{"id":84,"date":"2021-12-15T09:53:20","date_gmt":"2021-12-15T09:53:20","guid":{"rendered":"https:\/\/wp-prd.let.ethz.ch\/analysis19\/chapter\/weitere-lernmaterialien-7\/"},"modified":"2021-12-15T09:53:20","modified_gmt":"2021-12-15T09:53:20","slug":"weitere-lernmaterialien-7","status":"publish","type":"chapter","link":"https:\/\/wp-prd.let.ethz.ch\/analysis19\/chapter\/weitere-lernmaterialien-7\/","title":{"raw":"Weitere Lernmaterialien","rendered":"Weitere Lernmaterialien"},"content":{"raw":"\n<style>.cmr-5{font-size:50%;}\n.cmr-7{font-size:70%;}\n.cmmi-5{font-size:50%;font-style: italic;}\n.cmmi-7{font-size:70%;font-style: italic;}\n.cmmi-10{font-style: italic;}\n.cmsy-5{font-size:50%;}\n.cmsy-7{font-size:70%;}\n.cmbx-10{ font-weight: bold;}\n.cmbsy-10{font-weight: bold;}\n.cmbsy-10{font-weight: bold;}\n.cmbsy-10{font-weight: bold;}\n.cmbsy-7{font-size:70%;font-weight: bold;}\n.cmbsy-7{font-weight: bold;}\n.cmbsy-7{font-weight: bold;}\n.cmbsy-5{font-size:50%;font-weight: bold;}\n.cmbsy-5{font-weight: bold;}\n.cmbsy-5{font-weight: bold;}\n.cmex-7{font-size:70%;}\n.cmex-7x-x-71{font-size:49%;}\n.msam-7{font-size:70%;}\n.msam-5{font-size:50%;}\n.msbm-7{font-size:70%;}\n.msbm-5{font-size:50%;}\n.cmr-17{font-size:170%;}\n.cmr-12{font-size:120%;}\n.cmti-10{ font-style: italic;}\np{margin-top:0;margin-bottom:0}\np.indent{text-indent:0;}\np + p{margin-top:1em;}\np + div, p + pre {margin-top:1em;}\ndiv + p, pre + p {margin-top:1em;}\n@media print {div.crosslinks {visibility:hidden;}}\na img { border-top: 0; 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\n}\ndiv.proof p:first-of-type {\n\tmargin: 0px;\n}\ndiv.qed {\n\tmargin-top: -25px;\n\tmargin-bottom: -7px;\n\ttext-align: right;\n}\ntable.equation+div.qed {\n\tmargin-top: -65px;\n}\n\n\/* The following is making also math-formulas inside the headers of Lemmas, etc., white. *\/\ndiv.melemma h4 span {\n    color: white;\n}\ndiv.metheorem h4 span {\n    color: white;\n}\n\n\/* The following are used to avoid fullstop, period, colon, semicolon, and endquote (broader) to move by itself to the next line after a formula.\n   The math-environment before needs to be wrapped in span.maperiod and the fullstop etc. in a span.period --- together they achieve what we want.  *\/\nspan.maperiod {\n       margin-right: 5px;\n}\nspan.period {\n       display: inline-block;\n       width: 0px;\n       margin-left: -5px;\n       margin-right: 4.9px;\n\t   text-indent: 0px;\n}\nspan.maendquote {\n       margin-right: 8px;\n}\nspan.endquote {\n       display: inline-block;\n       width: 0px;\n       margin-left: -8px;\n       margin-right: 7.9px;\n}\n\n\n\/* The following is removing an extra space left of the equation side in aligned equations *\/\nspan.mjx-mtd {\n    padding-left: 0em !important;\n}\n\n\/* The following fixes the weird problem that math appears smaller if it was rendered while the details tag was closed. *\/\ndetails span.mjx-chtml, details span.MathJax_CHTML {\n font-size: 100% !important;\n}\n\n\/* trying to fix line breaks in verbatim, new lines are missing *\/\npre.verbatim {\n\twhite-space: pre-wrap;\n\tfont-size: small;\n}\n<\/style><h3 id=\"z6f8a0471ddc2\" class=\"sectionHead\"><span class=\"titlemark\">7.9 <\/span> <a id=\"x1-2220009\"><\/a>Weitere Lernmaterialien<\/h3> <a id=\"x1-222001r221\"><\/a> <h4 id=\"zbd77bd63a8cf\" class=\"subsectionHead\"><span class=\"titlemark\">7.9.1 <\/span> <a id=\"x1-2230001\"><\/a>Verwendung des Kapitels<\/h4> <p class=\"noindent\">Wie wir gesehen haben, sind Potenzreihen, deren Konvergenzradius und Konvergenzverhalten fundamentale Werkzeuge f\u00fcr die Definition von vielen Ihnen bereits bekannten Funktionen (und auch weiteren). Wir werden also ab nun sowohl die komplexe Exponentialfunktion, die trigonometrischen Funktionen auf <math display=\"inline\"><mi>\u211d<\/mi><\/math> und auf <span class=\"maperiod\"><math display=\"inline\"><mi>\u2102<\/mi><\/math><\/span><span class=\"period\">,<\/span> als auch die hyperbolischen Funktionen gemeinsam mit den wichtigsten Eigenschaften dieser Funktionen (meist ohne Verweise) verwenden. (Die Umkehrfunktionen der trigonometrischen Funktionen werden wir erst im n\u00e4chsten Kapitel einf\u00fchren.) <\/p><p class=\"indent\">F\u00fcr den Begriff der Potenzreihe ben\u00f6tigten wir die grundlegenden Definitionen der Reihe und der Funktionenfolgen. F\u00fcr Reihen ist die Unterscheidung der bedingten und absoluten Konvergenz fundamental, da gewisse Operationen (Umordnen, Cauchy-Produkt) nur f\u00fcr den letzteren Konvergenzbegriff erlaubt sind. Dabei ist es sehr hilfreich, dass f\u00fcr Potenzreihen im Inneren des Konvergenzbereichs absolute Konvergenz vorliegt und damit alle Operationen erlaubt sind. Die folgenden Konvergenzkriterien sind f\u00fcr Beispiele aber auch f\u00fcr die Theorie unabdingbar: <\/p> <div class=\"custom-itemize\"><div class=\"item-head\"> <span class=\"tcrm-1095\">\u2022<\/span><\/div><div class=\"item-content\">die geometrische Reihe in Beispiel <a href=\"..