=\"MathClass-punc\">.<\/mo><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\" \/> <mtd class=\"align-label\"> <mspace width=\"2em\" \/><\/mtd><\/mtr><\/mtable><\/math> <div class=\"me melemma\"> <p class=\"indent\"><\/p><h4 id=\"z5f5d8aa5e667\"> <a id=\"x1-79001r5\"><\/a> <span class=\"ecbx-1095\">Lemma 3.5 <\/span>(Bernoulli\u2019sche Ungleichung)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">F<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r alle reellen Zahlen <\/span><math display=\"inline\"><mi>a<\/mi> <mo class=\"MathClass-rel\">\u2265<\/mo><mo class=\"MathClass-bin\">\u2212<\/mo><mn>1<\/mn><\/math> <span class=\"ecti-1095\">und <\/span><math display=\"inline\"><mi>n<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <msub><mrow><mi>\u2115<\/mi><\/mrow><mrow><mn>0<\/mn> <\/mrow> <\/msub> <\/math> <span class=\"ecti-1095\">gilt <\/span><span class=\"maperiod\"><math display=\"inline\"><msup><mrow><mo class=\"MathClass-open\">(<\/mo><mn>1<\/mn> <mo class=\"MathClass-bin\">+<\/mo> <mi>a<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mi>n<\/mi> <\/mrow> <\/msup> <mo class=\"MathClass-rel\">\u2265<\/mo> <mn>1<\/mn> <mo class=\"MathClass-bin\">+<\/mo> <mi>n<\/mi><mi>a<\/mi><\/math><\/span><span class=\"period\">.<\/span> <\/p> <\/div> <p class=\"indent\"> <\/p> <div class=\"proof\"> <p class=\"indent\"><span class=\"head\"><\/span><\/p><details open><summary><b>Beweis.<\/b><\/summary><p class=\"indent\" style=\"margin-top: 10\">Wir verwenden vollst\u00e4ndige Induktion. F\u00fcr <math display=\"inline\"><mi>n<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mn>0<\/mn><\/math> haben wir <span class=\"maperiod\"><math display=\"inline\"><msup><mrow><mo class=\"MathClass-open\">(<\/mo><mn>1<\/mn> <mo class=\"MathClass-bin\">+<\/mo> <mi>a<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mi>n<\/mi> <\/mrow> <\/msup> <mo class=\"MathClass-rel\">=<\/mo> <mn>1<\/mn> <mo class=\"MathClass-rel\">=<\/mo> <mn>1<\/mn> <mo class=\"MathClass-bin\">+<\/mo> <mi>n<\/mi><mi>a<\/mi><\/math><\/span><span class=\"period\">.<\/span> Angenommen die Ungleichung <math display=\"inline\"><msup><mrow><mo class=\"MathClass-open\">(<\/mo><mn>1<\/mn> <mo class=\"MathClass-bin\">+<\/mo> <mi>a<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msup> <mo class=\"MathClass-rel\">\u2265<\/mo> <mn>1<\/mn> <mo class=\"MathClass-bin\">+<\/mo> <mi>n<\/mi><mi>a<\/mi><\/math> gilt f\u00fcr ein <span class=\"maperiod\"><math display=\"inline\"><mi>n<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <msub><mrow><mi>\u2115<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><\/math><\/span><span class=\"period\">.<\/span> Nach Annahme an <math display=\"inline\"><mi>a<\/mi><\/math> ist <span class=\"maperiod\"><math display=\"inline\"><mi>a<\/mi> <mo class=\"MathClass-rel\">\u2265<\/mo> <mo class=\"MathClass-bin\">\u2212<\/mo><mn>1<\/mn><\/math><\/span><span class=\"period\">,<\/span> was in Kombination mit der Annahme an <math display=\"inline\"><mi>n<\/mi><\/math> <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><msup><mrow><mo class=\"MathClass-open\">(<\/mo><mn>1<\/mn> <mo class=\"MathClass-bin\">+<\/mo> <mi>a<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mi>n<\/mi><mo class=\"MathClass-bin\">+<\/mo><mn>1<\/mn><\/mrow><\/msup><\/mtd> <mtd class=\"align-even\"> <mo class=\"MathClass-rel\">=<\/mo> <msup><mrow><mo class=\"MathClass-open\">(<\/mo><mn>1<\/mn> <mo class=\"MathClass-bin\">+<\/mo> <mi>a<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msup><mo class=\"MathClass-open\">(<\/mo><mn>1<\/mn> <mo class=\"MathClass-bin\">+<\/mo> <mi>a<\/mi><mo class=\"MathClass-close\">)<\/mo><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\" \/> <mtd class=\"align-label\"> <mspace width=\"2em\" \/><\/mtd><\/mtr><mtr><mtd class=\"align-odd\" columnalign=\"right\" \/> <mtd class=\"align-even\"> <mo class=\"MathClass-rel\">\u2265<\/mo> <mo class=\"MathClass-open\">(<\/mo><mn>1<\/mn> <mo class=\"MathClass-bin\">+<\/mo> <mi>n<\/mi><mi>a<\/mi><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-open\">(<\/mo><mn>1<\/mn> <mo class=\"MathClass-bin\">+<\/mo> <mi>a<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <mn>1<\/mn> <mo class=\"MathClass-bin\">+<\/mo> <mi>n<\/mi><mi>a<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mi>a<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mi>n<\/mi><msup><mrow><mi>a<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\" \/> <mtd class=\"align-label\"> <mspace width=\"2em\" \/><\/mtd><\/mtr><mtr><mtd class=\"align-odd\" columnalign=\"right\" \/> <mtd class=\"align-even\"> <mo class=\"MathClass-rel\">\u2265<\/mo> <mn>1<\/mn> <mo class=\"MathClass-bin\">+<\/mo> <mo class=\"MathClass-open\">(<\/mo><mi>n<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mn>1<\/mn><mo class=\"MathClass-close\">)<\/mo><mi>a<\/mi><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\" \/> <mtd class=\"align-label\"> <mspace width=\"2em\" \/><\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">ergibt und damit den Induktionsschritt zeigt. Das Lemma folgt. <span>&nbsp;&nbsp;<\/span><\/p><div class=\"qed\">\u25a0<\/div><\/details><\/div> <div class=\"me meexample\"> <p class=\"indent\"><\/p><h4 id=\"zc34830504c28\"> <a id=\"x1-79002r6\"><\/a> <span class=\"ecbx-1095\">\u00dc<\/span><span class=\"ecbx-1095\">bung 3.6 <\/span>(Archimedisches Prinzip f\u00fcr Potenzen)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Verwenden Sie die Bernoulli\u2019sche Ungleichung und das Archimedische Prinzip (Satz <\/span><a href=\"..\/..\/chapter\/erste-konsequenzen-der-vollstaendigkeit#x1-68001r68\"><span class=\"ecti-1095\">2.68<\/span><\/a><span class=\"ecti-1095\">),<\/span> <span class=\"ecti-1095\">um folgende Aussage zu beweisen. F<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r alle <\/span><math display=\"inline\"><mi>x<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>y<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u211d<\/mi><\/math> <span class=\"ecti-1095\">mit<\/span><span class=\"ecti-1095\">&nbsp;<\/span><math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">&gt;<\/mo> <mn>1<\/mn><\/math> <span class=\"ecti-1095\">existiert ein <\/span><span class=\"maperiod\"><math display=\"inline\"><mi>n<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <msub><mrow><mi>\u2115<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><\/math><\/span><span class=\"period\">,<\/span> <span class=\"ecti-1095\">so dass <\/span><span class=\"maperiod\"><math display=\"inline\"><msup><mrow><mi>x<\/mi><\/mrow><mrow><mi>n<\/mi> <\/mrow> <\/msup> <mo class=\"MathClass-rel\">\u2265<\/mo> <mi>y<\/mi><\/math><\/span><span class=\"period\">.<\/span> <\/p><details><summary style=\"color:#FF7F00\"><span class=\"ecti-1095\">L<\/span><span class=\"ecti-1095\">\u00f6<\/span><span class=\"ecti-1095\">sung.<\/span><\/summary><p class=\"indent\" style=\"margin-top: 0\"> <span class=\"ecti-1095\">Sei <\/span><span class=\"maperiod\"><math display=\"inline\"><mi>a<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mi>x<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <mn>1<\/mn> <mo class=\"MathClass-rel\">&gt;<\/mo> <mn>0<\/mn><\/math><\/span><span class=\"period\">.<\/span> <span class=\"ecti-1095\">Dann gilt <\/span><math display=\"inline\"><msup><mrow><mi>x<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msup> <mo class=\"MathClass-rel\">=<\/mo> <msup><mrow><mo class=\"MathClass-open\">(<\/mo><mn>1<\/mn> <mo class=\"MathClass-bin\">+<\/mo> <mi>a<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msup> <mo class=\"MathClass-rel\">\u2265<\/mo> <mi>n<\/mi><mi>a<\/mi><\/math> <span class=\"ecti-1095\">auf Grund der Bernoulli\u2019schen Ungleichung. Auf Grund des Archimedischen Prinzip hat <\/span><math display=\"inline\"><mi>\u2115<\/mi><\/math> <span class=\"ecti-1095\">keine obere Schranke in <\/span><span class=\"maperiod\"><math display=\"inline\"><mi>\u211d<\/mi><\/math><\/span><span class=\"period\">.<\/span> <span class=\"ecti-1095\">Daher und wegen <\/span><math display=\"inline\"><mi>a<\/mi> <mo class=\"MathClass-rel\">&gt;<\/mo> <mn>0<\/mn><\/math> <span class=\"ecti-1095\">existiert f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r jedes <\/span><math display=\"inline\"><mi>y<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u211d<\/mi><\/math> <span class=\"ecti-1095\">ein <\/span><math display=\"inline\"><mi>n<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2115<\/mi><\/math> <span class=\"ecti-1095\">mit <\/span><span class=\"maperiod\"><math display=\"inline\"><mi>n<\/mi><mi>a<\/mi> <mo class=\"MathClass-rel\">&gt;<\/mo> <mi>y<\/mi><\/math><\/span><span class=\"period\">.<\/span> <span class=\"ecti-1095\">Zusammen erhalten wir daraus <\/span><span class=\"maperiod\"><math display=\"inline\"><msup><mrow><mi>x<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msup> <mo class=\"MathClass-rel\">&gt;<\/mo> <mi>y<\/mi><\/math><\/span><span class=\"period\">.<\/span> <\/p><\/details>  <\/div> <div class=\"me meexample\"> <p class=\"indent\"><\/p><h4 id=\"z996788b02822\"> <a id=\"x1-79003r7\"><\/a> <span class=\"ecbx-1095\">\u00dc<\/span><span class=\"ecbx-1095\">bung 3.7 <\/span>(Zifferndarstellungen nat\u00fcrlicher Zahlen)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Sei <\/span><math display=\"inline\"><mi>q<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2115<\/mi><\/math> <span class=\"ecti-1095\">eine nat<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">rliche Zahl. Zeigen Sie, dass sich jede nat<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">rliche Zahl <\/span><math display=\"inline\"><mi>m<\/mi><\/math> <span class=\"ecti-1095\">als Summe der Form <\/span><math display=\"inline\"><mi>m<\/mi> <mo class=\"MathClass-rel\">=<\/mo><msubsup><mrow><mi class=\"MathClass-op\"> \u2211<\/mi><mo> <\/mo> <\/mrow><mrow><mi>k<\/mi><mo class=\"MathClass-rel\">=<\/mo><mn>0<\/mn><\/mrow><mrow><mi>\u2113<\/mi><\/mrow><\/msubsup><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>k<\/mi><\/mrow><\/msub><msup><mrow><mi>q<\/mi><\/mrow><mrow><mi>k<\/mi><\/mrow><\/msup><\/math> <span class=\"ecti-1095\">schreiben l<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">sst wobei <\/span><math display=\"inline\"><mi>\u2113<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <msub><mrow><mi>\u2115<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><\/math> <span class=\"ecti-1095\">und die Koeffizienten <\/span><span class=\"maperiod\"><math display=\"inline\"><msub><mrow><mi>a<\/mi><\/mrow><mrow><mn>0<\/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>\u2113<\/mi><\/mrow><\/msub> <mo class=\"MathClass-rel\">\u2208<\/mo> <msub><mrow><mi>\u2115<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub> <mo class=\"MathClass-bin\">\u2229<\/mo> <mo class=\"MathClass-open\">[<\/mo><mn>0<\/mn><mo class=\"MathClass-punc\">,<\/mo><mi>q<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <mn>1<\/mn><mo class=\"MathClass-close\">]<\/mo><\/math><\/span><span class=\"period\">.<\/span> <span class=\"ecti-1095\">Diese Aussage kennen Sie schon f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r<\/span><span class=\"ecti-1095\">&nbsp;<\/span><math display=\"inline\"><mi>q<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mn>1<\/mn><mn>0<\/mn><\/math> <span class=\"ecti-1095\">wegen der Dezimaldarstellung nat<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">rlicher Zahlen und vielleicht auch f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r<\/span><span class=\"ecti-1095\">&nbsp;<\/span><math display=\"inline\"><mi>q<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mn>2<\/mn><\/math> <span class=\"ecti-1095\">wegen der Bin<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">rdarstellung. F<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r ein allgemeines <\/span><math display=\"inline\"><mi>q<\/mi><\/math> <span class=\"ecti-1095\">spricht man auch von der <\/span><math display=\"inline\"><mi>q<\/mi><\/math><span class=\"ecti-1095\">-n<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">ren<\/span> <span class=\"ecti-1095\">Darstellung.<\/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\">Nach <\/span><span class=\"ecti-1095\">\u00dc<\/span><span class=\"ecti-1095\">bung <\/span><a href=\"..\/..\/chapter\/summen-und-produkte#x1-79002r6\"><span class=\"ecti-1095\">3.6<\/span><\/a> <span class=\"ecti-1095\">existiert f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r alle <\/span><math display=\"inline\"><mi>m<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2115<\/mi><\/math> <span class=\"ecti-1095\">ein <\/span><math display=\"inline\"><mi>\u2113<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2115<\/mi><\/math> <span class=\"ecti-1095\">mit <\/span><span class=\"maperiod\"><math display=\"inline\"><mi>m<\/mi> <mo class=\"MathClass-rel\">&lt;<\/mo> <msup><mrow><mi>q<\/mi><\/mrow><mrow><mi>\u2113<\/mi> <\/mrow> <\/msup> <\/math><\/span><span class=\"period\">.<\/span> <span class=\"ecti-1095\">Sei nun <\/span><math display=\"inline\"><mi>A<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>\u2113<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math> <span class=\"ecti-1095\">die Aussage, dass sich jede nat<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">rliche Zahl <\/span><math display=\"inline\"><mi>m<\/mi><\/math> <span class=\"ecti-1095\">mit <\/span><math display=\"inline\"><msup><mrow><mi>q<\/mi><\/mrow><mrow><mi>\u2113<\/mi> <\/mrow> <\/msup> <mo class=\"MathClass-rel\">\u2264<\/mo> <mi>m<\/mi> <mo class=\"MathClass-rel\">&lt;<\/mo> <msup><mrow><mi>q<\/mi><\/mrow><mrow><mi>\u2113<\/mi><mo class=\"MathClass-bin\">+<\/mo><mn>1<\/mn><\/mrow><\/msup><\/math> <span class=\"ecti-1095\">in der gew<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">nschten Form schreiben l<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">sst. Verwenden Sie vollst<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">ndige Induktion <\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">ber <\/span><math display=\"inline\"><mi>\u2113<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <msub><mrow><mi>\u2115<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><\/math> <span class=\"ecti-1095\">und Division mit Rest f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r den Induktionsschritt.<\/span><\/p><\/details>  <\/div> <a id=\"x1-79004r79\"><\/a> <h4 id=\"za8cda99b01e8\" class=\"subsectionHead\"><span class=\"titlemark\">3.1.3 <\/span> <a id=\"x1-800003\"><\/a>Die geometrische Summe<\/h4> <p class=\"noindent\">In diesem kurzen Abschnitt m\u00f6chten wir folgende, vermutlich schon bekannte und f\u00fcr uns sp\u00e4ter sehr wichtige Formel beweisen. <\/p> <div class=\"me metheorem\"> <p class=\"indent\"><\/p><h4 id=\"z49fdf5cec16f\"> <a id=\"x1-80001r8\"><\/a> <span class=\"ecbx-1095\">Proposition 3.8 <\/span>(Geometrische Summenformel)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Sei <\/span><math display=\"inline\"><mi>n<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <msub><mrow><mi>\u2115<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><\/math> <span class=\"ecti-1095\">und <\/span><span class=\"maperiod\"><math display=\"inline\"><mi>q<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2102<\/mi><\/math><\/span><span class=\"period\">.<\/span> <span class=\"ecti-1095\">Dann gilt<\/span> <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><munderover accent=\"false\" accentunder=\"false\"><mrow><mo>\u2211<\/mo> <\/mrow><mrow><mi>k<\/mi><mo class=\"MathClass-rel\">=<\/mo><mn>0<\/mn><\/mrow><mrow><mi>n<\/mi><\/mrow><\/munderover><msup><mrow><mi>q<\/mi><\/mrow><mrow><mi>k<\/mi><\/mrow><\/msup> <mo class=\"MathClass-rel\">=<\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> {<\/mo><mrow> <mtable align=\"axis\" class=\"array\" columnlines=\"none\" equalcolumns=\"false\" equalrows=\"false\"> <mtr><mtd class=\"array\" columnalign=\"center\"> <mi>n<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mn>1<\/mn> <\/mtd><mtd class=\"array\" columnalign=\"center\"><mstyle class=\"text\"><mtext>falls&nbsp;<\/mtext><\/mstyle><mi>q<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mn>1<\/mn><\/mtd> <\/mtr> <mtr><mtd class=\"array\" columnalign=\"center\"><mfrac><mrow><msup><mrow><mi>q<\/mi><\/mrow><mrow><mi>n<\/mi><mo class=\"MathClass-bin\">+<\/mo><mn>1<\/mn><\/mrow><\/msup><mo class=\"MathClass-bin\">\u2212<\/mo><mn>1<\/mn><\/mrow> <mrow><mi>q<\/mi><mo class=\"MathClass-bin\">\u2212<\/mo><mn>1<\/mn><\/mrow><\/mfrac> <\/mtd><mtd class=\"array\" columnalign=\"center\"> <mstyle class=\"text\"><mtext>falls&nbsp;<\/mtext><\/mstyle><mi>q<\/mi><mo class=\"MathClass-rel\">\u2260<\/mo><mn>1<\/mn> <\/mtd><\/mtr> <\/mtable> <\/mrow><mo fence=\"true\" form=\"postfix\" \/><\/mrow><mo class=\"MathClass-punc\">.<\/mo><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <\/div> <p class=\"indent\">Der direkte (aber sicher nicht eleganteste) Beweis verwendet vollst\u00e4ndige Induktion: <\/p><p class=\"indent\"> <\/p> <div class=\"proof\"> <p class=\"indent\"><span class=\"head\"><\/span><\/p><details open><summary><b>Beweis.<\/b><\/summary><p class=\"indent\" style=\"margin-top: 10\">F\u00fcr&nbsp;<math display=\"inline\"><mi>q<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mn>1<\/mn><\/math> ist&nbsp;<math display=\"inline\"><msup><mrow><mi>q<\/mi><\/mrow><mrow><mi>k<\/mi> <\/mrow> <\/msup> <mo class=\"MathClass-rel\">=<\/mo> <mn>1<\/mn><\/math> f\u00fcr alle&nbsp;<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> und die Aussage folgt aus den Eigenschaften der Summe. Sei nun&nbsp;<span class=\"maperiod\"><math display=\"inline\"><mi>q<\/mi><mo class=\"MathClass-rel\">\u2260<\/mo> <mn>1<\/mn><\/math><\/span><span class=\"period\">.