\/..\/chapter\/reihen#x1-187003r3\">7.3<\/a>, <\/div><div class=\"item-head\"> <span class=\"tcrm-1095\">\u2022<\/span><\/div><div class=\"item-content\">Majoranten-   und   Minorantenkriterium   f\u00fcr   Reihen   mit   positiven   Gliedern   in Korollar&nbsp;<a href=\"..\/..\/chapter\/reihen#x1-188002r12\">7.12<\/a> und Korollar <a href=\"..\/..\/chapter\/absolute-konvergenz#x1-193001r29\">7.29<\/a>, <\/div><div class=\"item-head\"> <span class=\"tcrm-1095\">\u2022<\/span><\/div><div class=\"item-content\">Verdichtung in Proposition <a href=\"..\/..\/chapter\/reihen#x1-188006r16\">7.16<\/a>, <\/div><div class=\"item-head\"> <span class=\"tcrm-1095\">\u2022<\/span><\/div><div class=\"item-content\"><math display=\"inline\"><mi>p<\/mi><\/math>-Test in Beispiel <a href=\"..\/..\/chapter\/reihen#x1-188007r17\">7.17<\/a>, <\/div><div class=\"item-head\"> <span class=\"tcrm-1095\">\u2022<\/span><\/div><div class=\"item-content\">Leibniz-Kriterium in Proposition <a href=\"..\/..\/chapter\/reihen#x1-190001r25\">7.25<\/a> (welches vor allem f\u00fcr bedingt konvergente aber wegen der  Fehlerabsch\u00e4tzung  auch  f\u00fcr  absolut  konvergente  Reihen  n\u00fctzlich  sein kann), <\/div><div class=\"item-head\"> <span class=\"tcrm-1095\">\u2022<\/span><\/div><div class=\"item-content\">Cauchy-Kriterium in Satz <a href=\"..\/..\/chapter\/reihen#x1-191001r26\">7.26<\/a> (meist als theoretisches Hilfsmittel),                                                                                                                                                                           <\/div><div class=\"item-head\"> <span class=\"tcrm-1095\">\u2022<\/span><\/div><div class=\"item-content\">Wurzelkriterium in Korollar <a href=\"..\/..\/chapter\/absolute-konvergenz#x1-193002r30\">7.30<\/a> (als theoretisches und praktisches Hilfsmittel), <\/div><div class=\"item-head\"> <span class=\"tcrm-1095\">\u2022<\/span><\/div><div class=\"item-content\">Quotientenkriterium  in  Korollar  <a href=\"..\/..\/chapter\/absolute-konvergenz#x1-193004r32\">7.32<\/a>  (meist  als  praktisches  Hilfsmittel,  da  es  oft einfacher anwendbar ist, aber im Gegensatz zu dem Wurzelkriterium zum Beispiel f\u00fcr Potenzreihen weniger allgemein einsetzbar ist), <\/div><div class=\"item-head\"> <span class=\"tcrm-1095\">\u2022<\/span><\/div><div class=\"item-content\">aber wenn sonst nichts zum Erfolg f\u00fchrt, sollte man nicht vergessen, dass auf Grund von Proposition <a href=\"..\/..\/chapter\/reihen#x1-187002r2\">7.2<\/a> die Folgenglieder einer konvergenten Reihe eine Nullfolge bilden.<\/div><\/div> <p class=\"noindent\">Wir bemerken noch, dass diese Kriterien sehr hilfreich sind f\u00fcr die Entscheidung ob Konvergenz oder Divergenz bei einer Reihe vorliegt, doch haben wir sehr wenige allgemeine Gesetze um den Grenzwert von Reihen zu bestimmen. <\/p><p class=\"indent\">Wie bereits erw\u00e4hnt war der Begriff der Funktionenfolge auch f\u00fcr die Besprechung der Potenzreihen notwendig. F\u00fcr Funktionenfolgen haben wir zwei unterschiedliche Konvergenzbegriffe besprochen. Der Begriff der punktweisen Konvergenz mag zwar als der nat\u00fcrliche Konvergenzbegriff f\u00fcr Funktionen betrachtet werden, doch hat dieser keine guten Eigenschaften (weder f\u00fcr Stetigkeit noch f\u00fcr das Riemann-Integral). Sie sollten die entsprechenden Gegenbeispiele im Ged\u00e4chtnis behalten. Dies motivierte die Definition der gleichm\u00e4ssigen Konvergenz, welche wegen den guten Eigenschaften f\u00fcr Stetigkeit und das Riemann-Integral f\u00fcr uns immer wieder wichtig sein wird. Die Unterscheidung dieser Konvergenzbegriffe ist wohlgemerkt keine Spitzfindigkeit. <a id=\"x1-223001r223\"><\/a> <\/p> <h4 id=\"zcbbfaabbe411\" class=\"subsectionHead\"><span class=\"titlemark\">7.9.2 <\/span> <a id=\"x1-2240002\"><\/a>\u00dcbungen<\/h4> <div class=\"me meexample\"> <p class=\"indent\"><\/p><h4 id=\"z2a25b2aa8160\"> <span class=\"ecbx-1095\">\u00dc<\/span><span class=\"ecbx-1095\">bung.