<\/span> F\u00fcr <math display=\"inline\"><mi>n<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mn>0<\/mn><\/math> gilt <span class=\"maperiod\"><math display=\"inline\"><msubsup><mrow><mi class=\"MathClass-op\"> \u2211<\/mi><mo> <\/mo> <\/mrow><mrow><mi>k<\/mi><mo class=\"MathClass-rel\">=<\/mo><mn>0<\/mn><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msubsup><msup><mrow><mi>q<\/mi><\/mrow><mrow><mi>k<\/mi><\/mrow><\/msup> <mo class=\"MathClass-rel\">=<\/mo> <msup><mrow><mi>q<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msup> <mo class=\"MathClass-rel\">=<\/mo> <mn>1<\/mn> <mo class=\"MathClass-rel\">=<\/mo> <mfrac><mrow><mi>q<\/mi><mo class=\"MathClass-bin\">\u2212<\/mo><mn>1<\/mn><\/mrow> <mrow><mi>q<\/mi><mo class=\"MathClass-bin\">\u2212<\/mo><mn>1<\/mn><\/mrow><\/mfrac><\/math><\/span><span class=\"period\">,<\/span> was also den Induktionsanfang zeigt. Angenommen die Formel in der Proposition gilt bereits f\u00fcr                                                                                                                                                                           <span class=\"maperiod\"><math display=\"inline\"><mi>n<\/mi><\/math><\/span><span class=\"period\">.<\/span> Dann ist <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><munderover accent=\"false\" accentunder=\"false\"><mrow><mo> \u2211<\/mo> <\/mrow><mrow><mi>k<\/mi><mo class=\"MathClass-rel\">=<\/mo><mn>0<\/mn><\/mrow><mrow><mi>n<\/mi><mo class=\"MathClass-bin\">+<\/mo><mn>1<\/mn><\/mrow><\/munderover><msup><mrow><mi>q<\/mi><\/mrow><mrow><mi>k<\/mi><\/mrow><\/msup> <mo class=\"MathClass-rel\">=<\/mo><munderover accent=\"false\" accentunder=\"false\"><mrow><mo> \u2211<\/mo> <\/mrow><mrow><mi>k<\/mi><mo class=\"MathClass-rel\">=<\/mo><mn>0<\/mn><\/mrow><mrow><mi>n<\/mi><\/mrow><\/munderover><msup><mrow><mi>q<\/mi><\/mrow><mrow><mi>k<\/mi><\/mrow><\/msup> <mo class=\"MathClass-bin\">+<\/mo> <msup><mrow><mi>q<\/mi><\/mrow><mrow><mi>n<\/mi><mo class=\"MathClass-bin\">+<\/mo><mn>1<\/mn><\/mrow><\/msup> <mo class=\"MathClass-rel\">=<\/mo> <mfrac><mrow><msup><mrow><mi>q<\/mi><\/mrow><mrow><mi>n<\/mi><mo class=\"MathClass-bin\">+<\/mo><mn>1<\/mn><\/mrow><\/msup> <mo class=\"MathClass-bin\">\u2212<\/mo> <mn>1<\/mn><\/mrow> <mrow><mi>q<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <mn>1<\/mn><\/mrow><\/mfrac> <mo class=\"MathClass-bin\">+<\/mo> <msup><mrow><mi>q<\/mi><\/mrow><mrow><mi>n<\/mi><mo class=\"MathClass-bin\">+<\/mo><mn>1<\/mn><\/mrow><\/msup> <mo class=\"MathClass-rel\">=<\/mo> <mfrac><mrow><msup><mrow><mi>q<\/mi><\/mrow><mrow><mi>n<\/mi><mo class=\"MathClass-bin\">+<\/mo><mn>1<\/mn><\/mrow><\/msup> <mo class=\"MathClass-bin\">\u2212<\/mo> <mn>1<\/mn><\/mrow> <mrow><mi>q<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <mn>1<\/mn><\/mrow><\/mfrac> <mo class=\"MathClass-bin\">+<\/mo> <mfrac><mrow><msup><mrow><mi>q<\/mi><\/mrow><mrow><mi>n<\/mi><mo class=\"MathClass-bin\">+<\/mo><mn>2<\/mn><\/mrow><\/msup> <mo class=\"MathClass-bin\">\u2212<\/mo> <msup><mrow><mi>q<\/mi><\/mrow><mrow><mi>n<\/mi><mo class=\"MathClass-bin\">+<\/mo><mn>1<\/mn><\/mrow><\/msup><\/mrow> <mrow><mi>q<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <mn>1<\/mn><\/mrow><\/mfrac> <mo class=\"MathClass-rel\">=<\/mo> <mfrac><mrow><msup><mrow><mi>q<\/mi><\/mrow><mrow><mi>n<\/mi><mo class=\"MathClass-bin\">+<\/mo><mn>2<\/mn><\/mrow><\/msup> <mo class=\"MathClass-bin\">\u2212<\/mo> <mn>1<\/mn><\/mrow> <mrow><mi>q<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <mn>1<\/mn><\/mrow><\/mfrac> <mo class=\"MathClass-punc\">,<\/mo><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">womit der Induktionsschritt gezeigt ist und die Proposition folgt. <span>&nbsp;&nbsp;<\/span><\/p><div class=\"qed\">\u25a0<\/div><\/details><\/div> <p class=\"indent\">Wir laden Sie dazu ein, in folgender \u00dcbung einen eleganteren Beweis zu finden. <\/p> <div class=\"me meexample\"> <p class=\"indent\"><\/p><h4 id=\"ze2a5432680ff\"> <a id=\"x1-80002r9\"><\/a> <span class=\"ecbx-1095\">\u00dc<\/span><span class=\"ecbx-1095\">bung 3.9 <\/span>(Geometrische Summenformel)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Verwenden Sie eine Teleskopsumme um die geometrische Summenformel (Proposition<\/span><span class=\"ecti-1095\">&nbsp;<\/span><a href=\"..\/..\/chapter\/summen-und-produkte#x1-80001r8\"><span class=\"ecti-1095\">3.8<\/span><\/a><span class=\"ecti-1095\">)<\/span> <span class=\"ecti-1095\">f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r<\/span><span class=\"ecti-1095\">&nbsp;<\/span><math display=\"inline\"><mi>q<\/mi><mo class=\"MathClass-rel\">\u2260<\/mo> <mn>1<\/mn><\/math> <span class=\"ecti-1095\">zu beweisen.<\/span> <\/p><p class=\"indent\"><\/p><details><summary style=\"color:#FF7F00\"><span class=\"ecti-1095\">Hinweis.<\/span><\/summary><p class=\"indent\" style=\"margin-top: 0\"><span class=\"ecti-1095\">Multiplizieren Sie <\/span><math display=\"inline\"><msubsup><mrow><mi class=\"MathClass-op\">\u2211<\/mi><mo> <\/mo> <\/mrow><mrow><mi>k<\/mi><mo class=\"MathClass-rel\">=<\/mo><mn>0<\/mn><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msubsup><msup><mrow><mi>q<\/mi><\/mrow><mrow><mi>k<\/mi><\/mrow><\/msup><\/math> <span class=\"ecti-1095\">mit <\/span><span class=\"maperiod\"><math display=\"inline\"><mi>q<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <mn>1<\/mn><\/math><\/span><span class=\"period\">.<\/span><\/p><\/details>  <\/div> <div class=\"me meexample\"> <p class=\"indent\"><\/p><h4 id=\"zefec752ac269\"> <a id=\"x1-80003r10\"><\/a> <span class=\"ecbx-1095\">\u00dc<\/span><span class=\"ecbx-1095\">bung 3.10 <\/span>(Eindeutigkeit der Ziffernentwickung nat\u00fcrlicher Zahlen)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Zeigen Sie, dass die <\/span><math display=\"inline\"><mi>q<\/mi><\/math><span class=\"ecti-1095\">-n<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">re<\/span> <span class=\"ecti-1095\">Darstellung einer nat<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">rlichen Zahl in <\/span><span class=\"ecti-1095\">\u00dc<\/span><span class=\"ecti-1095\">bung<\/span><span class=\"ecti-1095\">&nbsp;<\/span><a href=\"..\/..\/chapter\/summen-und-produkte#x1-79003r7\"><span class=\"ecti-1095\">3.7<\/span><\/a> <span class=\"ecti-1095\">eindeutig bestimmt ist. Das heisst, f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r<\/span> <span class=\"ecti-1095\">jedes<\/span><span class=\"ecti-1095\">&nbsp;<\/span><math display=\"inline\"><mi>m<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2115<\/mi><\/math> <span class=\"ecti-1095\">mit<\/span><span class=\"ecti-1095\">&nbsp;<\/span><math display=\"inline\"><mi>m<\/mi> <mo class=\"MathClass-rel\">=<\/mo><msubsup><mrow><mi class=\"MathClass-op\"> \u2211<\/mi><mo> <\/mo> <\/mrow><mrow><mi>k<\/mi><mo class=\"MathClass-rel\">=<\/mo><mn>0<\/mn><\/mrow><mrow><mi>\u2113<\/mi><\/mrow><\/msubsup><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>k<\/mi><\/mrow><\/msub><msup><mrow><mi>q<\/mi><\/mrow><mrow><mi>k<\/mi><\/mrow><\/msup><\/math> <span class=\"ecti-1095\">und <\/span><math display=\"inline\"><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>\u2113<\/mi> <\/mrow> <\/msub> <mo class=\"MathClass-rel\">\u2260<\/mo> <mn>0<\/mn><\/math> <span class=\"ecti-1095\">sind <\/span><math display=\"inline\"><mi>\u2113<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <msub><mrow><mi>\u2115<\/mi><\/mrow><mrow><mn>0<\/mn> <\/mrow> <\/msub> <\/math> <span class=\"ecti-1095\">und die Koeffizienten <\/span><math display=\"inline\"><msub><mrow><mi>a<\/mi><\/mrow><mrow><mn>0<\/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>\u2113<\/mi><\/mrow><\/msub> <mo class=\"MathClass-rel\">\u2208<\/mo> <msub><mrow><mi>\u2115<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub> <mo class=\"MathClass-bin\">\u2229<\/mo> <mo class=\"MathClass-open\">[<\/mo><mn>0<\/mn><mo class=\"MathClass-punc\">,<\/mo><mi>q<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <mn>1<\/mn><mo class=\"MathClass-close\">]<\/mo><\/math> <span class=\"ecti-1095\">eindeutig durch<\/span><span class=\"ecti-1095\">&nbsp;<\/span><math display=\"inline\"><mi>m<\/mi><\/math> <span class=\"ecti-1095\">bestimmt.<\/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\">Zeigen Sie zuerst f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r <\/span><math display=\"inline\"><msub><mrow><mi>b<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><mo class=\"MathClass-punc\">,<\/mo><mi class=\"MathClass-op\">\u2026<\/mi><mo> <\/mo><mo class=\"MathClass-punc\">,<\/mo><msub><mrow><mi>b<\/mi><\/mrow><mrow><mi>\u2113<\/mi><\/mrow><\/msub> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2124<\/mi><\/math> <span class=\"ecti-1095\">mit <\/span><math display=\"inline\"><mo class=\"MathClass-rel\">|<\/mo><msub><mrow><mi>b<\/mi><\/mrow><mrow><mi>k<\/mi> <\/mrow> <\/msub> <mo class=\"MathClass-rel\">|<\/mo> <mo class=\"MathClass-rel\">&lt;<\/mo> <mi>q<\/mi><\/math> <span class=\"ecti-1095\">f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r <\/span><math display=\"inline\"><mi>k<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mn>1<\/mn><mo class=\"MathClass-punc\">,<\/mo> <mi class=\"MathClass-op\">\u2026<\/mi><mo> <\/mo><mo class=\"MathClass-punc\">,<\/mo><mi>\u2113<\/mi><\/math> <span class=\"ecti-1095\">die Absch<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">tzung <\/span><span class=\"maperiod\"><math display=\"inline\"><mo class=\"MathClass-rel\">|<\/mo><msubsup><mrow><mi class=\"MathClass-op\">\u2211<\/mi><mo> <\/mo> <\/mrow><mrow><mi>k<\/mi><mo class=\"MathClass-rel\">=<\/mo><mn>0<\/mn><\/mrow><mrow><mi>\u2113<\/mi><\/mrow><\/msubsup><msub><mrow><mi>b<\/mi><\/mrow><mrow><mi>k<\/mi><\/mrow><\/msub><msup><mrow><mi>q<\/mi><\/mrow><mrow><mi>k<\/mi><\/mrow><\/msup><mo class=\"MathClass-rel\">|<\/mo> <mo class=\"MathClass-rel\">&lt;<\/mo> <msup><mrow><mi>q<\/mi><\/mrow><mrow><mi>\u2113<\/mi><mo class=\"MathClass-bin\">+<\/mo><mn>1<\/mn><\/mrow><\/msup><\/math><\/span><span class=\"period\">.<\/span> <span class=\"ecti-1095\">Nehmen Sie indirekt an, dass es eine nat<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">rliche Zahl <\/span><math display=\"inline\"><mi>m<\/mi><\/math> <span class=\"ecti-1095\">mit zwei Darstellungen gibt und w<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">hlen Sie eine minimale derartige Zahl. Sei nun <\/span><math display=\"inline\"><mi>m<\/mi> <mo class=\"MathClass-rel\">=<\/mo><msubsup><mrow><mi class=\"MathClass-op\"> \u2211<\/mi><mo> <\/mo> <\/mrow><mrow><mi>k<\/mi><mo class=\"MathClass-rel\">=<\/mo><mn>0<\/mn><\/mrow><mrow><mi>\u2113<\/mi><\/mrow><\/msubsup><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>k<\/mi><\/mrow><\/msub><msup><mrow><mi>q<\/mi><\/mrow><mrow><mi>k<\/mi><\/mrow><\/msup> <mo class=\"MathClass-rel\">=<\/mo><msubsup><mrow><mi class=\"MathClass-op\"> \u2211<\/mi><mo> <\/mo> <\/mrow><mrow><mi>k<\/mi><mo class=\"MathClass-rel\">=<\/mo><mn>0<\/mn><\/mrow><mrow><msup><mrow><mi>\u2113<\/mi><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup> <\/mrow><\/msubsup><msubsup><mrow><mi>a<\/mi><\/mrow><mrow><mi>k<\/mi><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msubsup><msup><mrow><mi>q<\/mi><\/mrow><mrow><mi>k<\/mi><\/mrow><\/msup><\/math> <span class=\"ecti-1095\">zwei Darstellungen von <\/span><span class=\"maperiod\"><math display=\"inline\"><mi>m<\/mi><\/math><\/span><span class=\"period\">.<\/span> <span class=\"ecti-1095\">Auf Grund der Minimalit<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">t von <\/span><math display=\"inline\"><mi>m<\/mi><\/math> <span class=\"ecti-1095\">muss <\/span><math display=\"inline\"><mi>\u2113<\/mi><mo class=\"MathClass-rel\">\u2260<\/mo> <msup><mrow><mi>\u2113<\/mi><\/mrow><mrow><mo>\u2032<\/mo> <\/mrow> <\/msup> <\/math> <span class=\"ecti-1095\">gelten, doch auf Grund der Absch<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">tzung kann dies zu einem Widerspruch gef<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">hrt werden.<\/span><\/p><\/details>  <\/div> <a id=\"x1-80004r77\"><\/a> \n","rendered":"\n<style scoped=\"scoped\">.cmr-5{font-size:50%;}\n.cmr-7{font-size:70%;}\n.cmmi-5{font-size:50%;font-style: italic;}\n.cmmi-7{font-size:70%;font-style: italic;}\n.cmmi-10{font-style: italic;}\n.cmsy-5{font-size:50%;}\n.cmsy-7{font-size:70%;}\n.cmbx-10{ font-weight: bold;}\n.cmbsy-10{font-weight: bold;}\n.cmbsy-10{font-weight: bold;}\n.cmbsy-10{font-weight: bold;}\n.cmbsy-7{font-size:70%;font-weight: bold;}\n.cmbsy-7{font-weight: bold;}\n.cmbsy-7{font-weight: bold;}\n.cmbsy-5{font-size:50%;font-weight: bold;}\n.cmbsy-5{font-weight: bold;}\n.cmbsy-5{font-weight: bold;}\n.cmex-7{font-size:70%;}\n.cmex-7x-x-71{font-size:49%;}\n.msam-7{font-size:70%;}\n.msam-5{font-size:50%;}\n.msbm-7{font-size:70%;}\n.msbm-5{font-size:50%;}\n.cmr-17{font-size:170%;}\n.cmr-12{font-size:120%;}\n.cmti-10{ font-style: italic;}\np{margin-top:0;margin-bottom:0}\np.indent{text-indent:0;}\np + p{margin-top:1em;}\np + div, p + pre {margin-top:1em;}\ndiv + p, pre + p {margin-top:1em;}\n@media print {div.crosslinks {visibility:hidden;}}\na img { border-top: 0; 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}\n.hline hr, .cline hr{border:none;border-top:1px solid black;}\n.equation-star td{text-align:center; vertical-align:middle; }\ntable.equation-star { width:100%; 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; 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width:125%;}\ndt {text-align:right; font-weight:bold; clear:left; float:left;}\ndd {width:100%; padding-left:1em; padding-top: 0px; clear:right;}\ndd + dd {float:right; clear:both;}\ndd + dt {clear:both;}\ndt + dt {width: 100%; float: none; padding: 0 70% 0 0;}\ndt + dt + dd {margin-top: -2em;}\ndt + dt + dd + dt {margin-top: 2em;}\n<\/style>\n<style scoped=\"scoped\">\n\/* CSS Analysis-Skript D-Math ETHZ *\/\n\n\/* Uniform Font, also for headers *\/\nh3 {\n\tfont-family: \"Times New Roman\", serif;\n\tmargin-bottom: 35px;\n}\nh4 {\n\tfont-family: \"Times New Roman\", serif;\n}\nh5 {\n\tfont-family: \"Times New Roman\", serif;\n}\n\n\/* Bold font, e.g. for definitions *\/\n.ecbx-1095 {font-weight: 550 ;}\n\n\n\/* Uniform spacing, indent: larger, noindent, enumerate, itemize *\/\np.indent {\n\tmargin: 25px 0px 0px 0px;\n\ttext-indent: 0px; \n}\np.noindent {\n\tmargin: 15px 0px 0px 0px;\n\ttext-indent: 0px; \n}\ndl.enumerate {\n\tmargin: 0px 0px 0px 0px;\n}\ndl.enumerate dt, dl.enumerate dd {\n\tmargin-top: 15px;\n\tmargin-bottom: 0px;\n}\ndiv.custom-itemize {\n\tmargin: 0px 0px 0px 0px;\n}\ndiv.custom-itemize div.item-head {\n\tmargin-top: 15px;\n\tmargin-bottom: 0px;\n\ttext-align: center;\n}\ndiv.custom-itemize div.item-head:first-of-type {\n\tmargin-top: 0px;\n} \ndiv.custom-itemize div.item-content {\n\tmargin-top: 15px;\n\tmargin-bottom: 0px;\n}\n.MJXc-display {\n\tmargin: 15px 0px 0px 0px;\n}\n\n\n\n\/* green metheorem\/melemma CSS class for more\/medium important latex-theorem-environments *\/\n\/* metheorem box+header *\/\ndiv.metheorem {\n    margin-bottom: 40px;\n    margin-top: 40px;\n\tpadding: 0px 15px 15px 15px;\n    border: 1px solid #333;\n    border-color: #4eb79e;\n    background: #c7e4da;\n}\ndiv.metheorem h4 {\n    background: #4eb79e;\n    color: white;\n\tmargin-top: 12px;\n\tmargin-left: -15px;\n\tmargin-right: -15px;\n\tpadding: 0px 15px 0px 15px;\n}\n\/* melemma box+header *\/\ndiv.melemma {\n    margin-bottom: 40px;\n    margin-top: 40px;\n\tpadding: 0px 15px 15px 15px;\n    border: 1px solid #333;\n    border-color: #4eb79e;\n    background: #F2F2F2;\n}\ndiv.melemma h4 {\n    background: #4eb79e;\n    color: white;\n\tmargin-top: 12px;\n\tmargin-left: -15px;\n\tmargin-right: -15px;\n\tpadding: 0px 15px 0px 15px;\n}\n\/* meexample box+header *\/\ndiv.meexample {\n    margin-bottom: 30px;\n    margin-top: 30px;\n\tpadding: 0px 15px 15px 15px;\n\tborder-color: gainsboro;\n\tborder-style: solid;\n\tborder-width: thin;\n}\ndiv.meexample h4 {\n\tfont-size: inherit;\n\tfont-weight: bold;\n    padding: 15px 0px 0px 0px;\n\tmargin-top: 0px;\n\tmargin-bottom: 5px;\n}\ndiv.meexample h4+p.noindent, div.meexample h4+p.indent {\n\tmargin-top: 5px;\n\ttext-indent: 0px;\n}\n\/* padding and margins for stuff inside these boxes, CSS-selector &gt; doesn't work in WP *\/\ndiv.me details {\n\tmargin: 10px 0px 0px 0px;\n}\ndiv.me dd {\n    width: calc(100% - 30px);\n}\t\n\n\n\/* fixing background of pictures *\/\nimg {\n\tbackground: white;\n}\n\n\/* div-container for centered geoapplet *\/\ndiv.geoapplet {\n\tmargin-left: auto;\n\tmargin-right: auto;\n\tmargin-top: 15px;\n\tmax-width: 100%;\n}\ndiv.geoapplet iframe {\n\tborder-style: none;\n\tmax-height: 110vw;\n}\n\n\/* div-container for centered squeezed tables *\/\ndiv.websqueeze {\n\tmargin-left: auto;\n\tmargin-right: auto;\n}\n\n\/* two containers for squeezing text sizes *\/\ndiv.mesmalltext, div.mesmalltext * {\n\tfont-size: 15px;\n}\nspan.metinytext, span.metinytext * {\n\tfont-size: 12px;\n}\n\n\n\/* removing grid lines in equations *\/\n#content table.equation tr td, #content table.equation tr th {\n    border: none;\n}\n#content table.equation {\n    border: none;\n}\n\n\/* hover\/click-solution for short inline explanations and footnotes *\/\n.hover-text {    \/* hidden part *\/\n    display: none;\n}\n.marginpar {     \/* style for footnote as marginpar *\/\n\ttext-decoration: none;\n\tborder: solid;\n\tborder-width: 1pt;\n\tpadding: 3pt;\t\n\twidth: 30%;\n\tbackground: white;\n}\n.hover-trigger { \/* style for hover\/click-trigger text\/symbol *\/\n\tbackground: none;\n\tborder: none;\n\tpadding: 0;\n\toutline: inherit;\t\n\ttext-transform: none;\n\tfont: inherit;\n\tposition: inherit;\n\tvertical-align: baseline;\n    color: #FF7F00;\n\tcursor: help;\n}\n.hover-trigger:hover +.hover-text{\n    display: inline;\n}\n.hover-trigger:active +.hover-text{\n    display: inline;\n}\n\n\/* simplifying style of details\/summary, removing triangle *\/\ndetails summary {\n  background: none;\n  list-style: none;\n  outline: none;\n  cursor: pointer;\n}\ndetails summary::-webkit-details-marker { \n  display: inline;\n  display: none;\n}\n\n\/* MC-True\/False as inline details\/summary *\/\ndetails.mcquest, div.me details.mcquest {\n\tdisplay: inline;\n\tmargin-top: 0px;\n}\nsummary.mcquest {\n\tdisplay: inline;\n\tcolor: #FF7F00;\n\tcursor: help;\n}\n\n\/* proof style: simple black box with gray background \n                little black square at the end on the right *\/\ndiv.proof {\n\tborder-color: black;\n\tborder-style: solid;\n\tborder-width: thin;\n\tbackground-color: #F2F2F2;\n\tpadding: 15px;\n\tmargin-top: 1em; \n}\ndiv.proof p:first-of-type {\n\tmargin: 0px;\n}\ndiv.qed {\n\tmargin-top: -25px;\n\tmargin-bottom: -7px;\n\ttext-align: right;\n}\ntable.equation+div.qed {\n\tmargin-top: -65px;\n}\n\n\/* The following is making also math-formulas inside the headers of Lemmas, etc., white. *\/\ndiv.melemma h4 span {\n    color: white;\n}\ndiv.metheorem h4 span {\n    color: white;\n}\n\n\/* The following are used to avoid fullstop, period, colon, semicolon, and endquote (broader) to move by itself to the next line after a formula.