<\/span><\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Sei <\/span><math display=\"inline\"><msubsup><mrow><mi class=\"MathClass-op\"> \u2211<\/mi><mo> <\/mo> <\/mrow><mrow><mi>k<\/mi><mo class=\"MathClass-rel\">=<\/mo><mn>1<\/mn><\/mrow><mrow><mi>\u221e<\/mi><\/mrow><\/msubsup><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>k<\/mi><\/mrow><\/msub><\/math> <span class=\"ecti-1095\">eine konvergente Reihe. Falls <\/span><math display=\"inline\"><msub><mrow><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>k<\/mi><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mi>k<\/mi><\/mrow><\/msub><\/math> <span class=\"ecti-1095\">eine monoton fallende Folge ist, so ist nicht nur <\/span><span class=\"maperiod\"><math display=\"inline\"><msub><mrow><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>k<\/mi><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mi>k<\/mi><\/mrow><\/msub><\/math><\/span><span class=\"period\">,<\/span> <span class=\"ecti-1095\">sondern auch <\/span><math display=\"inline\"><msub><mrow><mo class=\"MathClass-open\">(<\/mo><mi>k<\/mi><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>k<\/mi><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mi>k<\/mi><\/mrow><\/msub><\/math> <span class=\"ecti-1095\">eine Nullfolge. Beweisen Sie dies.<\/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\">\u00dc<\/span><span class=\"ecti-1095\">berzeugen Sie sich von der Ungleichung<\/span> <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mo class=\"MathClass-open\">(<\/mo><mi>n<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>m<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mn>1<\/mn><mo class=\"MathClass-close\">)<\/mo><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub> <mo class=\"MathClass-rel\">\u2264<\/mo><munderover accent=\"false\" accentunder=\"false\"><mrow><mo>\u2211<\/mo> <\/mrow><mrow><mi>k<\/mi><mo class=\"MathClass-rel\">=<\/mo><mi>m<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/munderover><msub><mrow><mi>a<\/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\"><span class=\"ecti-1095\">f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r <\/span><span class=\"maperiod\"><math display=\"inline\"><mi>n<\/mi> <mo class=\"MathClass-rel\">&gt;<\/mo> <mi>m<\/mi><\/math><\/span><span class=\"period\">.<\/span> <\/p><\/details>  <\/div> <div class=\"me meexample\"> <p class=\"indent\"><\/p><h4 id=\"z926c9b97e4d5\"> <span class=\"ecbx-1095\">\u00dc<\/span><span class=\"ecbx-1095\">bung.<\/span><\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Sei <\/span><math display=\"inline\"><mi>f<\/mi> <mo class=\"MathClass-punc\">:<\/mo> <mi>\u2115<\/mi> <mo class=\"MathClass-rel\">\u2192<\/mo> <mi>\u2115<\/mi><\/math> <span class=\"ecti-1095\">eine bijektive Abbildung. Konvergiert die Reihe <\/span><span class=\"maendquote\"><math display=\"inline\"><msubsup><mrow><mi class=\"MathClass-op\">\u2211<\/mi><mo> <\/mo> <\/mrow><mrow><mi>n<\/mi><mo class=\"MathClass-rel\">=<\/mo><mn>1<\/mn><\/mrow><mrow><mi>\u221e<\/mi><\/mrow><\/msubsup><mfrac><mrow><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>n<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow> <mrow><msup><mrow><mi>n<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><\/mrow><\/mfrac> <\/math><\/span><span class=\"endquote\">?<\/span> <\/p><p class=\"indent\"><\/p><details><summary style=\"color:#FF7F00\"><span class=\"ecti-1095\">Hinweis.<\/span><\/summary><p class=\"indent\" style=\"margin-top: 0\"><span class=\"ecti-1095\">F<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r jedes <\/span><math display=\"inline\"><mi>n<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2115<\/mi><\/math> <span class=\"ecti-1095\">gilt <\/span><span class=\"maperiod\"><math display=\"inline\"><mo class=\"MathClass-rel\">|<\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> {<\/mo><mrow><mi>k<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2115<\/mi><mo class=\"MathClass-rel\">\u2223<\/mo><msup><mrow><mn>2<\/mn><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msup> <mo class=\"MathClass-rel\">\u2264<\/mo> <mi>k<\/mi> <mo class=\"MathClass-rel\">&lt;<\/mo> <msup><mrow><mn>2<\/mn><\/mrow><mrow><mi>n<\/mi><mo class=\"MathClass-bin\">+<\/mo><mn>1<\/mn><\/mrow><\/msup><mstyle class=\"text\"><mtext>&nbsp;und&nbsp;<\/mtext><\/mstyle><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>k<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">\u2265<\/mo> <msup><mrow><mn>2<\/mn><\/mrow><mrow><mi>n<\/mi><mo class=\"MathClass-bin\">\u2212<\/mo><mn>1<\/mn><\/mrow><\/msup><\/mrow><mo fence=\"true\" form=\"postfix\">}<\/mo><\/mrow> <mo class=\"MathClass-rel\">|<\/mo><mo class=\"MathClass-rel\">\u2265<\/mo> <msup><mrow><mn>2<\/mn><\/mrow><mrow><mi>n<\/mi><mo class=\"MathClass-bin\">\u2212<\/mo><mn>1<\/mn><\/mrow><\/msup><\/math><\/span><span class=\"period\">.