\n   The math-environment before needs to be wrapped in span.maperiod and the fullstop etc. in a span.period --- together they achieve what we want.  *\/\nspan.maperiod {\n       margin-right: 5px;\n}\nspan.period {\n       display: inline-block;\n       width: 0px;\n       margin-left: -5px;\n       margin-right: 4.9px;\n\t   text-indent: 0px;\n}\nspan.maendquote {\n       margin-right: 8px;\n}\nspan.endquote {\n       display: inline-block;\n       width: 0px;\n       margin-left: -8px;\n       margin-right: 7.9px;\n}\n\n\n\/* The following is removing an extra space left of the equation side in aligned equations *\/\nspan.mjx-mtd {\n    padding-left: 0em !important;\n}\n\n\/* The following fixes the weird problem that math appears smaller if it was rendered while the details tag was closed. *\/\ndetails span.mjx-chtml, details span.MathJax_CHTML {\n font-size: 100% !important;\n}\n\n\/* trying to fix line breaks in verbatim, new lines are missing *\/\npre.verbatim {\n\twhite-space: pre-wrap;\n\tfont-size: small;\n}\n<\/style><h3 id=\"z759f256f5507\" class=\"sectionHead\"><span class=\"titlemark\">3.1 <\/span> <a id=\"x1-770001\"><\/a>Summen und Produkte<\/h3> <p class=\"noindent\">Sei <math display=\"inline\"><mi>n<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2115<\/mi><\/math> und seien <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>\u2102<\/mi><\/math> oder <math display=\"inline\"><msub><mrow><mi>a<\/mi><\/mrow><mrow><mn>1<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-punc\">,<\/mo> <mi class=\"MathClass-op\">\u2026<\/mi><mo> <\/mo> <mo class=\"MathClass-punc\">,<\/mo> <msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi> <\/mrow> <\/msub> <\/math> Elemente eines Vektorraums&nbsp;<math display=\"inline\"><mi>V<\/mi> <\/math> (wie zum Beispiel <math display=\"inline\"><msup><mrow><mi>\u211d<\/mi><\/mrow><mrow><mi>d<\/mi><\/mrow><\/msup><\/math> f\u00fcr ein <math display=\"inline\"><mi>d<\/mi> <mo class=\"MathClass-rel\">\u2265<\/mo> <mn>1<\/mn><\/math>). Wir wollen hier f\u00fcr eine nat\u00fcrliche Zahl&nbsp;<math display=\"inline\"><mi>n<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2115<\/mi><\/math> die <span class=\"ecbx-1095\">Summe <\/span>von <math display=\"inline\"><msub><mrow><mi>a<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><\/math> bis <span class=\"maperiod\"><math display=\"inline\"><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi> <\/mrow> <\/msub> <\/math><\/span><span class=\"period\">,<\/span> also <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><munderover accent=\"false\" accentunder=\"false\"><mrow><mo> \u2211<\/mo> <\/mrow><mrow><mi>j<\/mi><mo class=\"MathClass-rel\">=<\/mo><mn>1<\/mn><\/mrow><mrow><mi>n<\/mi><\/mrow><\/munderover><msub><mrow><mi>a<\/mi><\/mrow><mrow> <mi>j<\/mi><\/mrow><\/msub> <mo class=\"MathClass-rel\">=<\/mo> <msub><mrow><mi>a<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub> <mo class=\"MathClass-bin\">+<\/mo> <mo>\u2026<\/mo> <mo class=\"MathClass-bin\">+<\/mo> <msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><mo class=\"MathClass-punc\">,<\/mo><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">besprechen und formal korrekt definieren. <\/p><p class=\"indent\">Vom formalen Standpunkt her gesehen ist <math display=\"inline\"><mi>j<\/mi><mo class=\"MathClass-rel\">\u21a6<\/mo><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>j<\/mi><\/mrow><\/msub> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>V<\/mi> <\/math> eine Funktion, (die oft auch durch eine konkrete Formel gegeben sein wird und) deren Definitionsbereich die Menge <\/p><table id=\"zd14d9022f276\" class=\"equation-star\"><tr><td> <math class=\"equation\" display=\"block\"> <mrow><mo fence=\"true\" form=\"prefix\"> {<\/mo><mrow><mi>j<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2115<\/mi><mo class=\"MathClass-rel\">\u2223<\/mo><mn>1<\/mn> <mo class=\"MathClass-rel\">\u2264<\/mo> <mi>j<\/mi> <mo class=\"MathClass-rel\">\u2264<\/mo> <mi>n<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">}<\/mo><\/mrow> <\/math><\/td><\/tr><\/table> <p class=\"indent\">enthalten muss. Wir k\u00f6nnen <math display=\"inline\"><msubsup><mrow><mi class=\"MathClass-op\"> \u2211<\/mi><mo> <\/mo> <\/mrow><mrow><mi>i<\/mi><mo class=\"MathClass-rel\">=<\/mo><mn>1<\/mn><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msubsup><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>j<\/mi><\/mrow><\/msub><\/math> rekursiv definieren durch <\/p><table id=\"z4d3f6041c5d4\" class=\"equation-star\"><tr><td> <math class=\"equation\" display=\"block\"> <munderover accent=\"false\" accentunder=\"false\"><mrow><mo>\u2211<\/mo> <\/mrow><mrow><mi>j<\/mi><mo class=\"MathClass-rel\">=<\/mo><mn>1<\/mn><\/mrow><mrow><mn>1<\/mn><\/mrow><\/munderover><msub><mrow><mi>a<\/mi><\/mrow><mrow> <mi>j<\/mi><\/mrow><\/msub> <mo class=\"MathClass-rel\">=<\/mo> <msub><mrow><mi>a<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><mstyle class=\"text\"><mtext>&nbsp;und&nbsp;<\/mtext><\/mstyle><munderover accent=\"false\" accentunder=\"false\"><mrow><mo> \u2211<\/mo> <\/mrow><mrow><mi>j<\/mi><mo class=\"MathClass-rel\">=<\/mo><mn>1<\/mn><\/mrow><mrow><mi>k<\/mi><mo class=\"MathClass-bin\">+<\/mo><mn>1<\/mn><\/mrow><\/munderover><msub><mrow><mi>a<\/mi><\/mrow><mrow> <mi>j<\/mi><\/mrow><\/msub> <mo class=\"MathClass-rel\">=<\/mo> <mstyle><mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow \/><mo fence=\"true\" form=\"postfix\" \/><\/mrow><\/mstyle><munderover accent=\"false\" accentunder=\"false\"><mrow><mo>\u2211<\/mo> <\/mrow><mrow><mi>j<\/mi><mo class=\"MathClass-rel\">=<\/mo><mn>1<\/mn><\/mrow><mrow><mi>k<\/mi><\/mrow><\/munderover><msub><mrow><mi>a<\/mi><\/mrow><mrow> <mi>j<\/mi><\/mrow><\/msub><mstyle><mrow><mo fence=\"true\" form=\"prefix\"> )<\/mo><mrow \/><mo fence=\"true\" form=\"postfix\" \/><\/mrow><\/mstyle> <mo class=\"MathClass-bin\">+<\/mo> <msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>k<\/mi><mo class=\"MathClass-bin\">+<\/mo><mn>1<\/mn><\/mrow><\/msub> <\/math><\/td><\/tr><\/table> <p class=\"indent\">f\u00fcr <span class=\"maperiod\"><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> <mo class=\"MathClass-bin\">\u2212<\/mo> <mn>1<\/mn><\/mrow><mo fence=\"true\" form=\"postfix\">}<\/mo><\/mrow><\/math><\/span><span class=\"period\">.<\/span> Diese Definition entspricht einem einfachen rekursiven Algorithmus, um die Summe <math display=\"inline\"><msubsup><mrow><mi class=\"MathClass-op\">\u2211<\/mi><mo> <\/mo> <\/mrow><mrow><mi>i<\/mi><mo class=\"MathClass-rel\">=<\/mo><mn>1<\/mn><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msubsup><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>j<\/mi><\/mrow><\/msub><\/math> zu berechnen. Allgemeiner ist die Summe <math display=\"inline\"><msubsup><mrow><mi class=\"MathClass-op\"> \u2211<\/mi><mo> <\/mo> <\/mrow><mrow><mi>i<\/mi><mo class=\"MathClass-rel\">=<\/mo><mi>m<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msubsup><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>j<\/mi><\/mrow><\/msub><\/math> f\u00fcr ganze Zahlen <math display=\"inline\"><mi>m<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>n<\/mi><\/math> ebenso rekursiv durch <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><munderover accent=\"false\" accentunder=\"false\"><mrow><mo> \u2211<\/mo> <\/mrow><mrow><mi>j<\/mi><mo class=\"MathClass-rel\">=<\/mo><mi>m<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/munderover><msub><mrow><mi>a<\/mi><\/mrow><mrow> <mi>j<\/mi><\/mrow><\/msub> <mo class=\"MathClass-rel\">=<\/mo> <mrow class=\"cases\"> <mrow><mo fence=\"true\" form=\"prefix\"> {<\/mo><mrow> <mtable align=\"axis\" class=\"array\" columnlines=\"none\" equalcolumns=\"false\" equalrows=\"false\"> <mtr><mtd class=\"array\" columnalign=\"left\"><mn>0<\/mn> <mspace class=\"quad\" width=\"1em\" \/><\/mtd><mtd class=\"array\" columnalign=\"left\"><mstyle class=\"text\"><mtext>falls&nbsp;<\/mtext><\/mstyle><mi>m<\/mi> <mo class=\"MathClass-rel\">&gt;<\/mo> <mi>n<\/mi><mo class=\"MathClass-punc\">,<\/mo> <\/mtd> <\/mtr> <mtr><mtd class=\"array\" columnalign=\"left\"><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>m<\/mi><\/mrow><\/msub> <mspace class=\"quad\" width=\"1em\" \/><\/mtd><mtd class=\"array\" columnalign=\"left\"><mstyle class=\"text\"><mtext>falls&nbsp;<\/mtext><\/mstyle><mi>m<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mi>n<\/mi><mstyle class=\"text\"><mtext>&nbsp;und<\/mtext><\/mstyle><\/mtd> <\/mtr> <mtr><mtd class=\"array\" columnalign=\"left\"><mstyle><mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow \/><mo fence=\"true\" form=\"postfix\" \/><\/mrow><\/mstyle><munderover accent=\"false\" accentunder=\"false\"><mrow><mo> \u2211<\/mo> <\/mrow><mrow><mi>j<\/mi><mo class=\"MathClass-rel\">=<\/mo><mi>m<\/mi><\/mrow><mrow><mi>n<\/mi><mo class=\"MathClass-bin\">\u2212<\/mo><mn>1<\/mn><\/mrow><\/munderover><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>j<\/mi><\/mrow><\/msub><mstyle><mrow><mo fence=\"true\" form=\"prefix\"> )<\/mo><mrow \/><mo fence=\"true\" form=\"postfix\" \/><\/mrow><\/mstyle> <mo class=\"MathClass-bin\">+<\/mo> <msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><mspace class=\"quad\" width=\"1em\" \/><\/mtd><mtd class=\"array\" columnalign=\"left\"><mstyle class=\"text\"><mtext>falls&nbsp;<\/mtext><\/mstyle><mi>m<\/mi> <mo class=\"MathClass-rel\">&lt;<\/mo> <mi>n<\/mi> <\/mtd><\/mtr> <\/mtable> <\/mrow><mo fence=\"true\" form=\"postfix\" \/><\/mrow><\/mrow><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">definiert. Wir werden&nbsp;<math display=\"inline\"><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>j<\/mi><\/mrow><\/msub><\/math> als die <span class=\"ecbx-1095\">Summanden <\/span>und&nbsp;<math display=\"inline\"><mi>j<\/mi><\/math> als den <span class=\"ecbx-1095\">Index <\/span>der Summe <math display=\"inline\"><msubsup><mrow><mi class=\"MathClass-op\"> \u2211<\/mi><mo> <\/mo> <\/mrow><mrow><mi>j<\/mi><mo class=\"MathClass-rel\">=<\/mo><mn>1<\/mn><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msubsup><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>j<\/mi><\/mrow><\/msub><\/math> bezeichnen. <\/p><p class=\"indent\">Falls nun <math display=\"inline\"><mi>m<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>n<\/mi><\/math> ganze Zahlen und <math display=\"inline\"><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>m<\/mi> <\/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>\u2102<\/mi><\/math> sind, dann k\u00f6nnen wir auch das <span class=\"ecbx-1095\">Produkt <\/span><math display=\"inline\"><msubsup><mrow><mi class=\"MathClass-op\">\u220f<\/mi><mo> <\/mo> <\/mrow><mrow><mi>j<\/mi><mo class=\"MathClass-rel\">=<\/mo><mi>m<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msubsup><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>j<\/mi><\/mrow><\/msub><\/math> von <math display=\"inline\"><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>m<\/mi> <\/mrow> <\/msub> <\/math> bis <math display=\"inline\"><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi> <\/mrow> <\/msub> <\/math> rekursiv durch <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><munderover accent=\"false\" accentunder=\"false\"><mrow><mo> \u220f<\/mo> <\/mrow><mrow><mi>j<\/mi><mo class=\"MathClass-rel\">=<\/mo><mi>m<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/munderover><msub><mrow><mi>a<\/mi><\/mrow><mrow> <mi>j<\/mi><\/mrow><\/msub> <mo class=\"MathClass-rel\">=<\/mo> <mrow class=\"cases\"> <mrow><mo fence=\"true\" form=\"prefix\"> {<\/mo><mrow> <mtable align=\"axis\" class=\"array\" columnlines=\"none\" equalcolumns=\"false\" equalrows=\"false\"> <mtr><mtd class=\"array\" columnalign=\"left\"><mn>1<\/mn> <mspace class=\"quad\" width=\"1em\" \/><\/mtd><mtd class=\"array\" columnalign=\"left\"><mstyle class=\"text\"><mtext>falls&nbsp;<\/mtext><\/mstyle><mi>m<\/mi> <mo class=\"MathClass-rel\">&gt;<\/mo> <mi>n<\/mi><mo class=\"MathClass-punc\">,<\/mo> <\/mtd> <\/mtr> <mtr><mtd class=\"array\" columnalign=\"left\"><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>m<\/mi><\/mrow><\/msub> <mspace class=\"quad\" width=\"1em\" \/><\/mtd><mtd class=\"array\" columnalign=\"left\"><mstyle class=\"text\"><mtext>falls&nbsp;<\/mtext><\/mstyle><mi>m<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mi>n<\/mi><mstyle class=\"text\"><mtext>&nbsp;und<\/mtext><\/mstyle><\/mtd> <\/mtr> <mtr><mtd class=\"array\" columnalign=\"left\"><mstyle><mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow \/><mo fence=\"true\" form=\"postfix\" \/><\/mrow><\/mstyle><munderover accent=\"false\" accentunder=\"false\"><mrow><mo> \u220f<\/mo> <\/mrow><mrow><mi>j<\/mi><mo class=\"MathClass-rel\">=<\/mo><mi>m<\/mi><\/mrow><mrow><mi>n<\/mi><mo class=\"MathClass-bin\">\u2212<\/mo><mn>1<\/mn><\/mrow><\/munderover><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>j<\/mi><\/mrow><\/msub><mstyle><mrow><mo fence=\"true\" form=\"prefix\"> )<\/mo><mrow \/><mo fence=\"true\" form=\"postfix\" \/><\/mrow><\/mstyle> <mo class=\"MathClass-bin\">\u22c5<\/mo> <msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><mspace class=\"quad\" width=\"1em\" \/><\/mtd><mtd class=\"array\" columnalign=\"left\"><mstyle class=\"text\"><mtext>falls&nbsp;<\/mtext><\/mstyle><mi>m<\/mi> <mo class=\"MathClass-rel\">&lt;<\/mo> <mi>n<\/mi> <\/mtd><\/mtr> <\/mtable> <\/mrow><mo fence=\"true\" form=\"postfix\" \/><\/mrow><\/mrow><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">definieren. Wir werden&nbsp;<math display=\"inline\"><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>j<\/mi><\/mrow><\/msub><\/math> als die <span class=\"ecbx-1095\">Faktoren <\/span>und&nbsp;<math display=\"inline\"><mi>j<\/mi><\/math> als den <span class=\"ecbx-1095\">Index <\/span>des Produkts <math display=\"inline\"><msubsup><mrow><mi class=\"MathClass-op\"> \u220f<\/mi><mo> <\/mo> <\/mrow><mrow><mi>j<\/mi><mo class=\"MathClass-rel\">=<\/mo><mi>m<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msubsup><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>j<\/mi><\/mrow><\/msub><\/math> bezeichnen. <\/p><p class=\"indent\">Der Index <math display=\"inline\"><mi>j<\/mi><\/math> in der Summe <math display=\"inline\"><msubsup><mrow><mi class=\"MathClass-op\"> \u2211<\/mi><mo> <\/mo> <\/mrow><mrow><mi>j<\/mi><mo class=\"MathClass-rel\">=<\/mo><mn>1<\/mn><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msubsup><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>j<\/mi><\/mrow><\/msub><\/math> oder dem Produkt <math display=\"inline\"><msubsup><mrow><mi class=\"MathClass-op\"> \u220f<\/mi><mo> <\/mo> <\/mrow><mrow><mi>j<\/mi><mo class=\"MathClass-rel\">=<\/mo><mn>1<\/mn><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msubsup><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>j<\/mi><\/mrow><\/msub><\/math> hat ausserhalb der Summe oder dem Produkt keinerlei Bedeutung; er ist sozusagen eine interne Variable f\u00fcr das rekursive Teilprogramm und wird von einem Programm, welches das Teilprogramm aufruft, nicht gesehen. Insbesondere gilt                                                                                                                                                                           <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><munderover accent=\"false\" accentunder=\"false\"><mrow><mo> \u2211<\/mo> <\/mrow><mrow><mi>j<\/mi><mo class=\"MathClass-rel\">=<\/mo><mi>m<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/munderover><msub><mrow><mi>a<\/mi><\/mrow><mrow> <mi>j<\/mi><\/mrow><\/msub> <mo class=\"MathClass-rel\">=<\/mo><munderover accent=\"false\" accentunder=\"false\"><mrow><mo> \u2211<\/mo> <\/mrow><mrow><mi>k<\/mi><mo class=\"MathClass-rel\">=<\/mo><mi>m<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/munderover><msub><mrow><mi>a<\/mi><\/mrow><mrow> <mi>k<\/mi><\/mrow><\/msub> <mo class=\"MathClass-rel\">=<\/mo><munderover accent=\"false\" accentunder=\"false\"><mrow><mo> \u2211<\/mo> <\/mrow><mrow><mi>\u2113<\/mi><mo class=\"MathClass-rel\">=<\/mo><mi>m<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/munderover><msub><mrow><mi>a<\/mi><\/mrow><mrow> <mi>\u2113<\/mi><\/mrow><\/msub><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">und analog f\u00fcr das Produkt. Manchmal werden wir auch eine <span class=\"ecbx-1095\">Indexverschiebung <\/span>anwenden, wie zum Beispiel in <\/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>j<\/mi><mo class=\"MathClass-rel\">=<\/mo><mi>m<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/munderover><msub><mrow><mi>a<\/mi><\/mrow><mrow> <mi>j<\/mi><\/mrow><\/msub> <mo class=\"MathClass-rel\">=<\/mo><munderover accent=\"false\" accentunder=\"false\"><mrow><mo> \u2211<\/mo> <\/mrow><mrow><mi>k<\/mi><mo class=\"MathClass-rel\">=<\/mo><mi>m<\/mi><mo class=\"MathClass-bin\">\u2212<\/mo><mn>1<\/mn><\/mrow><mrow><mi>n<\/mi><mo class=\"MathClass-bin\">\u2212<\/mo><mn>1<\/mn><\/mrow><\/munderover><msub><mrow><mi>a<\/mi><\/mrow><mrow> <mi>k<\/mi><mo class=\"MathClass-bin\">+<\/mo><mn>1<\/mn><\/mrow><\/msub> <mo class=\"MathClass-rel\">=<\/mo><munderover accent=\"false\" accentunder=\"false\"><mrow><mo> \u2211<\/mo> <\/mrow><mrow><mi>\u2113<\/mi><mo class=\"MathClass-rel\">=<\/mo><mi>m<\/mi><mo class=\"MathClass-bin\">+<\/mo><mn>1<\/mn><\/mrow><mrow><mi>n<\/mi><mo class=\"MathClass-bin\">+<\/mo><mn>1<\/mn><\/mrow><\/munderover><msub><mrow><mi>a<\/mi><\/mrow><mrow> <mi>\u2113<\/mi><mo class=\"MathClass-bin\">\u2212<\/mo><mn>1<\/mn><\/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=\"x1-77001r1\" \/><mstyle class=\"maketag\"><mtext>(3.1)<\/mtext><\/mstyle><mspace class=\"nbsp\" width=\"0.33em\" \/> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">Dies l\u00e4sst sich direkt mittels vollst\u00e4ndiger Induktion beweisen (siehe die folgende \u00dcbung), doch wollen wir bemerken, dass es leicht ist, sich diese Formeln zu merken. Statt diese auswendig zu lernen, \u00fcberpr\u00fcfen Sie einfach bei Auftreten von Indexverschiebungen dieser Form bei beiden Summen, ob jeweils diesselben Ausdr\u00fccke f\u00fcr den ersten und den letzten Summanden auftreten. <\/p> <div class=\"me meexample\"> <div class=\"wp-nocaption \"><\/div><h4 id=\"z591d29b41c63\"> <a id=\"x1-77002r1\"><\/a> <span class=\"ecbx-1095\">\u00dc<\/span><span class=\"ecbx-1095\">bung 3.1 <\/span>(Indexverschiebung)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Beweisen Sie  Gleichung<\/span>  (<a href=\"..