<\/span><\/p><\/details>  <\/div> <div class=\"me meexample\"> <p class=\"indent\"><\/p><h4 id=\"zeb59e7c92ab8\"> <span class=\"ecbx-1095\">\u00dc<\/span><span class=\"ecbx-1095\">bung <\/span>(Raabes Quotientenkriterium)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Sei <\/span><math display=\"inline\"><msub><mrow><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi> <\/mrow> <\/msub> <mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><\/math> <span class=\"ecti-1095\">eine Folge<\/span> <span class=\"ecti-1095\">komplexer Zahlen mit <\/span><math display=\"inline\"><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><mo class=\"MathClass-rel\">\u2260<\/mo><mn>0<\/mn><\/math> <span class=\"ecti-1095\">f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r alle <\/span><span class=\"maperiod\"><math display=\"inline\"><mi>n<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2115<\/mi><\/math><\/span><span class=\"period\">,<\/span> <span class=\"ecti-1095\">so dass <\/span><math display=\"inline\"><munder class=\"msub\"><mrow><mi class=\"qopname\"> lim<\/mi><mo>  <\/mo> <\/mrow><mrow><mi>n<\/mi><mo class=\"MathClass-rel\">\u2192<\/mo><mi>\u221e<\/mi><\/mrow><\/munder><mfrac><mrow><mo class=\"MathClass-rel\">|<\/mo><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi><mo class=\"MathClass-bin\">+<\/mo><mn>1<\/mn><\/mrow><\/msub><mo class=\"MathClass-rel\">|<\/mo><\/mrow> <mrow><mo class=\"MathClass-rel\">|<\/mo><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><mo class=\"MathClass-rel\">|<\/mo><\/mrow><\/mfrac> <mo class=\"MathClass-rel\">=<\/mo> <mn>1<\/mn><\/math> <span class=\"ecti-1095\">und<\/span> <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mi>Q<\/mi> <mo class=\"MathClass-rel\">=<\/mo><munder class=\"msub\"><mrow><mi class=\"qopname\"> lim<\/mi><mo>  <\/mo><\/mrow><mrow><mi>n<\/mi><mo class=\"MathClass-rel\">\u2192<\/mo><mi>\u221e<\/mi><\/mrow><\/munder><mi>n<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mn>1<\/mn> <mo class=\"MathClass-bin\">\u2212<\/mo><mfrac><mrow><mo class=\"MathClass-rel\">|<\/mo><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi><mo class=\"MathClass-bin\">+<\/mo><mn>1<\/mn><\/mrow><\/msub><mo class=\"MathClass-rel\">|<\/mo><\/mrow> <mrow><mo class=\"MathClass-rel\">|<\/mo><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><mo class=\"MathClass-rel\">|<\/mo><\/mrow><\/mfrac> <\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\"><span class=\"ecti-1095\">existiert. Zeigen Sie, dass die Reihe <\/span><math display=\"inline\"><msubsup><mrow><mi class=\"MathClass-op\">\u2211<\/mi><mo> <\/mo> <\/mrow><mrow><mi>n<\/mi><mo class=\"MathClass-rel\">=<\/mo><mn>1<\/mn><\/mrow><mrow><mi>\u221e<\/mi><\/mrow><\/msubsup><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><\/math> <span class=\"ecti-1095\">konvergiert, falls <\/span><span class=\"maperiod\"><math display=\"inline\"><mi>Q<\/mi> <mo class=\"MathClass-rel\">&gt;<\/mo> <mn>1<\/mn><\/math><\/span><span class=\"period\">.<\/span> <span class=\"ecti-1095\">(J.<\/span><span class=\"ecti-1095\">&nbsp;Raabe war einer der ersten Mathematikprofessoren an der ETH Z<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">rich.)<\/span> <\/p><p class=\"indent\"><span class=\"ecti-1095\">Hinweis: Finden Sie ein <\/span><math display=\"inline\"><mi>p<\/mi> <mo class=\"MathClass-rel\">&gt;<\/mo> <mn>1<\/mn><\/math><span class=\"ecti-1095\">, so<\/span> <span class=\"ecti-1095\">dass f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r alle bis auf endlich viele <\/span><math display=\"inline\"><mi>n<\/mi><\/math> <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mfrac><mrow><mo class=\"MathClass-rel\">|<\/mo><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi><mo class=\"MathClass-bin\">+<\/mo><mn>1<\/mn><\/mrow><\/msub><mo class=\"MathClass-rel\">|<\/mo><\/mrow> <mrow><mo class=\"MathClass-rel\">|<\/mo><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><mo class=\"MathClass-rel\">|<\/mo><\/mrow><\/mfrac> <mo class=\"MathClass-rel\">&lt;<\/mo> <mn>1<\/mn> <mo class=\"MathClass-bin\">\u2212<\/mo><mfrac><mrow><mi>p<\/mi><\/mrow> <mrow><mi>n<\/mi><\/mrow><\/mfrac><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\"><span class=\"ecti-1095\">gilt. Verwenden Sie die kontinuierliche Bernoulli-Ungleichung aus <\/span><span class=\"ecti-1095\">\u00dc<\/span><span class=\"ecti-1095\">bung <\/span><a href=\"..\/..\/chapter\/die-exponentialfunktion#x1-173006r36\"><span class=\"ecti-1095\">6.36<\/span><\/a><span class=\"ecti-1095\">, um zu zeigen,<\/span> <span class=\"ecti-1095\">dass<\/span> <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mfrac><mrow><mo class=\"MathClass-rel\">|<\/mo><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi><mo class=\"MathClass-bin\">+<\/mo><mn>1<\/mn><\/mrow><\/msub><mo class=\"MathClass-rel\">|<\/mo><\/mrow> <mrow><mo class=\"MathClass-rel\">|<\/mo><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><mo class=\"MathClass-rel\">|<\/mo><\/mrow><\/mfrac> <mo class=\"MathClass-rel\">\u2264<\/mo><msup><mrow> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mn>1<\/mn> <mo class=\"MathClass-bin\">\u2212<\/mo> <mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mi>n<\/mi><\/mrow><\/mfrac> <\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><\/mrow><mrow><mi>p<\/mi><\/mrow><\/msup> <mo class=\"MathClass-rel\">=<\/mo> <mfrac><mrow><msup><mrow><mo class=\"MathClass-open\">(<\/mo><mi>n<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <mn>1<\/mn><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mi>p<\/mi><\/mrow><\/msup><\/mrow> <mrow><msup><mrow><mi>n<\/mi><\/mrow><mrow><mi>p<\/mi><\/mrow><\/msup><\/mrow><\/mfrac> <mo class=\"MathClass-punc\">.<\/mo><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\"><span class=\"ecti-1095\">Schliessen Sie nun auf die Aussage unter Verwendung von Korollar <\/span><a href=\"..\/..\/chapter\/absolute-konvergenz#x1-193001r29\"><span class=\"ecti-1095\">7.29<\/span><\/a> <span class=\"ecti-1095\">und Beispiel<\/span> <a href=\"..\/..\/chapter\/reihen#x1-188007r17\"><span class=\"ecti-1095\">7.17<\/span><\/a><span class=\"ecti-1095\">.<\/span> <\/p> <\/div> <div class=\"me meexample\"> <p class=\"indent\"><\/p><h4 id=\"z1b0d38f7ef06\"> <span class=\"ecbx-1095\">\u00dc<\/span><span class=\"ecbx-1095\">bung <\/span>(Kronecker\u2019s Lemma)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Sei <\/span><math display=\"inline\"><msubsup><mrow><mi class=\"MathClass-op\"> \u2211<\/mi><mo> <\/mo> <\/mrow><mrow><mi>k<\/mi><mo class=\"MathClass-rel\">=<\/mo><mn>1<\/mn><\/mrow><mrow><mi>\u221e<\/mi><\/mrow><\/msubsup><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>k<\/mi><\/mrow><\/msub><\/math> <span class=\"ecti-1095\">eine konvergente<\/span> <span class=\"ecti-1095\">Reihe und sei <\/span><math display=\"inline\"><msub><mrow><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>b<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><\/math> <span class=\"ecti-1095\">eine divergente monoton wachsende Folge positiver Zahlen. Dann gilt<\/span> <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><munder class=\"msub\"><mrow><mi class=\"qopname\">lim<\/mi><mo>  <\/mo><\/mrow><mrow><mi>n<\/mi><mo class=\"MathClass-rel\">\u2192<\/mo><mi>\u221e<\/mi><\/mrow><\/munder> <mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><msub><mrow><mi>b<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><\/mrow><\/mfrac><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> <mn>0<\/mn><mo class=\"MathClass-punc\">.<\/mo><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\"><span class=\"ecti-1095\">Beweisen Sie Kronecker\u2019s Lemma unter Verwendung von Abel-Summation (<\/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#x1-78001r3\"><span class=\"ecti-1095\">3.3<\/span><\/a><span class=\"ecti-1095\">).<\/span> <\/p> <\/div> <div class=\"me meexample\"> <p class=\"indent\"><\/p><h4 id=\"z8c69610e1c6f\"> <span class=\"ecbx-1095\">\u00dc<\/span><span class=\"ecbx-1095\">bung <\/span>(Vertauschung der Summationsreihenfolge)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Wie schon vor dem Beweis des Produktsatzes (Satz <\/span><a href=\"..\/..\/chapter\/absolute-konvergenz#x1-195001r36\"><span class=\"ecti-1095\">7.36<\/span><\/a><span class=\"ecti-1095\">) angedeutet, ist der<\/span> <span class=\"ecti-1095\">Produktsatz stark mit Vertauschbarkeit von Summationsreihenfolge verwandt. Wir<\/span> <span class=\"ecti-1095\">wollen dies hier genauer formulieren. Dazu betrachten wir eine doppelt indizierte Folge<\/span> <span class=\"maperiod\"><math display=\"inline\"><msub><mrow><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>a<\/mi><\/mrow><mrow><mo class=\"MathClass-open\">(<\/mo><mi>m<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>n<\/mi><mo class=\"MathClass-close\">)<\/mo> <\/mrow> <\/msub> <mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mo class=\"MathClass-open\">(<\/mo><mi>m<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>n<\/mi><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-rel\">\u2208<\/mo><msup><mrow><mi>\u2115<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><\/mrow><\/msub><\/math><\/span><span class=\"period\">.