\/..\/chapter\/summen-und-produkte#x1-77001r1\">3.1<\/a>)  <span class=\"ecti-1095\">und  eine  analoge  Formel  f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r  das  Produkt  mittels<\/span> <span class=\"ecti-1095\">vollst<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">ndiger Induktion.<\/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\">Bei         dem         Induktionsbeweis         bez<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">glich         der         Variable<\/span> <math display=\"inline\"><mi>n<\/mi> <mo class=\"MathClass-rel\">\u2265<\/mo> <mi>m<\/mi><\/math> <span class=\"ecti-1095\">werden                                            Sie                                            feststellen,<\/span> <span class=\"ecti-1095\">dass man genau die Eigenschaften in obiger Merkregel verwendet: F<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r den Induktionsanfang<\/span> <math display=\"inline\"><mi>n<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mi>m<\/mi><\/math> <span class=\"ecti-1095\">m<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">ssen die   ersten   Summanden   <\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">bereinstimmen.   F<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r   den   Induktionsschritt   von<\/span> <math display=\"inline\"><mi>n<\/mi><\/math> <span class=\"ecti-1095\">nach<\/span> <math display=\"inline\"><mi>n<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mn>1<\/mn><\/math> <span class=\"ecti-1095\">m<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">ssen die zus<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">tzlichen (also die letzten) Summanden <\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">bereinstimmen.<\/span><\/p><\/details>  <\/div> <p class=\"indent\">Der einfachste Fall einer Funktion <math display=\"inline\"><mi>j<\/mi><mo class=\"MathClass-rel\">\u21a6<\/mo><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>j<\/mi><\/mrow><\/msub><\/math> ist der Fall der konstanten Funktion <math display=\"inline\"><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>j<\/mi><\/mrow><\/msub> <mo class=\"MathClass-rel\">=<\/mo> <mi>z<\/mi><\/math> f\u00fcr ein <math display=\"inline\"><mi>z<\/mi><\/math> und f\u00fcr alle <span class=\"maperiod\"><math display=\"inline\"><mi>j<\/mi><\/math><\/span><span class=\"period\">.<\/span> In diesem Fall ergibt sich die Summe zu&nbsp;<math display=\"inline\"><msubsup><mrow><mi class=\"MathClass-op\"> \u2211<\/mi><mo> <\/mo> <\/mrow><mrow><mi>j<\/mi><mo class=\"MathClass-rel\">=<\/mo><mn>1<\/mn><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msubsup><mi>z<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mi>n<\/mi><mi>z<\/mi><\/math> f\u00fcr alle <math display=\"inline\"><mi>n<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2115<\/mi><\/math> und <math display=\"inline\"><mi>z<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2102<\/mi><\/math> (oder <math display=\"inline\"><mi>z<\/mi><\/math> in einem Vektorraum). Im Falle des Produkts erhalten wir aber die Definition der <span class=\"ecbx-1095\">Potenzfunktion<\/span> f\u00fcr <math display=\"inline\"><mi>z<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2102<\/mi><\/math> und <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\"><msup><mrow><mi>z<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msup> <mo class=\"MathClass-rel\">=<\/mo><munderover accent=\"false\" accentunder=\"false\"><mrow><mo> \u220f<\/mo> <\/mrow><mrow><mi>j<\/mi><mo class=\"MathClass-rel\">=<\/mo><mn>1<\/mn><\/mrow><mrow><mi>n<\/mi><\/mrow><\/munderover><mi>z<\/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\">die somit rekursiv durch <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><msup><mrow><mi>z<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msup> <mo class=\"MathClass-rel\">=<\/mo> <mi>z<\/mi><mstyle class=\"text\"><mtext>&nbsp;und&nbsp;<\/mtext><\/mstyle><msup><mrow><mi>z<\/mi><\/mrow><mrow><mi>n<\/mi><mo class=\"MathClass-bin\">+<\/mo><mn>1<\/mn><\/mrow><\/msup> <mo class=\"MathClass-rel\">=<\/mo> <msup><mrow><mi>z<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msup><mi>z<\/mi><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">f\u00fcr alle <math display=\"inline\"><mi>n<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2115<\/mi><\/math> definiert ist. Wir nennen&nbsp;<math display=\"inline\"><mi>z<\/mi><\/math> die <span class=\"ecbx-1095\">Basis <\/span>und&nbsp;<math display=\"inline\"><mi>n<\/mi><\/math> den <span class=\"ecbx-1095\">Exponenten<\/span>. Wir erweitern diese Definition durch <\/p><math display=\"block\"><mtable class=\"align\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><msup><mrow><mi>z<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msup> <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=\"x1-77003r2\" \/><mstyle class=\"maketag\"><mtext>(3.2)<\/mtext><\/mstyle><mspace class=\"nbsp\" width=\"0.33em\" \/> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">f\u00fcr alle <math display=\"inline\"><mi>z<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2102<\/mi><\/math> (insbesondere<button class=\"hover-trigger\" style=\"vertical-align: super;font: smaller\">\u2020<\/button><span class=\"hover-text\"><span class=\"marginpar\">\u2020 Da <math display=\"inline\"><mi>z<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2102<\/mi><mo class=\"MathClass-rel\">\u21a6<\/mo> <msup><mrow><mi>z<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msup><\/math> die konstante Funktion mit Wert <math display=\"inline\"><mn>1<\/mn><\/math> darstellt, w\u00e4re es sehr eigenartig (und im n\u00e4chsten Abschnitt bei der Diskussion von Polynomen extrem st\u00f6rend), wenn wir diese Funktion f\u00fcr <math display=\"inline\"><mi>z<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mn>0<\/mn><\/math> undefiniert lassen oder mit einem anderen Wert versehen. Trotzdem ist der Ausdruck <math display=\"inline\"><msup><mrow><mn>0<\/mn><\/mrow><mrow><mn>0<\/mn> <\/mrow> <\/msup> <\/math> undefiniert, wenn dieser losgel\u00f6st von der Diskussion der Potenzfunktion <math display=\"inline\"><mi>z<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2102<\/mi><mo class=\"MathClass-rel\">\u21a6<\/mo> <msup><mrow><mi>z<\/mi><\/mrow><mrow><mi>n<\/mi> <\/mrow> <\/msup> <\/math> f\u00fcr <math display=\"inline\"><mi>n<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mn>0<\/mn><\/math> auftritt.<\/span><\/span> f\u00fcr <math display=\"inline\"><mi>z<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mn>0<\/mn><\/math>) und <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><msup><mrow><mi>z<\/mi><\/mrow><mrow><mo class=\"MathClass-bin\">\u2212<\/mo><mi>n<\/mi><\/mrow><\/msup> <mo class=\"MathClass-rel\">=<\/mo> <msup><mrow><mo class=\"MathClass-open\">(<\/mo><msup><mrow><mi>z<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msup><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mo class=\"MathClass-bin\">\u2212<\/mo><mn>1<\/mn><\/mrow><\/msup><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">f\u00fcr alle <math display=\"inline\"><mi>z<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <msup><mrow><mi>\u2102<\/mi><\/mrow><mrow><mo class=\"MathClass-bin\">\u00d7<\/mo><\/mrow><\/msup> <mo class=\"MathClass-punc\">:<\/mo><mo class=\"MathClass-rel\">=<\/mo> <mi>\u2102<\/mi> <mo class=\"MathClass-bin\">\u2216<\/mo><mrow><mo fence=\"true\" form=\"prefix\"> {<\/mo><mrow><mn>0<\/mn><\/mrow><mo fence=\"true\" form=\"postfix\">}<\/mo><\/mrow><\/math> und <span class=\"maperiod\"><math display=\"inline\"><mi>n<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2115<\/mi><\/math><\/span><span class=\"period\">.<\/span> Die n\u00e4chste \u00dcbung zeigt, dass die so definierte Potenzfunktion die \u00fcblichen Rechenregeln erf\u00fcllt. <\/p> <div class=\"me melemma\"> <div class=\"wp-nocaption \"><\/div><h4 id=\"zd9cc817b329e\"> <a id=\"x1-77004r2\"><\/a> <span class=\"ecbx-1095\">Wichtige <\/span><span class=\"ecbx-1095\">\u00dc<\/span><span class=\"ecbx-1095\">bung 3.2.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Beweisen Sie <\/span><span class=\"maperiod\"><math display=\"inline\"><msup><mrow><mo class=\"MathClass-open\">(<\/mo><mi>z<\/mi><mi>w<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mi>m<\/mi><\/mrow><\/msup> <mo class=\"MathClass-rel\">=<\/mo> <msup><mrow><mi>z<\/mi><\/mrow><mrow><mi>m<\/mi><\/mrow><\/msup><msup><mrow><mi>w<\/mi><\/mrow><mrow><mi>m<\/mi><\/mrow><\/msup><\/math><\/span><span class=\"period\">,<\/span> <math display=\"inline\"><msup><mrow><mi>z<\/mi><\/mrow><mrow><mi>m<\/mi><mo class=\"MathClass-bin\">+<\/mo><mi>n<\/mi> <\/mrow> <\/msup> <mo class=\"MathClass-rel\">=<\/mo> <msup><mrow><mi>z<\/mi><\/mrow><mrow><mi>m<\/mi> <\/mrow> <\/msup> <msup><mrow><mi>z<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msup><\/math> <span class=\"ecti-1095\">und<\/span> <math display=\"inline\"><msup><mrow><mo class=\"MathClass-open\">(<\/mo><msup><mrow><mi>z<\/mi><\/mrow><mrow><mi>m<\/mi> <\/mrow> <\/msup> <mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mi>n<\/mi> <\/mrow> <\/msup> <mo class=\"MathClass-rel\">=<\/mo> <msup><mrow><mi>z<\/mi><\/mrow><mrow><mi>m<\/mi><mi>n<\/mi><\/mrow><\/msup><\/math> <span class=\"ecti-1095\">zuerst f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r<\/span> <span class=\"ecti-1095\">alle <\/span><math display=\"inline\"><mi>z<\/mi><mo class=\"MathClass-punc\">,<\/mo> <mi>w<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2102<\/mi><\/math> <span class=\"ecti-1095\">und<\/span> <math display=\"inline\"><mi>m<\/mi><mo class=\"MathClass-punc\">,<\/mo> <mi>n<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <msub><mrow><mi>\u2115<\/mi><\/mrow><mrow><mn>0<\/mn> <\/mrow> <\/msub> <\/math> <span class=\"ecti-1095\">mit vollst<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">ndiger<\/span> <span class=\"ecti-1095\">Induktion und dann f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r alle <\/span><math display=\"inline\"><mi>z<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>w<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <msup><mrow><mi>\u2102<\/mi><\/mrow><mrow><mo class=\"MathClass-bin\">\u00d7<\/mo><\/mrow><\/msup><\/math> <span class=\"ecti-1095\">und <\/span><math display=\"inline\"><mi>m<\/mi><mo class=\"MathClass-punc\">,<\/mo> <mi>n<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2124<\/mi><\/math><span class=\"ecti-1095\">. <\/span><\/p><details><summary style=\"color:#FF7F00\"><span class=\"ecti-1095\">Teill<\/span><span class=\"ecti-1095\">\u00f6<\/span><span class=\"ecti-1095\">sung.<\/span><\/summary><p class=\"indent\" style=\"margin-top: 0\"> <span class=\"ecti-1095\">Wir beweisen <\/span><math display=\"inline\"><msup><mrow><mo class=\"MathClass-open\">(<\/mo><mi>z<\/mi><mi>w<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mi>m<\/mi><\/mrow><\/msup> <mo class=\"MathClass-rel\">=<\/mo> <msup><mrow><mi>z<\/mi><\/mrow><mrow><mi>m<\/mi><\/mrow><\/msup><msup><mrow><mi>w<\/mi><\/mrow><mrow><mi>m<\/mi><\/mrow><\/msup><\/math> <span class=\"ecti-1095\">f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r alle <\/span><math display=\"inline\"><mi>z<\/mi><mo class=\"MathClass-punc\">,<\/mo> <mi>w<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2102<\/mi><\/math> <span class=\"ecti-1095\">und<\/span> <math display=\"inline\"><mi>m<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <msub><mrow><mi>\u2115<\/mi><\/mrow><mrow><mn>0<\/mn> <\/mrow> <\/msub> <\/math> <span class=\"ecti-1095\">mittels<\/span> <span class=\"ecti-1095\">Induktion nach <\/span><span class=\"maperiod\"><math display=\"inline\"><mi>m<\/mi><\/math><\/span><span class=\"period\">.<\/span> <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\">=<\/mo> <mn>0<\/mn><\/math> <span class=\"ecti-1095\">ergibt<\/span> <span class=\"ecti-1095\">sich <\/span><math display=\"inline\"><mn>1<\/mn> <mo class=\"MathClass-rel\">=<\/mo> <mn>1<\/mn> <mo class=\"MathClass-bin\">\u22c5<\/mo> <mn>1<\/mn><\/math> <span class=\"ecti-1095\">nach Definition. Angenommen die Aussage gilt bereits f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r ein<\/span> <math display=\"inline\"><mi>m<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <msub><mrow><mi>\u2115<\/mi><\/mrow><mrow><mn>0<\/mn> <\/mrow> <\/msub> <\/math><span class=\"ecti-1095\">, dann<\/span> <span class=\"ecti-1095\">gilt ebenso<\/span> <\/p> <table id=\"z7fa83074ab6a\" class=\"equation-star\"><tr><td> <math class=\"equation\" display=\"block\"> <msup><mrow><mo class=\"MathClass-open\">(<\/mo><mi>z<\/mi><mi>w<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mi>m<\/mi><mo class=\"MathClass-bin\">+<\/mo><mn>1<\/mn><\/mrow><\/msup> <mo class=\"MathClass-rel\">=<\/mo> <msup><mrow><mo class=\"MathClass-open\">(<\/mo><mi>z<\/mi><mi>w<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mi>m<\/mi><\/mrow><\/msup><mo class=\"MathClass-open\">(<\/mo><mi>z<\/mi><mi>w<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <msup><mrow><mi>z<\/mi><\/mrow><mrow><mi>m<\/mi><\/mrow><\/msup><msup><mrow><mi>w<\/mi><\/mrow><mrow><mi>m<\/mi><\/mrow><\/msup><mi>z<\/mi><mi>w<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <msup><mrow><mi>z<\/mi><\/mrow><mrow><mi>m<\/mi><mo class=\"MathClass-bin\">+<\/mo><mn>1<\/mn><\/mrow><\/msup><msup><mrow><mi>w<\/mi><\/mrow><mrow><mi>m<\/mi><mo class=\"MathClass-bin\">+<\/mo><mn>1<\/mn><\/mrow><\/msup><mo class=\"MathClass-punc\">.<\/mo> <\/math><\/td><\/tr><\/table> <p class=\"indent\"><span class=\"ecti-1095\">Dies beweist den Induktionsschritt und damit die gew<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">nschte Aussage. <\/span><\/p><\/details>  <\/div> <div class=\"me meexample\"> <div class=\"wp-nocaption \"><\/div><h4 id=\"zc9b857a8d07e\"> <span class=\"ecti-1095\">Bemerkung.