<\/span> <\/p><dl class=\"enumerate\"><dt class=\"enumerate\"> <span class=\"ecti-1095\">a)<\/span><\/dt><dd class=\"enumerate\"><span class=\"ecti-1095\">Um zu sehen, dass die Vertauschbarkeit der Summationsreihenfolge f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r Reihen nicht immer gilt,<\/span> <span class=\"ecti-1095\">definieren wir<\/span> <math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><msub><mrow><mi>a<\/mi><\/mrow><mrow><mo class=\"MathClass-open\">(<\/mo><mi>m<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>n<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow><\/msub> <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=\"left\"><mn>1<\/mn> <\/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> <\/mtd> <\/mtr> <mtr><mtd class=\"array\" columnalign=\"left\"> <mo class=\"MathClass-bin\">\u2212<\/mo> <mn>1<\/mn><\/mtd><mtd class=\"array\" columnalign=\"left\"><mstyle class=\"text\"><mtext>falls&nbsp;<\/mtext><\/mstyle><mi>m<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mn>1<\/mn> <mo class=\"MathClass-rel\">=<\/mo> <mi>n<\/mi><\/mtd> <\/mtr> <mtr><mtd class=\"array\" columnalign=\"left\"><mn>0<\/mn> <\/mtd><mtd class=\"array\" columnalign=\"left\"><mstyle class=\"text\"><mtext>sonst&nbsp;<\/mtext><\/mstyle><mo class=\"MathClass-punc\">.<\/mo> <\/mtd><\/mtr><\/mtable> <\/mrow><mo fence=\"true\" form=\"postfix\" \/><\/mrow><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\"><span class=\"ecti-1095\">f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r alle <\/span><span class=\"maperiod\"><math display=\"inline\"><mo class=\"MathClass-open\">(<\/mo><mi>m<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>n<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">\u2208<\/mo> <msup><mrow><mi>\u2115<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><\/math><\/span><span class=\"period\">.<\/span> <span class=\"ecti-1095\">Zeigen Sie, dass die Doppelreihen<\/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>m<\/mi><mo class=\"MathClass-rel\">=<\/mo><mn>1<\/mn><\/mrow><mrow><mi>\u221e<\/mi><\/mrow><\/munderover><mstyle><mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow \/><mo fence=\"true\" form=\"postfix\" \/><\/mrow><\/mstyle><munderover accent=\"false\" accentunder=\"false\"><mrow><mo>\u2211<\/mo> <\/mrow><mrow><mi>n<\/mi><mo class=\"MathClass-rel\">=<\/mo><mn>1<\/mn><\/mrow><mrow><mi>\u221e<\/mi><\/mrow><\/munderover><msub><mrow><mi>a<\/mi><\/mrow><mrow> <mo class=\"MathClass-open\">(<\/mo><mi>m<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>n<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow><\/msub><mstyle><mrow><mo fence=\"true\" form=\"prefix\"> )<\/mo><mrow \/><mo fence=\"true\" form=\"postfix\" \/><\/mrow><\/mstyle><mo class=\"MathClass-punc\">,<\/mo><mspace class=\"quad\" width=\"1em\" \/><munderover accent=\"false\" accentunder=\"false\"><mrow><mo>\u2211<\/mo> <\/mrow><mrow><mi>n<\/mi><mo class=\"MathClass-rel\">=<\/mo><mn>1<\/mn><\/mrow><mrow><mi>\u221e<\/mi><\/mrow><\/munderover><mstyle><mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow \/><mo fence=\"true\" form=\"postfix\" \/><\/mrow><\/mstyle><munderover accent=\"false\" accentunder=\"false\"><mrow><mo>\u2211<\/mo> <\/mrow><mrow><mi>m<\/mi><mo class=\"MathClass-rel\">=<\/mo><mn>1<\/mn><\/mrow><mrow><mi>\u221e<\/mi><\/mrow><\/munderover><msub><mrow><mi>a<\/mi><\/mrow><mrow> <mo class=\"MathClass-open\">(<\/mo><mi>m<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>n<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow><\/msub><mstyle><mrow><mo fence=\"true\" form=\"prefix\"> )<\/mo><mrow \/><mo fence=\"true\" form=\"postfix\" \/><\/mrow><\/mstyle><\/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\">konvergieren, aber verschieden sind.<\/span> <\/p><\/dd><dt class=\"enumerate\"> <span class=\"ecti-1095\">b)<\/span><\/dt><dd class=\"enumerate\"><span class=\"ecti-1095\">Angenommen <\/span><math display=\"inline\"><msub><mrow><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>a<\/mi><\/mrow><mrow><mo class=\"MathClass-open\">(<\/mo><mi>m<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>n<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mo class=\"MathClass-open\">(<\/mo><mi>m<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>n<\/mi><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-rel\">\u2208<\/mo><msup><mrow><mi>\u2115<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><\/mrow><\/msub><\/math> <span class=\"ecti-1095\">erf<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">llt<\/span> <math display=\"inline\"><msubsup><mrow><mi class=\"MathClass-op\">\u2211<\/mi><mo> <\/mo> <\/mrow><mrow><mi>m<\/mi><mo class=\"MathClass-rel\">=<\/mo><mn>1<\/mn><\/mrow><mrow><mi>\u221e<\/mi><\/mrow><\/msubsup><mstyle><mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow \/><mo fence=\"true\" form=\"postfix\" \/><\/mrow><\/mstyle><msubsup><mrow><mi class=\"MathClass-op\">\u2211<\/mi><mo> <\/mo> <\/mrow><mrow><mi>n<\/mi><mo class=\"MathClass-rel\">=<\/mo><mn>1<\/mn><\/mrow><mrow><mi>\u221e<\/mi><\/mrow><\/msubsup><mo class=\"MathClass-rel\">|<\/mo><msub><mrow><mi>a<\/mi><\/mrow><mrow><mo class=\"MathClass-open\">(<\/mo><mi>m<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>n<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow><\/msub><mo class=\"MathClass-rel\">|<\/mo><mstyle><mrow><mo fence=\"true\" form=\"prefix\"> )<\/mo><mrow \/><mo fence=\"true\" form=\"postfix\" \/><\/mrow><\/mstyle> <mo class=\"MathClass-rel\">&lt;<\/mo> <mi>\u221e<\/mi><\/math><span class=\"ecti-1095\">. Zeigen Sie, dass<\/span> <span class=\"ecti-1095\">die beiden Reihen <\/span><math display=\"inline\"><msubsup><mrow><mi class=\"MathClass-op\">\u2211<\/mi><mo> <\/mo> <\/mrow><mrow><mi>m<\/mi><mo class=\"MathClass-rel\">=<\/mo><mn>1<\/mn><\/mrow><mrow><mi>\u221e<\/mi><\/mrow><\/msubsup><mstyle><mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow \/><mo fence=\"true\" form=\"postfix\" \/><\/mrow><\/mstyle><msubsup><mrow><mi class=\"MathClass-op\">\u2211<\/mi><mo> <\/mo> <\/mrow><mrow><mi>n<\/mi><mo class=\"MathClass-rel\">=<\/mo><mn>1<\/mn><\/mrow><mrow><mi>\u221e<\/mi><\/mrow><\/msubsup><msub><mrow><mi>a<\/mi><\/mrow><mrow><mo class=\"MathClass-open\">(<\/mo><mi>m<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>n<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow><\/msub><mstyle><mrow><mo fence=\"true\" form=\"prefix\"> )<\/mo><mrow \/><mo fence=\"true\" form=\"postfix\" \/><\/mrow><\/mstyle><\/math> <span class=\"ecti-1095\">und <\/span><math display=\"inline\"><msubsup><mrow><mi class=\"MathClass-op\"> \u2211<\/mi><mo> <\/mo> <\/mrow><mrow><mi>n<\/mi><mo class=\"MathClass-rel\">=<\/mo><mn>1<\/mn><\/mrow><mrow><mi>\u221e<\/mi><\/mrow><\/msubsup><mstyle><mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow \/><mo fence=\"true\" form=\"postfix\" \/><\/mrow><\/mstyle><msubsup><mrow><mi class=\"MathClass-op\">\u2211<\/mi><mo> <\/mo> <\/mrow><mrow><mi>m<\/mi><mo class=\"MathClass-rel\">=<\/mo><mn>1<\/mn><\/mrow><mrow><mi>\u221e<\/mi><\/mrow><\/msubsup><msub><mrow><mi>a<\/mi><\/mrow><mrow><mo class=\"MathClass-open\">(<\/mo><mi>m<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>n<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow><\/msub><mstyle><mrow><mo fence=\"true\" form=\"prefix\"> )<\/mo><mrow \/><mo fence=\"true\" form=\"postfix\" \/><\/mrow><\/mstyle><\/math> <span class=\"ecti-1095\">konvergent sind und den gleichen Wert haben.<\/span><\/dd><\/dl> <\/div> <div class=\"me meexample\"> <p class=\"indent\"><\/p><h4 id=\"z003dfb34fab8\"> <span class=\"ecbx-1095\">\u00dc<\/span><span class=\"ecbx-1095\">bung <\/span>(Zerlegung in gerade und ungerade Funktionen)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Sei<\/span> <math display=\"inline\"><mi>f<\/mi> <mo class=\"MathClass-punc\">:<\/mo> <mi>\u2102<\/mi> <mo class=\"MathClass-rel\">\u2192<\/mo> <mi>\u2102<\/mi><\/math> <span class=\"ecti-1095\">eine               Funktion.               Zeigen               Sie,               dass               sich<\/span> <math display=\"inline\"><mi>f<\/mi><\/math> <span class=\"ecti-1095\">als Summe einer geraden und einer ungeraden Funktion schreiben l<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">sst.<\/span> <\/p> <\/div> <div class=\"me meexample\"> <p class=\"indent\"><\/p><h4 id=\"z3f56df5a7d19\"> <span class=\"ecbx-1095\">\u00dc<\/span><span class=\"ecbx-1095\">bung.<\/span><\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Seien <\/span><math display=\"inline\"><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo> <mi>b<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u211d<\/mi><\/math> <span class=\"ecti-1095\">. Zeigen<\/span> <span class=\"ecti-1095\">Sie, dass es ein <\/span><math display=\"inline\"><mi>\ud835\udf03<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u211d<\/mi><\/math> <span class=\"ecti-1095\">gibt, so dass<\/span> <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mi>a<\/mi><mi class=\"qopname\">sin<\/mi><mo>  <\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>\u03c6<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-bin\">+<\/mo> <mi>b<\/mi><mi class=\"qopname\">cos<\/mi><mo>  <\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>\u03c6<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-rel\">=<\/mo> <msqrt><mrow><msup><mrow><mi>a<\/mi><\/mrow><mrow><mn>2<\/mn> <\/mrow> <\/msup> <mo class=\"MathClass-bin\">+<\/mo> <msup><mrow><mi>b<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><\/mrow><\/msqrt><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\"> sin<\/mi><mo>  <\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>\u03c6<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mi>\ud835\udf03<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\"><span class=\"ecti-1095\">f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r alle <\/span><span class=\"maperiod\"><math display=\"inline\"><mi>\u03c6<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u211d<\/mi><\/math><\/span><span class=\"period\">.