<\/span><\/h4> <p class=\"indent\">Formal gesehen  sollten  wir  auch  alle  weiteren  Rechenregeln  in  diesem  Abschnitt  mit vollst\u00e4ndiger Induktion beweisen. Da uns diese Beweise aber sehr wenig lehren, werden wir darauf verzichten. <\/p> <\/div> <a id=\"x1-77005r75\"><\/a> <h4 id=\"z8b67de9a107d\" class=\"subsectionHead\"><span class=\"titlemark\">3.1.1 <\/span> <a id=\"x1-780001\"><\/a>Rechenregeln f\u00fcr die Summe<\/h4> <p class=\"noindent\">Die Summe erf\u00fcllt f\u00fcr gegebene ganze Zahlen <math display=\"inline\"><mi>m<\/mi><mo class=\"MathClass-punc\">,<\/mo> <mi>n<\/mi><\/math> mit <math display=\"inline\"><mi>m<\/mi> <mo class=\"MathClass-rel\">\u2264<\/mo> <mi>n<\/mi><\/math> die Gleichungen <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><munderover accent=\"false\" accentunder=\"false\"><mrow><mo> \u2211<\/mo> <\/mrow><mrow><mi>k<\/mi><mo class=\"MathClass-rel\">=<\/mo><mi>m<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/munderover><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>a<\/mi><\/mrow><mrow> <mi>k<\/mi><\/mrow><\/msub> <mo class=\"MathClass-bin\">+<\/mo> <msub><mrow><mi>b<\/mi><\/mrow><mrow><mi>k<\/mi><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">=<\/mo><munderover accent=\"false\" accentunder=\"false\"><mrow><mo> \u2211<\/mo> <\/mrow><mrow><mi>k<\/mi><mo class=\"MathClass-rel\">=<\/mo><mi>m<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/munderover><msub><mrow><mi>a<\/mi><\/mrow><mrow> <mi>k<\/mi><\/mrow><\/msub> <mo class=\"MathClass-bin\">+<\/mo><munderover accent=\"false\" accentunder=\"false\"><mrow><mo> \u2211<\/mo> <\/mrow><mrow><mi>k<\/mi><mo class=\"MathClass-rel\">=<\/mo><mi>m<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/munderover><msub><mrow><mi>b<\/mi><\/mrow><mrow> <mi>k<\/mi><\/mrow><\/msub><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">und <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><munderover accent=\"false\" accentunder=\"false\"><mrow><mo> \u2211<\/mo> <\/mrow><mrow><mi>k<\/mi><mo class=\"MathClass-rel\">=<\/mo><mi>m<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/munderover><mo class=\"MathClass-open\">(<\/mo><mi>c<\/mi><msub><mrow><mi>a<\/mi><\/mrow><mrow> <mi>k<\/mi><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <mi>c<\/mi><munderover accent=\"false\" accentunder=\"false\"><mrow><mo>\u2211<\/mo> <\/mrow><mrow><mi>k<\/mi><mo class=\"MathClass-rel\">=<\/mo><mi>m<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/munderover><msub><mrow><mi>a<\/mi><\/mrow><mrow> <mi>k<\/mi><\/mrow><\/msub><mo class=\"MathClass-punc\">,<\/mo><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">wobei <span class=\"maperiod\"><math display=\"inline\"><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>m<\/mi> <\/mrow> <\/msub> <mo class=\"MathClass-punc\">,<\/mo> <mi class=\"MathClass-op\">\u2026<\/mi><mo> <\/mo> <mo class=\"MathClass-punc\">,<\/mo><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><\/math><\/span><span class=\"period\">,<\/span> <math display=\"inline\"><msub><mrow><mi>b<\/mi><\/mrow><mrow><mi>m<\/mi> <\/mrow> <\/msub> <mo class=\"MathClass-punc\">,<\/mo> <mi class=\"MathClass-op\">\u2026<\/mi><mo> <\/mo> <mo class=\"MathClass-punc\">,<\/mo> <msub><mrow><mi>b<\/mi><\/mrow><mrow><mi>n<\/mi> <\/mrow> <\/msub> <\/math> in einem reellen (respektive komplexen) Vektorraum&nbsp;<math display=\"inline\"><mi>V<\/mi> <\/math> liegen und <math display=\"inline\"><mi>c<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u211d<\/mi><\/math> (respektive <math display=\"inline\"><mi>c<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2102<\/mi><\/math>) ein Skalar ist. (Die erste Eigenschaft ist eine Mischung aus Assoziativgesetz und Kommutativgesetz f\u00fcr die Addition, und die zweite Eigenschaft ist eine Verallgemeinerung des Distributivgesetzes.) <\/p><p class=\"indent\">Diese beiden Eigenschaften (die Summe wird auf die Summe und das skalare Vielfache auf das skalare Vielfache abgebildet) werden auch als <span class=\"ecbx-1095\">Linearit<\/span><span class=\"ecbx-1095\">\u00e4<\/span><span class=\"ecbx-1095\">t der<\/span> <span class=\"ecbx-1095\">Abbildung<\/span>&nbsp;<math display=\"inline\"><mi class=\"MathClass-op\"> \u2211<\/mi><mo> <\/mo> <\/math> bezeichnet, wobei&nbsp;<math display=\"inline\"><mi class=\"MathClass-op\"> \u2211<\/mi><mo> <\/mo> <\/math> auf dem Vektorraum <math display=\"inline\"><msup><mrow><mi>V<\/mi> <\/mrow><mrow><mo class=\"MathClass-open\">{<\/mo><mi>m<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi class=\"MathClass-op\">\u2026<\/mi><mo> <\/mo><mo class=\"MathClass-punc\">,<\/mo><mi>n<\/mi><mo class=\"MathClass-close\">}<\/mo><\/mrow><\/msup><\/math> der Funktionen von&nbsp;<math display=\"inline\"><mo class=\"MathClass-open\">{<\/mo><mi>m<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi class=\"MathClass-op\">\u2026<\/mi><mo> <\/mo><mo class=\"MathClass-punc\">,<\/mo><mi>n<\/mi><mo class=\"MathClass-close\">}<\/mo><\/math> nach <math display=\"inline\"><mi>V<\/mi> <\/math> definiert ist, den Vektorraum <math display=\"inline\"><mi>V<\/mi> <\/math> als Zielbereich besitzt, und&nbsp;<math display=\"inline\"><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>m<\/mi><\/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-close\">)<\/mo> <mo class=\"MathClass-rel\">\u2208<\/mo> <msup><mrow><mi>V<\/mi> <\/mrow><mrow><mo class=\"MathClass-open\">{<\/mo><mi>m<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi class=\"MathClass-op\">\u2026<\/mi><mo> <\/mo><mo class=\"MathClass-punc\">,<\/mo><mi>n<\/mi><mo class=\"MathClass-close\">}<\/mo><\/mrow><\/msup><\/math> auf&nbsp;<math display=\"inline\"><msubsup><mrow><mi class=\"MathClass-op\"> \u2211<\/mi><mo> <\/mo> <\/mrow><mrow><mi>k<\/mi><mo class=\"MathClass-rel\">=<\/mo><mi>m<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msubsup><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>k<\/mi><\/mrow><\/msub><\/math> abbildet. Wie der Name sagt, wird Linearit\u00e4t ausf\u00fchrlicher in der Linearen Algebra besprochen. Es handelt sich dabei aber auch um eine wichtige Eigenschaft f\u00fcr die Analysis, welche also h\u00e4ufig auftreten. <\/p><p class=\"indent\">Des Weiteren gilt die Formel f\u00fcr die <span class=\"ecbx-1095\">Teleskopsumme<\/span> <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><munderover accent=\"false\" accentunder=\"false\"><mrow><mo>\u2211<\/mo> <\/mrow><mrow><mi>k<\/mi><mo class=\"MathClass-rel\">=<\/mo><mi>m<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/munderover><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>a<\/mi><\/mrow><mrow> <mi>k<\/mi><mo class=\"MathClass-bin\">+<\/mo><mn>1<\/mn><\/mrow><\/msub> <mo class=\"MathClass-bin\">\u2212<\/mo> <msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>k<\/mi><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><\/mtd> <mtd class=\"align-even\"> <mo class=\"MathClass-rel\">=<\/mo> <mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>m<\/mi><mo class=\"MathClass-bin\">+<\/mo><mn>1<\/mn><\/mrow><\/msub> <mo class=\"MathClass-bin\">\u2212<\/mo> <msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>m<\/mi><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">+<\/mo> <mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>m<\/mi><mo class=\"MathClass-bin\">+<\/mo><mn>2<\/mn><\/mrow><\/msub> <mo class=\"MathClass-bin\">\u2212<\/mo> <msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>m<\/mi><mo class=\"MathClass-bin\">+<\/mo><mn>1<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">+<\/mo> <mi class=\"MathClass-op\">\u2026<\/mi><mo> <\/mo> <mo class=\"MathClass-bin\">+<\/mo> <mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub> <mo class=\"MathClass-bin\">\u2212<\/mo> <msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi><mo class=\"MathClass-bin\">\u2212<\/mo><mn>1<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">+<\/mo> <mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi><mo class=\"MathClass-bin\">+<\/mo><mn>1<\/mn><\/mrow><\/msub> <mo class=\"MathClass-bin\">\u2212<\/mo> <msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\" \/> <mtd class=\"align-label\"> <mspace width=\"2em\" \/><\/mtd><\/mtr><mtr><mtd class=\"align-odd\" columnalign=\"right\" \/> <mtd class=\"align-even\"> <mo class=\"MathClass-rel\">=<\/mo> <msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi><mo class=\"MathClass-bin\">+<\/mo><mn>1<\/mn><\/mrow><\/msub> <mo class=\"MathClass-bin\">\u2212<\/mo> <msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>m<\/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\">wobei <math display=\"inline\"><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>m<\/mi> <\/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><mo class=\"MathClass-bin\">+<\/mo><mn>1<\/mn><\/mrow><\/msub><\/math> in einem reellen oder einem komplexen Vektorraum liegen. Formaler argumentiert gilt                                                                                                                                                                           <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><munderover accent=\"false\" accentunder=\"false\"><mrow><mo> \u2211<\/mo> <\/mrow><mrow><mi>k<\/mi><mo class=\"MathClass-rel\">=<\/mo><mi>m<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/munderover><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>a<\/mi><\/mrow><mrow> <mi>k<\/mi><mo class=\"MathClass-bin\">+<\/mo><mn>1<\/mn><\/mrow><\/msub> <mo class=\"MathClass-bin\">\u2212<\/mo> <msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>k<\/mi><\/mrow><\/msub><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><mi>m<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/munderover><msub><mrow><mi>a<\/mi><\/mrow><mrow> <mi>k<\/mi><mo class=\"MathClass-bin\">+<\/mo><mn>1<\/mn><\/mrow><\/msub> <mo class=\"MathClass-bin\">\u2212<\/mo><munderover accent=\"false\" accentunder=\"false\"><mrow><mo>\u2211<\/mo> <\/mrow><mrow><mi>k<\/mi><mo class=\"MathClass-rel\">=<\/mo><mi>m<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/munderover><msub><mrow><mi>a<\/mi><\/mrow><mrow> <mi>k<\/mi><\/mrow><\/msub> <mo class=\"MathClass-rel\">=<\/mo><munderover accent=\"false\" accentunder=\"false\"><mrow><mo> \u2211<\/mo> <\/mrow><mrow><mi>j<\/mi><mo class=\"MathClass-rel\">=<\/mo><mi>m<\/mi><mo class=\"MathClass-bin\">+<\/mo><mn>1<\/mn><\/mrow><mrow><mi>n<\/mi><mo class=\"MathClass-bin\">+<\/mo><mn>1<\/mn><\/mrow><\/munderover><msub><mrow><mi>a<\/mi><\/mrow><mrow> <mi>j<\/mi><\/mrow><\/msub> <mo class=\"MathClass-bin\">\u2212<\/mo><munderover accent=\"false\" accentunder=\"false\"><mrow><mo>\u2211<\/mo> <\/mrow><mrow><mi>k<\/mi><mo class=\"MathClass-rel\">=<\/mo><mi>m<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/munderover><msub><mrow><mi>a<\/mi><\/mrow><mrow> <mi>k<\/mi><\/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> <mstyle><mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow \/><mo fence=\"true\" form=\"postfix\" \/><\/mrow><\/mstyle><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi><mo class=\"MathClass-bin\">+<\/mo><mn>1<\/mn><\/mrow><\/msub> <mo class=\"MathClass-bin\">+<\/mo><munderover accent=\"false\" accentunder=\"false\"><mrow><mo> \u2211<\/mo> <\/mrow><mrow><mi>j<\/mi><mo class=\"MathClass-rel\">=<\/mo><mi>m<\/mi><mo class=\"MathClass-bin\">+<\/mo><mn>1<\/mn><\/mrow><mrow><mi>n<\/mi><\/mrow><\/munderover><msub><mrow><mi>a<\/mi><\/mrow><mrow> <mi>j<\/mi><\/mrow><\/msub><mstyle><mrow><mo fence=\"true\" form=\"prefix\"> )<\/mo><mrow \/><mo fence=\"true\" form=\"postfix\" \/><\/mrow><\/mstyle> <mo class=\"MathClass-bin\">\u2212<\/mo><mstyle><mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow \/><mo fence=\"true\" form=\"postfix\" \/><\/mrow><\/mstyle><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>m<\/mi><\/mrow><\/msub> <mo class=\"MathClass-bin\">+<\/mo><munderover accent=\"false\" accentunder=\"false\"><mrow><mo> \u2211<\/mo> <\/mrow><mrow><mi>k<\/mi><mo class=\"MathClass-rel\">=<\/mo><mi>m<\/mi><mo class=\"MathClass-bin\">+<\/mo><mn>1<\/mn><\/mrow><mrow><mi>n<\/mi><\/mrow><\/munderover><msub><mrow><mi>a<\/mi><\/mrow><mrow> <mi>k<\/mi><\/mrow><\/msub><mstyle><mrow><mo fence=\"true\" form=\"prefix\"> )<\/mo><mrow \/><mo fence=\"true\" form=\"postfix\" \/><\/mrow><\/mstyle> <mo class=\"MathClass-rel\">=<\/mo> <msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi><mo class=\"MathClass-bin\">+<\/mo><mn>1<\/mn><\/mrow><\/msub> <mo class=\"MathClass-bin\">\u2212<\/mo> <msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>m<\/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\">wie bereits behauptet. Die Formel f\u00fcr die Teleskopsumme l\u00e4sst sich zur Abel-Summationsformel verallgemeinern, welche \u00fcberraschend viele Anwendungen in der Analysis und Zahlentheorie findet. <\/p> <div class=\"me meexample\"> <div class=\"wp-nocaption \"><\/div><h4 id=\"zfd32429ac61d\"> <a id=\"x1-78001r3\"><\/a> <span class=\"ecbx-1095\">\u00dc<\/span><span class=\"ecbx-1095\">bung 3.3 <\/span>(Abel-Summation)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Seien <\/span><span class=\"maperiod\"><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-punc\">,<\/mo><msub><mrow><mi>b<\/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>b<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2102<\/mi><\/math><\/span><span class=\"period\">.<\/span> <span class=\"ecti-1095\">Wir setzen <\/span><math display=\"inline\"><msub><mrow><mi>A<\/mi><\/mrow><mrow><mi>k<\/mi><\/mrow><\/msub> <mo class=\"MathClass-rel\">=<\/mo><msubsup><mrow><mi class=\"MathClass-op\"> \u2211<\/mi><mo> <\/mo> <\/mrow><mrow><mi>j<\/mi><mo class=\"MathClass-rel\">=<\/mo><mn>1<\/mn><\/mrow><mrow><mi>k<\/mi><\/mrow><\/msubsup><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>j<\/mi><\/mrow><\/msub><\/math> <span class=\"ecti-1095\">f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r <\/span><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> <span class=\"ecti-1095\">mit <\/span><span class=\"maperiod\"><math display=\"inline\"><mi>k<\/mi> <mo class=\"MathClass-rel\">\u2264<\/mo> <mi>n<\/mi><\/math><\/span><span class=\"period\">.<\/span> <span class=\"ecti-1095\">Zeigen Sie die Abel-Summationsformel<\/span> <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><munderover accent=\"false\" accentunder=\"false\"><mrow><mo>\u2211<\/mo> <\/mrow><mrow><mi>k<\/mi><mo class=\"MathClass-rel\">=<\/mo><mn>1<\/mn><\/mrow><mrow><mi>n<\/mi><\/mrow><\/munderover><msub><mrow><mi>a<\/mi><\/mrow><mrow> <mi>k<\/mi><\/mrow><\/msub><msub><mrow><mi>b<\/mi><\/mrow><mrow><mi>k<\/mi><\/mrow><\/msub> <mo class=\"MathClass-rel\">=<\/mo> <msub><mrow><mi>A<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><msub><mrow><mi>b<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub> <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><mo class=\"MathClass-bin\">\u2212<\/mo><mn>1<\/mn><\/mrow><\/munderover><msub><mrow><mi>A<\/mi><\/mrow><mrow> <mi>k<\/mi><\/mrow><\/msub> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><msub><mrow><mi>b<\/mi><\/mrow><mrow><mi>k<\/mi><\/mrow><\/msub> <mo class=\"MathClass-bin\">\u2212<\/mo> <msub><mrow><mi>b<\/mi><\/mrow><mrow><mi>k<\/mi><mo class=\"MathClass-bin\">+<\/mo><mn>1<\/mn><\/mrow><\/msub><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-punc\">.<\/mo><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\"><span class=\"ecti-1095\">Verwenden Sie dazu die Gleichung <\/span><math display=\"inline\"><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>k<\/mi><\/mrow><\/msub> <mo class=\"MathClass-rel\">=<\/mo> <msub><mrow><mi>A<\/mi><\/mrow><mrow><mi>k<\/mi><\/mrow><\/msub> <mo class=\"MathClass-bin\">\u2212<\/mo> <msub><mrow><mi>A<\/mi><\/mrow><mrow><mi>k<\/mi><mo class=\"MathClass-bin\">\u2212<\/mo><mn>1<\/mn><\/mrow><\/msub><\/math> <span class=\"ecti-1095\">f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r alle <\/span><math display=\"inline\"><mi>k<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2115<\/mi><\/math> <span class=\"ecti-1095\">mit <\/span><span class=\"maperiod\"><math display=\"inline\"><mi>k<\/mi> <mo class=\"MathClass-rel\">\u2264<\/mo> <mi>n<\/mi><\/math><\/span><span class=\"period\">.<\/span> <span class=\"ecti-1095\">Wenden Sie des Weiteren die Abel-Summation auf die Summe<\/span> <math display=\"inline\"><msubsup><mrow><mi class=\"MathClass-op\">\u2211<\/mi><mo> <\/mo> <\/mrow><mrow><mi>k<\/mi><mo class=\"MathClass-rel\">=<\/mo><mn>1<\/mn><\/mrow><mrow><mn>2<\/mn><mi>n<\/mi><\/mrow><\/msubsup><mfrac><mrow><msup><mrow><mo class=\"MathClass-open\">(<\/mo><mo class=\"MathClass-bin\">\u2212<\/mo><mn>1<\/mn><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mi>k<\/mi><\/mrow><\/msup><\/mrow> <mrow><mi>k<\/mi><\/mrow><\/mfrac> <\/math> <span class=\"ecti-1095\">an.<\/span> <\/p><div class=\"wp-nocaption \"><\/div><details><summary style=\"color:#FF7F00\"><span class=\"ecti-1095\">L<\/span><span class=\"ecti-1095\">\u00f6<\/span><span class=\"ecti-1095\">sung.<\/span><\/summary><p class=\"indent\" style=\"margin-top: 0\"> <span class=\"ecti-1095\">Wir bemerken zuerst, dass <\/span><math display=\"inline\"><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>k<\/mi><\/mrow><\/msub> <mo class=\"MathClass-rel\">=<\/mo> <msub><mrow><mi>A<\/mi><\/mrow><mrow><mi>k<\/mi><\/mrow><\/msub> <mo class=\"MathClass-bin\">\u2212<\/mo> <msub><mrow><mi>A<\/mi><\/mrow><mrow><mi>k<\/mi><mo class=\"MathClass-bin\">\u2212<\/mo><mn>1<\/mn><\/mrow><\/msub><\/math> <span class=\"ecti-1095\">f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r alle <\/span><math display=\"inline\"><mi>k<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2115<\/mi><\/math> <span class=\"ecti-1095\">mit<\/span> <math display=\"inline\"><mi>k<\/mi> <mo class=\"MathClass-rel\">\u2264<\/mo> <mi>n<\/mi><\/math> <span class=\"ecti-1095\">auf Grund von<\/span> <math display=\"inline\"><msub><mrow><mi>A<\/mi><\/mrow><mrow><mn>0<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-rel\">=<\/mo> <mn>0<\/mn><\/math> <span class=\"ecti-1095\">und der rekursiven<\/span> <span class=\"ecti-1095\">Definition der Summe <\/span><math display=\"inline\"><msub><mrow><mi>A<\/mi><\/mrow><mrow><mi>k<\/mi><\/mrow><\/msub> <mo class=\"MathClass-rel\">=<\/mo><msubsup><mrow><mi class=\"MathClass-op\"> \u2211<\/mi><mo> <\/mo> <\/mrow><mrow><mi>j<\/mi><mo class=\"MathClass-rel\">=<\/mo><mn>1<\/mn><\/mrow><mrow><mi>k<\/mi><\/mrow><\/msubsup><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>j<\/mi><\/mrow><\/msub><\/math> <span class=\"ecti-1095\">gilt. Nun gehen wir wie vorgeschlagen vor und erhalten<\/span> <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><munderover accent=\"false\" accentunder=\"false\"><mrow><mo>\u2211<\/mo> <\/mrow><mrow><mi>k<\/mi><mo class=\"MathClass-rel\">=<\/mo><mn>1<\/mn><\/mrow><mrow><mi>n<\/mi><\/mrow><\/munderover><msub><mrow><mi>a<\/mi><\/mrow><mrow> <mi>k<\/mi><\/mrow><\/msub><msub><mrow><mi>b<\/mi><\/mrow><mrow><mi>k<\/mi><\/mrow><\/msub><\/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><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>A<\/mi><\/mrow><mrow> <mi>k<\/mi><\/mrow><\/msub> <mo class=\"MathClass-bin\">\u2212<\/mo> <msub><mrow><mi>A<\/mi><\/mrow><mrow><mi>k<\/mi><mo class=\"MathClass-bin\">\u2212<\/mo><mn>1<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><msub><mrow><mi>b<\/mi><\/mrow><mrow><mi>k<\/mi><\/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><munderover accent=\"false\" accentunder=\"false\"><mrow><mo> \u2211<\/mo> <\/mrow><mrow><mi>k<\/mi><mo class=\"MathClass-rel\">=<\/mo><mn>1<\/mn><\/mrow><mrow><mi>n<\/mi><\/mrow><\/munderover><msub><mrow><mi>A<\/mi><\/mrow><mrow> <mi>k<\/mi><\/mrow><\/msub><msub><mrow><mi>b<\/mi><\/mrow><mrow><mi>k<\/mi><\/mrow><\/msub> <mo class=\"MathClass-bin\">\u2212<\/mo><munderover accent=\"false\" accentunder=\"false\"><mrow><mo>\u2211<\/mo> <\/mrow><mrow><mi>j<\/mi><mo class=\"MathClass-rel\">=<\/mo><mn>0<\/mn><\/mrow><mrow><mi>n<\/mi><mo class=\"MathClass-bin\">\u2212<\/mo><mn>1<\/mn><\/mrow><\/munderover><msub><mrow><mi>A<\/mi><\/mrow><mrow> <mi>j<\/mi><\/mrow><\/msub><msub><mrow><mi>b<\/mi><\/mrow><mrow><mi>j<\/mi><mo class=\"MathClass-bin\">+<\/mo><mn>1<\/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> <msub><mrow><mi>A<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><msub><mrow><mi>b<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub> <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><mo class=\"MathClass-bin\">\u2212<\/mo><mn>1<\/mn><\/mrow><\/munderover><msub><mrow><mi>A<\/mi><\/mrow><mrow> <mi>k<\/mi><\/mrow><\/msub><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>b<\/mi><\/mrow><mrow><mi>k<\/mi><\/mrow><\/msub> <mo class=\"MathClass-bin\">\u2212<\/mo> <msub><mrow><mi>b<\/mi><\/mrow><mrow><mi>k<\/mi><mo class=\"MathClass-bin\">+<\/mo><mn>1<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-punc\">,<\/mo><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\" \/> <mtd class=\"align-label\"> <mspace width=\"2em\" \/><\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\"><span class=\"ecti-1095\">wobei wir die Linearit<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">t der Summe, die Rekursionsformel der Summe und Indexverschiebung<\/span> <span class=\"ecti-1095\">verwendet haben.<\/span> <\/p><p class=\"indent\"><span class=\"ecti-1095\">In dem Spezialfall <\/span><math display=\"inline\"><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>k<\/mi><\/mrow><\/msub> <mo class=\"MathClass-rel\">=<\/mo> <msup><mrow><mo class=\"MathClass-open\">(<\/mo><mo class=\"MathClass-bin\">\u2212<\/mo><mn>1<\/mn><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mi>k<\/mi><\/mrow><\/msup><\/math> <span class=\"ecti-1095\">f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r <\/span><math display=\"inline\"><mn>1<\/mn> <mo class=\"MathClass-rel\">\u2264<\/mo> <mi>k<\/mi> <mo class=\"MathClass-rel\">\u2264<\/mo> <mn>2<\/mn><mi>n<\/mi><\/math> <span class=\"ecti-1095\">ergibt sich (mittels<\/span> <span class=\"ecti-1095\">vollst<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">ndiger Induktion), dass <\/span><math display=\"inline\"><msub><mrow><mi>A<\/mi><\/mrow><mrow><mi>k<\/mi><\/mrow><\/msub> <mo class=\"MathClass-rel\">=<\/mo> <mn>0<\/mn><\/math> <span class=\"ecti-1095\">f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r alle geraden <\/span><span class=\"maperiod\"><math display=\"inline\"><mi>k<\/mi> <mo class=\"MathClass-rel\">\u2264<\/mo> <mn>2<\/mn><mi>n<\/mi><\/math><\/span><span class=\"period\">,<\/span> <span class=\"ecti-1095\">insbesondere <\/span><math display=\"inline\"><msub><mrow><mi>A<\/mi><\/mrow><mrow><mn>2<\/mn><mi>n<\/mi><\/mrow><\/msub> <mo class=\"MathClass-rel\">=<\/mo> <mn>0<\/mn><\/math><span class=\"ecti-1095\">, und<\/span> <math display=\"inline\"><msub><mrow><mi>A<\/mi><\/mrow><mrow><mi>k<\/mi> <\/mrow> <\/msub> <mo class=\"MathClass-rel\">=<\/mo> <mo class=\"MathClass-bin\">\u2212<\/mo><mn>1<\/mn><\/math> <span class=\"ecti-1095\">f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r alle ungeraden<\/span> <math display=\"inline\"><mi>k<\/mi> <mo class=\"MathClass-rel\">\u2264<\/mo> <mn>2<\/mn><mi>n<\/mi><\/math><span class=\"ecti-1095\">. Da die ungeraden<\/span> <span class=\"ecti-1095\">ganzen Zahlen <\/span><math display=\"inline\"><mi>k<\/mi><\/math> <span class=\"ecti-1095\">mit <\/span><math display=\"inline\"><mn>1<\/mn> <mo class=\"MathClass-rel\">\u2264<\/mo> <mi>k<\/mi> <mo class=\"MathClass-rel\">\u2264<\/mo> <mn>2<\/mn><mi>n<\/mi><\/math> <span class=\"ecti-1095\">genau<\/span> <span class=\"ecti-1095\">die Form <\/span><math display=\"inline\"><mi>k<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mn>2<\/mn><mi>\u2113<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <mn>1<\/mn><\/math> <span class=\"ecti-1095\">f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r <\/span><math display=\"inline\"><mi>\u2113<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mn>1<\/mn><mo class=\"MathClass-punc\">,<\/mo> <mi class=\"MathClass-op\">\u2026<\/mi><mo> <\/mo><mo class=\"MathClass-punc\">,<\/mo><mi>n<\/mi><\/math> <span class=\"ecti-1095\">haben, erhalten wir daraus<\/span> <\/p><table id=\"z3a544a809f19\" class=\"equation-star\"><tr><td> <math class=\"equation\" display=\"block\"> <munderover accent=\"false\" accentunder=\"false\"><mrow><mo>\u2211<\/mo> <\/mrow><mrow><mi>k<\/mi><mo class=\"MathClass-rel\">=<\/mo><mn>1<\/mn><\/mrow><mrow><mn>2<\/mn><mi>n<\/mi><\/mrow><\/munderover><msub><mrow><mi>b<\/mi><\/mrow><mrow> <mi>k<\/mi><\/mrow><\/msub> <mo class=\"MathClass-rel\">=<\/mo> <mo class=\"MathClass-bin\">\u2212<\/mo><munderover accent=\"false\" accentunder=\"false\"><mrow><mo>\u2211<\/mo> <\/mrow><mrow><mi>\u2113<\/mi><mo class=\"MathClass-rel\">=<\/mo><mn>1<\/mn><\/mrow><mrow><mi>n<\/mi><\/mrow><\/munderover> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><msub><mrow><mi>b<\/mi><\/mrow><mrow> <mn>2<\/mn><mi>\u2113<\/mi><mo class=\"MathClass-bin\">\u2212<\/mo><mn>1<\/mn><\/mrow><\/msub> <mo class=\"MathClass-bin\">\u2212<\/mo> <msub><mrow><mi>b<\/mi><\/mrow><mrow><mn>2<\/mn><mi>\u2113<\/mi><\/mrow><\/msub><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-punc\">.<\/mo> <\/math><\/td><\/tr><\/table> <p class=\"indent\"><span class=\"ecti-1095\">Also entspricht die Abel-Summation in diesem Fall einer Zusammenfassung von jeweils zwei benachbarten<\/span> <span class=\"ecti-1095\">Summanden. Falls <\/span><math display=\"inline\"><msub><mrow><mi>b<\/mi><\/mrow><mrow><mi>k<\/mi><\/mrow><\/msub> <mo class=\"MathClass-rel\">=<\/mo> <mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mi>k<\/mi><\/mrow><\/mfrac><\/math> <span class=\"ecti-1095\">f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r <\/span><math display=\"inline\"><mn>1<\/mn> <mo class=\"MathClass-rel\">\u2264<\/mo> <mi>k<\/mi> <mo class=\"MathClass-rel\">\u2264<\/mo> <mn>2<\/mn><mi>n<\/mi><\/math><span class=\"ecti-1095\">, k<\/span><span class=\"ecti-1095\">\u00f6<\/span><span class=\"ecti-1095\">nnen<\/span> <span class=\"ecti-1095\">wir noch <\/span><math display=\"inline\"><msub><mrow><mi>b<\/mi><\/mrow><mrow><mn>2<\/mn><mi>\u2113<\/mi><mo class=\"MathClass-bin\">\u2212<\/mo><mn>1<\/mn><\/mrow><\/msub> <mo class=\"MathClass-bin\">\u2212<\/mo> <msub><mrow><mi>b<\/mi><\/mrow><mrow><mn>2<\/mn><mi>\u2113<\/mi><\/mrow><\/msub> <mo class=\"MathClass-rel\">=<\/mo> <mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mn>2<\/mn><mi>\u2113<\/mi><mo class=\"MathClass-open\">(<\/mo><mn>2<\/mn><mi>\u2113<\/mi><mo class=\"MathClass-bin\">\u2212<\/mo><mn>1<\/mn><mo class=\"MathClass-close\">)<\/mo><\/mrow><\/mfrac><\/math> <span class=\"ecti-1095\">in diese Formel einsetzen.<\/span><\/p><\/details>  <\/div> <p class=\"indent\">Anstelle der Dreiecksungleichung werden wir oft auch folgende verallgemeinerte Dreiecksungleichung f\u00fcr Summen verwenden. <\/p> <div class=\"me melemma\"> <div class=\"wp-nocaption \"><\/div><h4 id=\"z095794f0b523\"> <a id=\"x1-78002r4\"><\/a> <span class=\"ecbx-1095\">Wichtige <\/span><span class=\"ecbx-1095\">\u00dc<\/span><span class=\"ecbx-1095\">bung 3.4 <\/span>(Verallgemeinerte Dreiecksungleichung)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Zeigen Sie, dass f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r alle Zahlen <\/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>\u2102<\/mi><\/math> <span class=\"ecti-1095\">die Ungleichung<\/span> <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mstyle><mrow><mo fence=\"true\" form=\"prefix\"> |<\/mo><mrow \/><mo fence=\"true\" form=\"postfix\" \/><\/mrow><\/mstyle><munderover accent=\"false\" accentunder=\"false\"><mrow><mo>\u2211<\/mo> <\/mrow><mrow><mi>i<\/mi><mo class=\"MathClass-rel\">=<\/mo><mn>1<\/mn><\/mrow><mrow><mi>n<\/mi><\/mrow><\/munderover><msub><mrow><mi>a<\/mi><\/mrow><mrow> <mi>i<\/mi><\/mrow><\/msub><mstyle><mrow><mo fence=\"true\" form=\"prefix\"> |<\/mo><mrow \/><mo fence=\"true\" form=\"postfix\" \/><\/mrow><\/mstyle> <mo class=\"MathClass-rel\">\u2264<\/mo><munderover accent=\"false\" accentunder=\"false\"><mrow><mo>\u2211<\/mo> <\/mrow><mrow><mi>i<\/mi><mo class=\"MathClass-rel\">=<\/mo><mn>1<\/mn><\/mrow><mrow><mi>n<\/mi><\/mrow><\/munderover><mo class=\"MathClass-rel\">|<\/mo><msub><mrow><mi>a<\/mi><\/mrow><mrow> <mi>i<\/mi><\/mrow><\/msub><mo class=\"MathClass-rel\">|<\/mo><mo class=\"MathClass-punc\">.<\/mo><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\"><span class=\"ecti-1095\">gilt.<\/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\">Wenden Sie Induktion nach <\/span><math display=\"inline\"><mi>n<\/mi><\/math> <span class=\"ecti-1095\">an und verwenden Sie die regul<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">re Dreiecksungleichung und die rekursive Definition der Summe f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r<\/span> <span class=\"ecti-1095\">den Induktionsschritt.<\/span><\/p><\/details>  <\/div> <p class=\"indent\">Manchmal wollen wir in einer Summe einen gewissen Summanden getrennt betrachten und dazu die Summe aufteilen. Dies kann dann zum Beispiel f\u00fcr&nbsp;<math display=\"inline\"><mn>1<\/mn> <mo class=\"MathClass-rel\">\u2264<\/mo> <mi>k<\/mi> <mo class=\"MathClass-rel\">\u2264<\/mo> <mi>n<\/mi><\/math> die Form <\/p> <table id=\"z7ec56ba905ce\" class=\"equation-star\"><tr><td> <math class=\"equation\" display=\"block\"> <munderover accent=\"false\" accentunder=\"false\"><mrow><mo>\u2211<\/mo> <\/mrow><mrow><mi>j<\/mi><mo class=\"MathClass-rel\">=<\/mo><mn>1<\/mn><\/mrow><mrow><mi>n<\/mi><\/mrow><\/munderover><msub><mrow><mi>a<\/mi><\/mrow><mrow> <mi>j<\/mi><\/mrow><\/msub> <mo class=\"MathClass-rel\">=<\/mo><munderover accent=\"false\" accentunder=\"false\"><mrow><mo> \u2211<\/mo> <\/mrow><mrow><mi>j<\/mi><mo class=\"MathClass-rel\">=<\/mo><mn>1<\/mn><\/mrow><mrow><mi>k<\/mi><\/mrow><\/munderover><msub><mrow><mi>a<\/mi><\/mrow><mrow> <mi>j<\/mi><\/mrow><\/msub> <mo class=\"MathClass-bin\">+<\/mo><munderover accent=\"false\" accentunder=\"false\"><mrow><mo> \u2211<\/mo> <\/mrow><mrow><mi>j<\/mi><mo class=\"MathClass-rel\">=<\/mo><mi>k<\/mi><mo class=\"MathClass-bin\">+<\/mo><mn>1<\/mn><\/mrow><mrow><mi>n<\/mi><\/mrow><\/munderover><msub><mrow><mi>a<\/mi><\/mrow><mrow> <mi>j<\/mi><\/mrow><\/msub> <mo class=\"MathClass-rel\">=<\/mo><munderover accent=\"false\" accentunder=\"false\"><mrow><mo> \u2211<\/mo> <\/mrow><mrow><mi>j<\/mi><mo class=\"MathClass-rel\">=<\/mo><mn>1<\/mn><\/mrow><mrow><mi>k<\/mi><mo class=\"MathClass-bin\">\u2212<\/mo><mn>1<\/mn><\/mrow><\/munderover><msub><mrow><mi>a<\/mi><\/mrow><mrow> <mi>j<\/mi><\/mrow><\/msub> <mo class=\"MathClass-bin\">+<\/mo> <msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>k<\/mi><\/mrow><\/msub> <mo class=\"MathClass-bin\">+<\/mo><munderover accent=\"false\" accentunder=\"false\"><mrow><mo> \u2211<\/mo> <\/mrow><mrow><mi>j<\/mi><mo class=\"MathClass-rel\">=<\/mo><mi>k<\/mi><mo class=\"MathClass-bin\">+<\/mo><mn>1<\/mn><\/mrow><mrow><mi>n<\/mi><\/mrow><\/munderover><msub><mrow><mi>a<\/mi><\/mrow><mrow> <mi>j<\/mi><\/mrow><\/msub> <\/math><\/td><\/tr><\/table> <p class=\"indent\">annehmen. In dem Spezialfall&nbsp;<math display=\"inline\"><mi>k<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mn>1<\/mn><\/math> sollte dies aber mit&nbsp;<math display=\"inline\"><msub><mrow><mi>a<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub> <mo class=\"MathClass-bin\">+<\/mo><msubsup><mrow><mi class=\"MathClass-op\"> \u2211<\/mi><mo> <\/mo> <\/mrow><mrow><mi>j<\/mi><mo class=\"MathClass-rel\">=<\/mo><mn>2<\/mn><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msubsup><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>j<\/mi><\/mrow><\/msub><\/math> und in dem Spezialfall&nbsp;<math display=\"inline\"><mi>k<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mi>n<\/mi><\/math> mit&nbsp;<math display=\"inline\"><msubsup><mrow><mi class=\"MathClass-op\"> \u2211<\/mi><mo> <\/mo> <\/mrow><mrow><mi>j<\/mi><mo class=\"MathClass-rel\">=<\/mo><mn>1<\/mn><\/mrow><mrow><mi>n<\/mi><mo class=\"MathClass-bin\">\u2212<\/mo><mn>1<\/mn><\/mrow><\/msubsup><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>j<\/mi><\/mrow><\/msub> <mo class=\"MathClass-bin\">+<\/mo> <msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><\/math> \u00fcbereinstimmen, was auf Grund unserer Definitionen&nbsp;<math display=\"inline\"><msubsup><mrow><mi class=\"MathClass-op\"> \u2211<\/mi><mo> <\/mo> <\/mrow><mrow><mi>j<\/mi><mo class=\"MathClass-rel\">=<\/mo><mn>1<\/mn><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msubsup><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>j<\/mi><\/mrow><\/msub> <mo class=\"MathClass-rel\">=<\/mo> <mn>0<\/mn><\/math> und&nbsp;<math display=\"inline\"><msubsup><mrow><mi class=\"MathClass-op\"> \u2211<\/mi><mo> <\/mo> <\/mrow><mrow><mi>j<\/mi><mo class=\"MathClass-rel\">=<\/mo><mi>n<\/mi><mo class=\"MathClass-bin\">+<\/mo><mn>1<\/mn><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msubsup><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>j<\/mi><\/mrow><\/msub> <mo class=\"MathClass-rel\">=<\/mo> <mn>0<\/mn><\/math> in der Tat gilt. (Der formale Beweis erfolgt wiederum mit Induktion nach <span class=\"maperiod\"><math display=\"inline\"><mi>n<\/mi> <mo class=\"MathClass-rel\">\u2265<\/mo> <mi>k<\/mi><\/math><\/span><span class=\"period\">.<\/span>) <a id=\"x1-78003r78\"><\/a> <\/p> <h4 id=\"z8611c8aa8059\" class=\"subsectionHead\"><span class=\"titlemark\">3.1.2 <\/span> <a id=\"x1-790002\"><\/a>Rechenregeln f\u00fcr das Produkt<\/h4> <p class=\"noindent\">F\u00fcr ganze Zahlen <math display=\"inline\"><mi>m<\/mi> <mo class=\"MathClass-rel\">\u2264<\/mo> <mi>n<\/mi><\/math> und <math display=\"inline\"><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>m<\/mi> <\/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-punc\">,<\/mo><msub><mrow><mi>b<\/mi><\/mrow><mrow><mi>m<\/mi><\/mrow><\/msub><mo class=\"MathClass-punc\">,<\/mo><mi class=\"MathClass-op\">\u2026<\/mi><mo> <\/mo><mo class=\"MathClass-punc\">,<\/mo><msub><mrow><mi>b<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2102<\/mi><\/math> gilt                                                                                                                                                                           <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><munderover accent=\"false\" accentunder=\"false\"><mrow><mo> \u220f<\/mo> <\/mrow><mrow><mi>k<\/mi><mo class=\"MathClass-rel\">=<\/mo><mi>m<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/munderover><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>a<\/mi><\/mrow><mrow> <mi>k<\/mi><\/mrow><\/msub><msub><mrow><mi>b<\/mi><\/mrow><mrow><mi>k<\/mi><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <mstyle><mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow \/><mo fence=\"true\" form=\"postfix\" \/><\/mrow><\/mstyle><munderover accent=\"false\" accentunder=\"false\"><mrow><mo>\u220f<\/mo> <\/mrow><mrow><mi>k<\/mi><mo class=\"MathClass-rel\">=<\/mo><mi>m<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/munderover><msub><mrow><mi>a<\/mi><\/mrow><mrow> <mi>k<\/mi><\/mrow><\/msub><mstyle><mrow><mo fence=\"true\" form=\"prefix\"> )<\/mo><mrow \/><mo fence=\"true\" form=\"postfix\" \/><\/mrow><\/mstyle><mstyle><mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow \/><mo fence=\"true\" form=\"postfix\" \/><\/mrow><\/mstyle><munderover accent=\"false\" accentunder=\"false\"><mrow><mo> \u220f<\/mo> <\/mrow><mrow><mi>k<\/mi><mo class=\"MathClass-rel\">=<\/mo><mi>m<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/munderover><msub><mrow><mi>b<\/mi><\/mrow><mrow> <mi>k<\/mi><\/mrow><\/msub><mstyle><mrow><mo fence=\"true\" form=\"prefix\"> )<\/mo><mrow \/><mo fence=\"true\" form=\"postfix\" \/><\/mrow><\/mstyle><mo class=\"MathClass-punc\">.<\/mo><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">Insbesondere ist f\u00fcr alle <math display=\"inline\"><mi>c<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2102<\/mi><\/math> <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><munderover accent=\"false\" accentunder=\"false\"><mrow><mo> \u220f<\/mo> <\/mrow><mrow><mi>k<\/mi><mo class=\"MathClass-rel\">=<\/mo><mi>m<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/munderover><mo class=\"MathClass-open\">(<\/mo><mi>c<\/mi><msub><mrow><mi>a<\/mi><\/mrow><mrow> <mi>k<\/mi><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <msup><mrow><mi>c<\/mi><\/mrow><mrow><mi>n<\/mi><mo class=\"MathClass-bin\">\u2212<\/mo><mi>m<\/mi><mo class=\"MathClass-bin\">+<\/mo><mn>1<\/mn><\/mrow><\/msup><mstyle><mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow \/><mo fence=\"true\" form=\"postfix\" \/><\/mrow><\/mstyle><munderover accent=\"false\" accentunder=\"false\"><mrow><mo> \u220f<\/mo> <\/mrow><mrow><mi>k<\/mi><mo class=\"MathClass-rel\">=<\/mo><mi>m<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/munderover><msub><mrow><mi>a<\/mi><\/mrow><mrow> <mi>k<\/mi><\/mrow><\/msub><mstyle><mrow><mo fence=\"true\" form=\"prefix\"> )<\/mo><mrow \/><mo fence=\"true\" form=\"postfix\" \/><\/mrow><\/mstyle><mo class=\"MathClass-punc\">.<\/mo><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">Des Weiteren gilt f\u00fcr alle <math display=\"inline\"><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>m<\/mi><\/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>\u2102<\/mi> <mo class=\"MathClass-bin\">\u2216<\/mo><mo class=\"MathClass-open\">{<\/mo><mn>0<\/mn><mo class=\"MathClass-close\">}<\/mo><\/math> die Formel f\u00fcr das <span class=\"ecbx-1095\">Teleskopprodukt<\/span> <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><munderover accent=\"false\" accentunder=\"false\"><mrow><mo>\u220f<\/mo> <\/mrow><mrow><mi>k<\/mi><mo class=\"MathClass-rel\">=<\/mo><mi>m<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/munderover><mfrac><mrow><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>k<\/mi><mo class=\"MathClass-bin\">+<\/mo><mn>1<\/mn><\/mrow><\/msub><\/mrow> <mrow><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>k<\/mi><\/mrow><\/msub><\/mrow><\/mfrac> <\/mtd> <mtd class=\"align-even\"> <mo class=\"MathClass-rel\">=<\/mo> <mstyle><mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow \/><mo fence=\"true\" form=\"postfix\" \/><\/mrow><\/mstyle><munderover accent=\"false\" accentunder=\"false\"><mrow><mo>\u220f<\/mo> <\/mrow><mrow><mi>k<\/mi><mo class=\"MathClass-rel\">=<\/mo><mi>m<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/munderover><msub><mrow><mi>a<\/mi><\/mrow><mrow> <mi>k<\/mi><mo class=\"MathClass-bin\">+<\/mo><mn>1<\/mn><\/mrow><\/msub><mstyle><mrow><mo fence=\"true\" form=\"prefix\"> )<\/mo><mrow \/><mo fence=\"true\" form=\"postfix\" \/><\/mrow><\/mstyle><mstyle><mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow \/><mo fence=\"true\" form=\"postfix\" \/><\/mrow><\/mstyle><munderover accent=\"false\" accentunder=\"false\"><mrow><mo> \u220f<\/mo> <\/mrow><mrow><mi>k<\/mi><mo class=\"MathClass-rel\">=<\/mo><mi>m<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/munderover> <mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>k<\/mi><\/mrow><\/msub><\/mrow><\/mfrac><mstyle><mrow><mo fence=\"true\" form=\"prefix\"> )<\/mo><mrow \/><mo fence=\"true\" form=\"postfix\" \/><\/mrow><\/mstyle> <mo class=\"MathClass-rel\">=<\/mo> <mstyle><mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow \/><mo fence=\"true\" form=\"postfix\" \/><\/mrow><\/mstyle><munderover accent=\"false\" accentunder=\"false\"><mrow><mo>\u220f<\/mo> <\/mrow><mrow><mi>k<\/mi><mo class=\"MathClass-rel\">=<\/mo><mi>m<\/mi><mo class=\"MathClass-bin\">+<\/mo><mn>1<\/mn><\/mrow><mrow><mi>n<\/mi><mo class=\"MathClass-bin\">+<\/mo><mn>1<\/mn><\/mrow><\/munderover><msub><mrow><mi>a<\/mi><\/mrow><mrow> <mi>k<\/mi><\/mrow><\/msub><mstyle><mrow><mo fence=\"true\" form=\"prefix\"> )<\/mo><mrow \/><mo fence=\"true\" form=\"postfix\" \/><\/mrow><\/mstyle><mstyle><mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow \/><mo fence=\"true\" form=\"postfix\" \/><\/mrow><\/mstyle><munderover accent=\"false\" accentunder=\"false\"><mrow><mo> \u220f<\/mo> <\/mrow><mrow><mi>k<\/mi><mo class=\"MathClass-rel\">=<\/mo><mi>m<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/munderover> <mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>k<\/mi><\/mrow><\/msub><\/mrow><\/mfrac><mstyle><mrow><mo fence=\"true\" form=\"prefix\"> )<\/mo><mrow \/><mo fence=\"true\" form=\"postfix\" \/><\/mrow><\/mstyle><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\" \/> <mtd class=\"align-label\"> <mspace width=\"2em\" \/><\/mtd><\/mtr><mtr><mtd class=\"align-odd\" columnalign=\"right\" \/> <mtd class=\"align-even\"> <mo class=\"MathClass-rel\">=<\/mo> <msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi><mo class=\"MathClass-bin\">+<\/mo><mn>1<\/mn><\/mrow><\/msub><mstyle><mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow \/><mo fence=\"true\" form=\"postfix\" \/><\/mrow><\/mstyle><munderover accent=\"false\" accentunder=\"false\"><mrow><mo> \u220f<\/mo> <\/mrow><mrow><mi>k<\/mi><mo class=\"MathClass-rel\">=<\/mo><mi>m<\/mi><mo class=\"MathClass-bin\">+<\/mo><mn>1<\/mn><\/mrow><mrow><mi>n<\/mi><\/mrow><\/munderover><msub><mrow><mi>a<\/mi><\/mrow><mrow> <mi>k<\/mi><\/mrow><\/msub><mstyle><mrow><mo fence=\"true\" form=\"prefix\"> )<\/mo><mrow \/><mo fence=\"true\" form=\"postfix\" \/><\/mrow><\/mstyle><mstyle><mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow \/><mo fence=\"true\" form=\"postfix\" \/><\/mrow><\/mstyle><munderover accent=\"false\" accentunder=\"false\"><mrow><mo> \u220f<\/mo> <\/mrow><mrow><mi>k<\/mi><mo class=\"MathClass-rel\">=<\/mo><mi>m<\/mi><mo class=\"MathClass-bin\">+<\/mo><mn>1<\/mn><\/mrow><mrow><mi>n<\/mi><\/mrow><\/munderover> <mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>k<\/mi><\/mrow><\/msub><\/mrow><\/mfrac><mstyle><mrow><mo fence=\"true\" form=\"prefix\"> )<\/mo><mrow \/><mo fence=\"true\" form=\"postfix\" \/><\/mrow><\/mstyle> <mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>m<\/mi><\/mrow><\/msub><\/mrow><\/mfrac> <mo class=\"MathClass-rel\">=<\/mo> <mfrac><mrow><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi><mo class=\"MathClass-bin\">+<\/mo><mn>1<\/mn><\/mrow><\/msub><\/mrow> <mrow><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>m<\/mi><\/mrow><\/msub><\/mrow><\/mfrac> <mo class=\"MathClass-punc\">.<\/mo><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\" \/> <mtd class=\"align-label\"> <mspace width=\"2em\" \/><\/mtd><\/mtr><\/mtable><\/math> <div class=\"me melemma\"> <div class=\"wp-nocaption \"><\/div><h4 id=\"z5f5d8aa5e667\"> <a id=\"x1-79001r5\"><\/a> <span class=\"ecbx-1095\">Lemma 3.5 <\/span>(Bernoulli\u2019sche Ungleichung)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">F<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r alle reellen Zahlen <\/span><math display=\"inline\"><mi>a<\/mi> <mo class=\"MathClass-rel\">\u2265<\/mo><mo class=\"MathClass-bin\">\u2212<\/mo><mn>1<\/mn><\/math> <span class=\"ecti-1095\">und <\/span><math display=\"inline\"><mi>n<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <msub><mrow><mi>\u2115<\/mi><\/mrow><mrow><mn>0<\/mn> <\/mrow> <\/msub> <\/math> <span class=\"ecti-1095\">gilt <\/span><span class=\"maperiod\"><math display=\"inline\"><msup><mrow><mo class=\"MathClass-open\">(<\/mo><mn>1<\/mn> <mo class=\"MathClass-bin\">+<\/mo> <mi>a<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mi>n<\/mi> <\/mrow> <\/msup> <mo class=\"MathClass-rel\">\u2265<\/mo> <mn>1<\/mn> <mo class=\"MathClass-bin\">+<\/mo> <mi>n<\/mi><mi>a<\/mi><\/math><\/span><span class=\"period\">.<\/span> <\/p> <\/div> <div class=\"wp-nocaption \"><\/div> <div class=\"proof\"> <p class=\"indent\"><span class=\"head\"><\/span><\/p><details open=\"open\"><summary><b>Beweis.<\/b><\/summary><p class=\"indent\" style=\"margin-top: 10\">Wir verwenden vollst\u00e4ndige Induktion. F\u00fcr <math display=\"inline\"><mi>n<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mn>0<\/mn><\/math> haben wir <span class=\"maperiod\"><math display=\"inline\"><msup><mrow><mo class=\"MathClass-open\">(<\/mo><mn>1<\/mn> <mo class=\"MathClass-bin\">+<\/mo> <mi>a<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mi>n<\/mi> <\/mrow> <\/msup> <mo class=\"MathClass-rel\">=<\/mo> <mn>1<\/mn> <mo class=\"MathClass-rel\">=<\/mo> <mn>1<\/mn> <mo class=\"MathClass-bin\">+<\/mo> <mi>n<\/mi><mi>a<\/mi><\/math><\/span><span class=\"period\">.<\/span> Angenommen die Ungleichung <math display=\"inline\"><msup><mrow><mo class=\"MathClass-open\">(<\/mo><mn>1<\/mn> <mo class=\"MathClass-bin\">+<\/mo> <mi>a<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msup> <mo class=\"MathClass-rel\">\u2265<\/mo> <mn>1<\/mn> <mo class=\"MathClass-bin\">+<\/mo> <mi>n<\/mi><mi>a<\/mi><\/math> gilt f\u00fcr ein <span class=\"maperiod\"><math display=\"inline\"><mi>n<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <msub><mrow><mi>\u2115<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><\/math><\/span><span class=\"period\">.<\/span> Nach Annahme an <math display=\"inline\"><mi>a<\/mi><\/math> ist <span class=\"maperiod\"><math display=\"inline\"><mi>a<\/mi> <mo class=\"MathClass-rel\">\u2265<\/mo> <mo class=\"MathClass-bin\">\u2212<\/mo><mn>1<\/mn><\/math><\/span><span class=\"period\">,<\/span> was in Kombination mit der Annahme an <math display=\"inline\"><mi>n<\/mi><\/math> <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><msup><mrow><mo class=\"MathClass-open\">(<\/mo><mn>1<\/mn> <mo class=\"MathClass-bin\">+<\/mo> <mi>a<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mi>n<\/mi><mo class=\"MathClass-bin\">+<\/mo><mn>1<\/mn><\/mrow><\/msup><\/mtd> <mtd class=\"align-even\"> <mo class=\"MathClass-rel\">=<\/mo> <msup><mrow><mo class=\"MathClass-open\">(<\/mo><mn>1<\/mn> <mo class=\"MathClass-bin\">+<\/mo> <mi>a<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msup><mo class=\"MathClass-open\">(<\/mo><mn>1<\/mn> <mo class=\"MathClass-bin\">+<\/mo> <mi>a<\/mi><mo class=\"MathClass-close\">)<\/mo><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\" \/> <mtd class=\"align-label\"> <mspace width=\"2em\" \/><\/mtd><\/mtr><mtr><mtd class=\"align-odd\" columnalign=\"right\" \/> <mtd class=\"align-even\"> <mo class=\"MathClass-rel\">\u2265<\/mo> <mo class=\"MathClass-open\">(<\/mo><mn>1<\/mn> <mo class=\"MathClass-bin\">+<\/mo> <mi>n<\/mi><mi>a<\/mi><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-open\">(<\/mo><mn>1<\/mn> <mo class=\"MathClass-bin\">+<\/mo> <mi>a<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <mn>1<\/mn> <mo class=\"MathClass-bin\">+<\/mo> <mi>n<\/mi><mi>a<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mi>a<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mi>n<\/mi><msup><mrow><mi>a<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\" \/> <mtd class=\"align-label\"> <mspace width=\"2em\" \/><\/mtd><\/mtr><mtr><mtd class=\"align-odd\" columnalign=\"right\" \/> <mtd class=\"align-even\"> <mo class=\"MathClass-rel\">\u2265<\/mo> <mn>1<\/mn> <mo class=\"MathClass-bin\">+<\/mo> <mo class=\"MathClass-open\">(<\/mo><mi>n<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mn>1<\/mn><mo class=\"MathClass-close\">)<\/mo><mi>a<\/mi><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\" \/> <mtd class=\"align-label\"> <mspace width=\"2em\" \/><\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">ergibt und damit den Induktionsschritt zeigt. Das Lemma folgt. <span>&nbsp;&nbsp;<\/span><\/p><div class=\"qed\">\u25a0<\/div><\/details><\/div> <div class=\"me meexample\"> <div class=\"wp-nocaption \"><\/div><h4 id=\"zc34830504c28\"> <a id=\"x1-79002r6\"><\/a> <span class=\"ecbx-1095\">\u00dc<\/span><span class=\"ecbx-1095\">bung 3.6 <\/span>(Archimedisches Prinzip f\u00fcr Potenzen)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Verwenden Sie die Bernoulli\u2019sche Ungleichung und das Archimedische Prinzip (Satz <\/span><a href=\"..\/..\/chapter\/erste-konsequenzen-der-vollstaendigkeit#x1-68001r68\"><span class=\"ecti-1095\">2.68<\/span><\/a><span class=\"ecti-1095\">),<\/span> <span class=\"ecti-1095\">um folgende Aussage zu beweisen. F<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r alle <\/span><math display=\"inline\"><mi>x<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>y<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u211d<\/mi><\/math> <span class=\"ecti-1095\">mit<\/span><span class=\"ecti-1095\">&nbsp;<\/span><math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">&gt;<\/mo> <mn>1<\/mn><\/math> <span class=\"ecti-1095\">existiert ein <\/span><span class=\"maperiod\"><math display=\"inline\"><mi>n<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <msub><mrow><mi>\u2115<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><\/math><\/span><span class=\"period\">,<\/span> <span class=\"ecti-1095\">so dass <\/span><span class=\"maperiod\"><math display=\"inline\"><msup><mrow><mi>x<\/mi><\/mrow><mrow><mi>n<\/mi> <\/mrow> <\/msup> <mo class=\"MathClass-rel\">\u2265<\/mo> <mi>y<\/mi><\/math><\/span><span class=\"period\">.<\/span> <\/p><details><summary style=\"color:#FF7F00\"><span class=\"ecti-1095\">L<\/span><span class=\"ecti-1095\">\u00f6<\/span><span class=\"ecti-1095\">sung.<\/span><\/summary><p class=\"indent\" style=\"margin-top: 0\"> <span class=\"ecti-1095\">Sei <\/span><span class=\"maperiod\"><math display=\"inline\"><mi>a<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mi>x<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <mn>1<\/mn> <mo class=\"MathClass-rel\">&gt;<\/mo> <mn>0<\/mn><\/math><\/span><span class=\"period\">.<\/span> <span class=\"ecti-1095\">Dann gilt <\/span><math display=\"inline\"><msup><mrow><mi>x<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msup> <mo class=\"MathClass-rel\">=<\/mo> <msup><mrow><mo class=\"MathClass-open\">(<\/mo><mn>1<\/mn> <mo class=\"MathClass-bin\">+<\/mo> <mi>a<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msup> <mo class=\"MathClass-rel\">\u2265<\/mo> <mi>n<\/mi><mi>a<\/mi><\/math> <span class=\"ecti-1095\">auf Grund der Bernoulli\u2019schen Ungleichung. Auf Grund des Archimedischen Prinzip hat <\/span><math display=\"inline\"><mi>\u2115<\/mi><\/math> <span class=\"ecti-1095\">keine obere Schranke in <\/span><span class=\"maperiod\"><math display=\"inline\"><mi>\u211d<\/mi><\/math><\/span><span class=\"period\">.<\/span> <span class=\"ecti-1095\">Daher und wegen <\/span><math display=\"inline\"><mi>a<\/mi> <mo class=\"MathClass-rel\">&gt;<\/mo> <mn>0<\/mn><\/math> <span class=\"ecti-1095\">existiert f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r jedes <\/span><math display=\"inline\"><mi>y<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u211d<\/mi><\/math> <span class=\"ecti-1095\">ein <\/span><math display=\"inline\"><mi>n<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2115<\/mi><\/math> <span class=\"ecti-1095\">mit <\/span><span class=\"maperiod\"><math display=\"inline\"><mi>n<\/mi><mi>a<\/mi> <mo class=\"MathClass-rel\">&gt;<\/mo> <mi>y<\/mi><\/math><\/span><span class=\"period\">.<\/span> <span class=\"ecti-1095\">Zusammen erhalten wir daraus <\/span><span class=\"maperiod\"><math display=\"inline\"><msup><mrow><mi>x<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msup> <mo class=\"MathClass-rel\">&gt;<\/mo> <mi>y<\/mi><\/math><\/span><span class=\"period\">.<\/span> <\/p><\/details>  <\/div> <div class=\"me meexample\"> <div class=\"wp-nocaption \"><\/div><h4 id=\"z996788b02822\"> <a id=\"x1-79003r7\"><\/a> <span class=\"ecbx-1095\">\u00dc<\/span><span class=\"ecbx-1095\">bung 3.7 <\/span>(Zifferndarstellungen nat\u00fcrlicher Zahlen)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Sei <\/span><math display=\"inline\"><mi>q<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2115<\/mi><\/math> <span class=\"ecti-1095\">eine nat<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">rliche Zahl. Zeigen Sie, dass sich jede nat<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">rliche Zahl <\/span><math display=\"inline\"><mi>m<\/mi><\/math> <span class=\"ecti-1095\">als Summe der Form <\/span><math display=\"inline\"><mi>m<\/mi> <mo class=\"MathClass-rel\">=<\/mo><msubsup><mrow><mi class=\"MathClass-op\"> \u2211<\/mi><mo> <\/mo> <\/mrow><mrow><mi>k<\/mi><mo class=\"MathClass-rel\">=<\/mo><mn>0<\/mn><\/mrow><mrow><mi>\u2113<\/mi><\/mrow><\/msubsup><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>k<\/mi><\/mrow><\/msub><msup><mrow><mi>q<\/mi><\/mrow><mrow><mi>k<\/mi><\/mrow><\/msup><\/math> <span class=\"ecti-1095\">schreiben l<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">sst wobei <\/span><math display=\"inline\"><mi>\u2113<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <msub><mrow><mi>\u2115<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><\/math> <span class=\"ecti-1095\">und die Koeffizienten <\/span><span class=\"maperiod\"><math display=\"inline\"><msub><mrow><mi>a<\/mi><\/mrow><mrow><mn>0<\/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>\u2113<\/mi><\/mrow><\/msub> <mo class=\"MathClass-rel\">\u2208<\/mo> <msub><mrow><mi>\u2115<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub> <mo class=\"MathClass-bin\">\u2229<\/mo> <mo class=\"MathClass-open\">[<\/mo><mn>0<\/mn><mo class=\"MathClass-punc\">,<\/mo><mi>q<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <mn>1<\/mn><mo class=\"MathClass-close\">]<\/mo><\/math><\/span><span class=\"period\">.<\/span> <span class=\"ecti-1095\">Diese Aussage kennen Sie schon f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r<\/span><span class=\"ecti-1095\">&nbsp;<\/span><math display=\"inline\"><mi>q<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mn>1<\/mn><mn>0<\/mn><\/math> <span class=\"ecti-1095\">wegen der Dezimaldarstellung nat<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">rlicher Zahlen und vielleicht auch f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r<\/span><span class=\"ecti-1095\">&nbsp;<\/span><math display=\"inline\"><mi>q<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mn>2<\/mn><\/math> <span class=\"ecti-1095\">wegen der Bin<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">rdarstellung. F<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r ein allgemeines <\/span><math display=\"inline\"><mi>q<\/mi><\/math> <span class=\"ecti-1095\">spricht man auch von der <\/span><math display=\"inline\"><mi>q<\/mi><\/math><span class=\"ecti-1095\">-n<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">ren<\/span> <span class=\"ecti-1095\">Darstellung.<\/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\">Nach <\/span><span class=\"ecti-1095\">\u00dc<\/span><span class=\"ecti-1095\">bung <\/span><a href=\"..\/..\/chapter\/summen-und-produkte#x1-79002r6\"><span class=\"ecti-1095\">3.6<\/span><\/a> <span class=\"ecti-1095\">existiert f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r alle <\/span><math display=\"inline\"><mi>m<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2115<\/mi><\/math> <span class=\"ecti-1095\">ein <\/span><math display=\"inline\"><mi>\u2113<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2115<\/mi><\/math> <span class=\"ecti-1095\">mit <\/span><span class=\"maperiod\"><math display=\"inline\"><mi>m<\/mi> <mo class=\"MathClass-rel\">&lt;<\/mo> <msup><mrow><mi>q<\/mi><\/mrow><mrow><mi>\u2113<\/mi> <\/mrow> <\/msup> <\/math><\/span><span class=\"period\">.<\/span> <span class=\"ecti-1095\">Sei nun <\/span><math display=\"inline\"><mi>A<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>\u2113<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math> <span class=\"ecti-1095\">die Aussage, dass sich jede nat<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">rliche Zahl <\/span><math display=\"inline\"><mi>m<\/mi><\/math> <span class=\"ecti-1095\">mit <\/span><math display=\"inline\"><msup><mrow><mi>q<\/mi><\/mrow><mrow><mi>\u2113<\/mi> <\/mrow> <\/msup> <mo class=\"MathClass-rel\">\u2264<\/mo> <mi>m<\/mi> <mo class=\"MathClass-rel\">&lt;<\/mo> <msup><mrow><mi>q<\/mi><\/mrow><mrow><mi>\u2113<\/mi><mo class=\"MathClass-bin\">+<\/mo><mn>1<\/mn><\/mrow><\/msup><\/math> <span class=\"ecti-1095\">in der gew<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">nschten Form schreiben l<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">sst. Verwenden Sie vollst<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">ndige Induktion <\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">ber <\/span><math display=\"inline\"><mi>\u2113<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <msub><mrow><mi>\u2115<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><\/math> <span class=\"ecti-1095\">und Division mit Rest f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r den Induktionsschritt.<\/span><\/p><\/details>  <\/div> <a id=\"x1-79004r79\"><\/a> <h4 id=\"za8cda99b01e8\" class=\"subsectionHead\"><span class=\"titlemark\">3.1.3 <\/span> <a id=\"x1-800003\"><\/a>Die geometrische Summe<\/h4> <p class=\"noindent\">In diesem kurzen Abschnitt m\u00f6chten wir folgende, vermutlich schon bekannte und f\u00fcr uns sp\u00e4ter sehr wichtige Formel beweisen. <\/p> <div class=\"me metheorem\"> <div class=\"wp-nocaption \"><\/div><h4 id=\"z49fdf5cec16f\"> <a id=\"x1-80001r8\"><\/a> <span class=\"ecbx-1095\">Proposition 3.8 <\/span>(Geometrische Summenformel)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Sei <\/span><math display=\"inline\"><mi>n<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <msub><mrow><mi>\u2115<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><\/math> <span class=\"ecti-1095\">und <\/span><span class=\"maperiod\"><math display=\"inline\"><mi>q<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2102<\/mi><\/math><\/span><span class=\"period\">.<\/span> <span class=\"ecti-1095\">Dann gilt<\/span> <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><munderover accent=\"false\" accentunder=\"false\"><mrow><mo>\u2211<\/mo> <\/mrow><mrow><mi>k<\/mi><mo class=\"MathClass-rel\">=<\/mo><mn>0<\/mn><\/mrow><mrow><mi>n<\/mi><\/mrow><\/munderover><msup><mrow><mi>q<\/mi><\/mrow><mrow><mi>k<\/mi><\/mrow><\/msup> <mo class=\"MathClass-rel\">=<\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> {<\/mo><mrow> <mtable align=\"axis\" class=\"array\" columnlines=\"none\" equalcolumns=\"false\" equalrows=\"false\"> <mtr><mtd class=\"array\" columnalign=\"center\"> <mi>n<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mn>1<\/mn> <\/mtd><mtd class=\"array\" columnalign=\"center\"><mstyle class=\"text\"><mtext>falls&nbsp;<\/mtext><\/mstyle><mi>q<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mn>1<\/mn><\/mtd> <\/mtr> <mtr><mtd class=\"array\" columnalign=\"center\"><mfrac><mrow><msup><mrow><mi>q<\/mi><\/mrow><mrow><mi>n<\/mi><mo class=\"MathClass-bin\">+<\/mo><mn>1<\/mn><\/mrow><\/msup><mo class=\"MathClass-bin\">\u2212<\/mo><mn>1<\/mn><\/mrow> <mrow><mi>q<\/mi><mo class=\"MathClass-bin\">\u2212<\/mo><mn>1<\/mn><\/mrow><\/mfrac> <\/mtd><mtd class=\"array\" columnalign=\"center\"> <mstyle class=\"text\"><mtext>falls&nbsp;<\/mtext><\/mstyle><mi>q<\/mi><mo class=\"MathClass-rel\">\u2260<\/mo><mn>1<\/mn> <\/mtd><\/mtr> <\/mtable> <\/mrow><mo fence=\"true\" form=\"postfix\" \/><\/mrow><mo class=\"MathClass-punc\">.<\/mo><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <\/div> <p class=\"indent\">Der direkte (aber sicher nicht eleganteste) Beweis verwendet vollst\u00e4ndige Induktion: <\/p><div class=\"wp-nocaption \"><\/div> <div class=\"proof\"> <p class=\"indent\"><span class=\"head\"><\/span><\/p><details open=\"open\"><summary><b>Beweis.<\/b><\/summary><p class=\"indent\" style=\"margin-top: 10\">F\u00fcr&nbsp;<math display=\"inline\"><mi>q<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mn>1<\/mn><\/math> ist&nbsp;<math display=\"inline\"><msup><mrow><mi>q<\/mi><\/mrow><mrow><mi>k<\/mi> <\/mrow> <\/msup> <mo class=\"MathClass-rel\">=<\/mo> <mn>1<\/mn><\/math> f\u00fcr alle&nbsp;<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> und die Aussage folgt aus den Eigenschaften der Summe. Sei nun&nbsp;<span class=\"maperiod\"><math display=\"inline\"><mi>q<\/mi><mo class=\"MathClass-rel\">\u2260<\/mo> <mn>1<\/mn><\/math><\/span><span class=\"period\">.<\/span> F\u00fcr <math display=\"inline\"><mi>n<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mn>0<\/mn><\/math> gilt <span class=\"maperiod\"><math display=\"inline\"><msubsup><mrow><mi class=\"MathClass-op\"> \u2211<\/mi><mo> <\/mo> <\/mrow><mrow><mi>k<\/mi><mo class=\"MathClass-rel\">=<\/mo><mn>0<\/mn><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msubsup><msup><mrow><mi>q<\/mi><\/mrow><mrow><mi>k<\/mi><\/mrow><\/msup> <mo class=\"MathClass-rel\">=<\/mo> <msup><mrow><mi>q<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msup> <mo class=\"MathClass-rel\">=<\/mo> <mn>1<\/mn> <mo class=\"MathClass-rel\">=<\/mo> <mfrac><mrow><mi>q<\/mi><mo class=\"MathClass-bin\">\u2212<\/mo><mn>1<\/mn><\/mrow> <mrow><mi>q<\/mi><mo class=\"MathClass-bin\">\u2212<\/mo><mn>1<\/mn><\/mrow><\/mfrac><\/math><\/span><span class=\"period\">,<\/span> was also den Induktionsanfang zeigt. Angenommen die Formel in der Proposition gilt bereits f\u00fcr                                                                                                                                                                           <span class=\"maperiod\"><math display=\"inline\"><mi>n<\/mi><\/math><\/span><span class=\"period\">.<\/span> Dann ist <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><munderover accent=\"false\" accentunder=\"false\"><mrow><mo> \u2211<\/mo> <\/mrow><mrow><mi>k<\/mi><mo class=\"MathClass-rel\">=<\/mo><mn>0<\/mn><\/mrow><mrow><mi>n<\/mi><mo class=\"MathClass-bin\">+<\/mo><mn>1<\/mn><\/mrow><\/munderover><msup><mrow><mi>q<\/mi><\/mrow><mrow><mi>k<\/mi><\/mrow><\/msup> <mo class=\"MathClass-rel\">=<\/mo><munderover accent=\"false\" accentunder=\"false\"><mrow><mo> \u2211<\/mo> <\/mrow><mrow><mi>k<\/mi><mo class=\"MathClass-rel\">=<\/mo><mn>0<\/mn><\/mrow><mrow><mi>n<\/mi><\/mrow><\/munderover><msup><mrow><mi>q<\/mi><\/mrow><mrow><mi>k<\/mi><\/mrow><\/msup> <mo class=\"MathClass-bin\">+<\/mo> <msup><mrow><mi>q<\/mi><\/mrow><mrow><mi>n<\/mi><mo class=\"MathClass-bin\">+<\/mo><mn>1<\/mn><\/mrow><\/msup> <mo class=\"MathClass-rel\">=<\/mo> <mfrac><mrow><msup><mrow><mi>q<\/mi><\/mrow><mrow><mi>n<\/mi><mo class=\"MathClass-bin\">+<\/mo><mn>1<\/mn><\/mrow><\/msup> <mo class=\"MathClass-bin\">\u2212<\/mo> <mn>1<\/mn><\/mrow> <mrow><mi>q<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <mn>1<\/mn><\/mrow><\/mfrac> <mo class=\"MathClass-bin\">+<\/mo> <msup><mrow><mi>q<\/mi><\/mrow><mrow><mi>n<\/mi><mo class=\"MathClass-bin\">+<\/mo><mn>1<\/mn><\/mrow><\/msup> <mo class=\"MathClass-rel\">=<\/mo> <mfrac><mrow><msup><mrow><mi>q<\/mi><\/mrow><mrow><mi>n<\/mi><mo class=\"MathClass-bin\">+<\/mo><mn>1<\/mn><\/mrow><\/msup> <mo class=\"MathClass-bin\">\u2212<\/mo> <mn>1<\/mn><\/mrow> <mrow><mi>q<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <mn>1<\/mn><\/mrow><\/mfrac> <mo class=\"MathClass-bin\">+<\/mo> <mfrac><mrow><msup><mrow><mi>q<\/mi><\/mrow><mrow><mi>n<\/mi><mo class=\"MathClass-bin\">+<\/mo><mn>2<\/mn><\/mrow><\/msup> <mo class=\"MathClass-bin\">\u2212<\/mo> <msup><mrow><mi>q<\/mi><\/mrow><mrow><mi>n<\/mi><mo class=\"MathClass-bin\">+<\/mo><mn>1<\/mn><\/mrow><\/msup><\/mrow> <mrow><mi>q<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <mn>1<\/mn><\/mrow><\/mfrac> <mo class=\"MathClass-rel\">=<\/mo> <mfrac><mrow><msup><mrow><mi>q<\/mi><\/mrow><mrow><mi>n<\/mi><mo class=\"MathClass-bin\">+<\/mo><mn>2<\/mn><\/mrow><\/msup> <mo class=\"MathClass-bin\">\u2212<\/mo> <mn>1<\/mn><\/mrow> <mrow><mi>q<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <mn>1<\/mn><\/mrow><\/mfrac> <mo class=\"MathClass-punc\">,<\/mo><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">womit der Induktionsschritt gezeigt ist und die Proposition folgt. <span>&nbsp;&nbsp;<\/span><\/p><div class=\"qed\">\u25a0<\/div><\/details><\/div> <p class=\"indent\">Wir laden Sie dazu ein, in folgender \u00dcbung einen eleganteren Beweis zu finden. <\/p> <div class=\"me meexample\"> <div class=\"wp-nocaption \"><\/div><h4 id=\"ze2a5432680ff\"> <a id=\"x1-80002r9\"><\/a> <span class=\"ecbx-1095\">\u00dc<\/span><span class=\"ecbx-1095\">bung 3.9 <\/span>(Geometrische Summenformel)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Verwenden Sie eine Teleskopsumme um die geometrische Summenformel (Proposition<\/span><span class=\"ecti-1095\">&nbsp;<\/span><a href=\"..\/..\/chapter\/summen-und-produkte#x1-80001r8\"><span class=\"ecti-1095\">3.8<\/span><\/a><span class=\"ecti-1095\">)<\/span> <span class=\"ecti-1095\">f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r<\/span><span class=\"ecti-1095\">&nbsp;<\/span><math display=\"inline\"><mi>q<\/mi><mo class=\"MathClass-rel\">\u2260<\/mo> <mn>1<\/mn><\/math> <span class=\"ecti-1095\">zu beweisen.<\/span> <\/p><div class=\"wp-nocaption \"><\/div><details><summary style=\"color:#FF7F00\"><span class=\"ecti-1095\">Hinweis.<\/span><\/summary><p class=\"indent\" style=\"margin-top: 0\"><span class=\"ecti-1095\">Multiplizieren Sie <\/span><math display=\"inline\"><msubsup><mrow><mi class=\"MathClass-op\">\u2211<\/mi><mo> <\/mo> <\/mrow><mrow><mi>k<\/mi><mo class=\"MathClass-rel\">=<\/mo><mn>0<\/mn><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msubsup><msup><mrow><mi>q<\/mi><\/mrow><mrow><mi>k<\/mi><\/mrow><\/msup><\/math> <span class=\"ecti-1095\">mit <\/span><span class=\"maperiod\"><math display=\"inline\"><mi>q<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <mn>1<\/mn><\/math><\/span><span class=\"period\">.<\/span><\/p><\/details>  <\/div> <div class=\"me meexample\"> <div class=\"wp-nocaption \"><\/div><h4 id=\"zefec752ac269\"> <a id=\"x1-80003r10\"><\/a> <span class=\"ecbx-1095\">\u00dc<\/span><span class=\"ecbx-1095\">bung 3.10 <\/span>(Eindeutigkeit der Ziffernentwickung nat\u00fcrlicher Zahlen)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Zeigen Sie, dass die <\/span><math display=\"inline\"><mi>q<\/mi><\/math><span class=\"ecti-1095\">-n<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">re<\/span> <span class=\"ecti-1095\">Darstellung einer nat<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">rlichen Zahl in <\/span><span class=\"ecti-1095\">\u00dc<\/span><span class=\"ecti-1095\">bung<\/span><span class=\"ecti-1095\">&nbsp;<\/span><a href=\"..\/..\/chapter\/summen-und-produkte#x1-79003r7\"><span class=\"ecti-1095\">3.7<\/span><\/a> <span class=\"ecti-1095\">eindeutig bestimmt ist. Das heisst, f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r<\/span> <span class=\"ecti-1095\">jedes<\/span><span class=\"ecti-1095\">&nbsp;<\/span><math display=\"inline\"><mi>m<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2115<\/mi><\/math> <span class=\"ecti-1095\">mit<\/span><span class=\"ecti-1095\">&nbsp;<\/span><math display=\"inline\"><mi>m<\/mi> <mo class=\"MathClass-rel\">=<\/mo><msubsup><mrow><mi class=\"MathClass-op\"> \u2211<\/mi><mo> <\/mo> <\/mrow><mrow><mi>k<\/mi><mo class=\"MathClass-rel\">=<\/mo><mn>0<\/mn><\/mrow><mrow><mi>\u2113<\/mi><\/mrow><\/msubsup><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>k<\/mi><\/mrow><\/msub><msup><mrow><mi>q<\/mi><\/mrow><mrow><mi>k<\/mi><\/mrow><\/msup><\/math> <span class=\"ecti-1095\">und <\/span><math display=\"inline\"><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>\u2113<\/mi> <\/mrow> <\/msub> <mo class=\"MathClass-rel\">\u2260<\/mo> <mn>0<\/mn><\/math> <span class=\"ecti-1095\">sind <\/span><math display=\"inline\"><mi>\u2113<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <msub><mrow><mi>\u2115<\/mi><\/mrow><mrow><mn>0<\/mn> <\/mrow> <\/msub> <\/math> <span class=\"ecti-1095\">und die Koeffizienten <\/span><math display=\"inline\"><msub><mrow><mi>a<\/mi><\/mrow><mrow><mn>0<\/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>\u2113<\/mi><\/mrow><\/msub> <mo class=\"MathClass-rel\">\u2208<\/mo> <msub><mrow><mi>\u2115<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub> <mo class=\"MathClass-bin\">\u2229<\/mo> <mo class=\"MathClass-open\">[<\/mo><mn>0<\/mn><mo class=\"MathClass-punc\">,<\/mo><mi>q<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <mn>1<\/mn><mo class=\"MathClass-close\">]<\/mo><\/math> <span class=\"ecti-1095\">eindeutig durch<\/span><span class=\"ecti-1095\">&nbsp;<\/span><math display=\"inline\"><mi>m<\/mi><\/math> <span class=\"ecti-1095\">bestimmt.<\/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\">Zeigen Sie zuerst f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r <\/span><math display=\"inline\"><msub><mrow><mi>b<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><mo class=\"MathClass-punc\">,<\/mo><mi class=\"MathClass-op\">\u2026<\/mi><mo> <\/mo><mo class=\"MathClass-punc\">,<\/mo><msub><mrow><mi>b<\/mi><\/mrow><mrow><mi>\u2113<\/mi><\/mrow><\/msub> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2124<\/mi><\/math> <span class=\"ecti-1095\">mit <\/span><math display=\"inline\"><mo class=\"MathClass-rel\">|<\/mo><msub><mrow><mi>b<\/mi><\/mrow><mrow><mi>k<\/mi> <\/mrow> <\/msub> <mo class=\"MathClass-rel\">|<\/mo> <mo class=\"MathClass-rel\">&lt;<\/mo> <mi>q<\/mi><\/math> <span class=\"ecti-1095\">f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r <\/span><math display=\"inline\"><mi>k<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mn>1<\/mn><mo class=\"MathClass-punc\">,<\/mo> <mi class=\"MathClass-op\">\u2026<\/mi><mo> <\/mo><mo class=\"MathClass-punc\">,<\/mo><mi>\u2113<\/mi><\/math> <span class=\"ecti-1095\">die Absch<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">tzung <\/span><span class=\"maperiod\"><math display=\"inline\"><mo class=\"MathClass-rel\">|<\/mo><msubsup><mrow><mi class=\"MathClass-op\">\u2211<\/mi><mo> <\/mo> <\/mrow><mrow><mi>k<\/mi><mo class=\"MathClass-rel\">=<\/mo><mn>0<\/mn><\/mrow><mrow><mi>\u2113<\/mi><\/mrow><\/msubsup><msub><mrow><mi>b<\/mi><\/mrow><mrow><mi>k<\/mi><\/mrow><\/msub><msup><mrow><mi>q<\/mi><\/mrow><mrow><mi>k<\/mi><\/mrow><\/msup><mo class=\"MathClass-rel\">|<\/mo> <mo class=\"MathClass-rel\">&lt;<\/mo> <msup><mrow><mi>q<\/mi><\/mrow><mrow><mi>\u2113<\/mi><mo class=\"MathClass-bin\">+<\/mo><mn>1<\/mn><\/mrow><\/msup><\/math><\/span><span class=\"period\">.<\/span> <span class=\"ecti-1095\">Nehmen Sie indirekt an, dass es eine nat<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">rliche Zahl <\/span><math display=\"inline\"><mi>m<\/mi><\/math> <span class=\"ecti-1095\">mit zwei Darstellungen gibt und w<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">hlen Sie eine minimale derartige Zahl. Sei nun <\/span><math display=\"inline\"><mi>m<\/mi> <mo class=\"MathClass-rel\">=<\/mo><msubsup><mrow><mi class=\"MathClass-op\"> \u2211<\/mi><mo> <\/mo> <\/mrow><mrow><mi>k<\/mi><mo class=\"MathClass-rel\">=<\/mo><mn>0<\/mn><\/mrow><mrow><mi>\u2113<\/mi><\/mrow><\/msubsup><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>k<\/mi><\/mrow><\/msub><msup><mrow><mi>q<\/mi><\/mrow><mrow><mi>k<\/mi><\/mrow><\/msup> <mo class=\"MathClass-rel\">=<\/mo><msubsup><mrow><mi class=\"MathClass-op\"> \u2211<\/mi><mo> <\/mo> <\/mrow><mrow><mi>k<\/mi><mo class=\"MathClass-rel\">=<\/mo><mn>0<\/mn><\/mrow><mrow><msup><mrow><mi>\u2113<\/mi><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup> <\/mrow><\/msubsup><msubsup><mrow><mi>a<\/mi><\/mrow><mrow><mi>k<\/mi><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msubsup><msup><mrow><mi>q<\/mi><\/mrow><mrow><mi>k<\/mi><\/mrow><\/msup><\/math> <span class=\"ecti-1095\">zwei Darstellungen von <\/span><span class=\"maperiod\"><math display=\"inline\"><mi>m<\/mi><\/math><\/span><span class=\"period\">.<\/span> <span class=\"ecti-1095\">Auf Grund der Minimalit<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">t von <\/span><math display=\"inline\"><mi>m<\/mi><\/math> <span class=\"ecti-1095\">muss <\/span><math display=\"inline\"><mi>\u2113<\/mi><mo class=\"MathClass-rel\">\u2260<\/mo> <msup><mrow><mi>\u2113<\/mi><\/mrow><mrow><mo>\u2032<\/mo> <\/mrow> <\/msup> <\/math> <span class=\"ecti-1095\">gelten, doch auf Grund der Absch<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">tzung kann dies zu einem Widerspruch gef<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">hrt werden.<\/span><\/p><\/details>  <\/div> <a id=\"x1-80004r77\"><\/a> \n","protected":false},"author":1089,"menu_order":1,"template":"","meta":{"pb_show_title":"","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-42","chapter","type-chapter","status-publish","hentry"],"part":41,"_links":{"self":[{"href":"https:\/\/wp-prd.let.ethz.ch\/analysis19\/wp-json\/pressbooks\/v2\/chapters\/42","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\/42\/revisions"}],"part":[{"href":"https:\/\/wp-prd.let.ethz.ch\/analysis19\/wp-json\/pressbooks\/v2\/parts\/41"}],"metadata":[{"href":"https:\/\/wp-prd.let.ethz.ch\/analysis19\/wp-json\/pressbooks\/v2\/chapters\/42\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/wp-prd.let.ethz.ch\/analysis19\/wp-json\/wp\/v2\/media?parent=42"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/wp-prd.let.ethz.ch\/analysis19\/wp-json\/pressbooks\/v2\/chapter-type?post=42"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/wp-prd.let.ethz.ch\/analysis19\/wp-json\/wp\/v2\/contributor?post=42"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/wp-prd.let.ethz.ch\/analysis19\/wp-json\/wp\/v2\/license?post=42"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}