<\/span> <\/p><div class=\"geoapplet\" style=\"width: 688px\"><iframe height=\"695px\" scrolling=\"no\" src=\"https:\/\/www.geogebra.org\/material\/iframe\/id\/SvkmUmkp\/width\/688\/height\/695\/border\/888888\/rc\/false\/ai\/false\/sdz\/false\/smb\/false\/stb\/false\/stbh\/false\/ld\/false\/sri\/false\" style=\"border:0px\"><\/iframe><\/div><p class=\"indent\"> <\/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\">Betrachten Sie <\/span><math display=\"inline\"><msqrt><mrow><msup><mrow><mi>a<\/mi><\/mrow><mrow><mn>2<\/mn> <\/mrow> <\/msup> <mo class=\"MathClass-bin\">+<\/mo> <msup><mrow><mi>b<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><\/mrow><\/msqrt> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow> <mfrac><mrow><mi>a<\/mi><\/mrow> <mrow><msqrt><mrow><msup><mrow><mi>a<\/mi><\/mrow><mrow><mn>2<\/mn> <\/mrow> <\/msup> <mo class=\"MathClass-bin\">+<\/mo><msup><mrow><mi>b<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><\/mrow><\/msqrt><\/mrow><\/mfrac><mi class=\"qopname\"> sin<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>\u03c6<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">+<\/mo> <mfrac><mrow><mi>b<\/mi><\/mrow> <mrow><msqrt><mrow><msup><mrow><mi>a<\/mi><\/mrow><mrow><mn>2<\/mn> <\/mrow> <\/msup> <mo class=\"MathClass-bin\">+<\/mo><msup><mrow><mi>b<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><\/mrow><\/msqrt><\/mrow><\/mfrac><mi class=\"qopname\"> cos<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>\u03c6<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><\/math> <span class=\"ecti-1095\">und finden Sie eine reelle Zahl<\/span><span class=\"ecti-1095\">&nbsp;<\/span><math display=\"inline\"><mi>\ud835\udf03<\/mi><\/math> <span class=\"ecti-1095\">mit<\/span><span class=\"ecti-1095\">&nbsp;<\/span><math display=\"inline\"><mi class=\"qopname\"> cos<\/mi><mo>  <\/mo> <mo class=\"MathClass-open\">(<\/mo><mi>\ud835\udf03<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <mfrac><mrow><mi>a<\/mi><\/mrow> <mrow><msqrt><mrow><msup><mrow><mi>a<\/mi><\/mrow><mrow><mn>2<\/mn> <\/mrow> <\/msup> <mo class=\"MathClass-bin\">+<\/mo><msup><mrow><mi>b<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><\/mrow><\/msqrt><\/mrow><\/mfrac><\/math> <span class=\"ecti-1095\">und<\/span><span class=\"ecti-1095\">&nbsp;<\/span><span class=\"maperiod\"><math display=\"inline\"><mi class=\"qopname\"> sin<\/mi><mo>  <\/mo> <mo class=\"MathClass-open\">(<\/mo><mi>\ud835\udf03<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <mfrac><mrow><mi>b<\/mi><\/mrow> <mrow><msqrt><mrow><msup><mrow><mi>a<\/mi><\/mrow><mrow><mn>2<\/mn> <\/mrow> <\/msup> <mo class=\"MathClass-bin\">+<\/mo><msup><mrow><mi>b<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><\/mrow><\/msqrt><\/mrow><\/mfrac><\/math><\/span><span class=\"period\">.<\/span><\/p><\/details>  <\/div> <div class=\"me meexample\"> <p class=\"indent\"><\/p><h4 id=\"z70c53ef216c1\"> <span class=\"ecbx-1095\">\u00dc<\/span><span class=\"ecbx-1095\">bung <\/span>(Irrationalit\u00e4t der Eulerschen Zahl)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Zeigen                                              Sie,                                              dass<\/span> <math display=\"inline\"><mi class=\"qopname\">e<\/mi><mo>  <\/mo><mo class=\"MathClass-rel\">=<\/mo><msubsup><mrow><mi class=\"qopname\"> \u2211<\/mi><mo>  <\/mo> <\/mrow><mrow><mi>k<\/mi><mo class=\"MathClass-rel\">=<\/mo><mn>0<\/mn><\/mrow><mrow><mi>\u221e<\/mi><\/mrow><\/msubsup><mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mi>k<\/mi><mo class=\"MathClass-punc\">!<\/mo><\/mrow><\/mfrac><\/math> <span class=\"ecti-1095\">irrational.              Nehmen              Sie              indirekt              an,              dass<\/span> <math display=\"inline\"><mi class=\"qopname\">e<\/mi><mo>  <\/mo><\/math> <span class=\"ecti-1095\">rational ist und verwenden Sie die Exponentialreihe.<\/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\">Nehmen Sie indirekt an, dass <\/span><math display=\"inline\"><msup><mrow><mi class=\"qopname\">e<\/mi><mo>  <\/mo><\/mrow><mrow><mo class=\"MathClass-bin\">\u2212<\/mo><mn>1<\/mn><\/mrow><\/msup> <mo class=\"MathClass-rel\">=<\/mo> <mfrac><mrow><mi>p<\/mi><\/mrow> <mrow><mi>q<\/mi><\/mrow><\/mfrac><\/math> <span class=\"ecti-1095\">f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r <\/span><span class=\"maperiod\"><math display=\"inline\"><mi>p<\/mi><mo class=\"MathClass-punc\">,<\/mo> <mi>q<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2115<\/mi><\/math><\/span><span class=\"period\">.<\/span> <span class=\"ecti-1095\">Nun vergleichen Sie, <\/span><math display=\"inline\"><mi class=\"qopname\">e<\/mi><mo>  <\/mo> <mo class=\"MathClass-rel\">=<\/mo> <mfrac><mrow><mi>p<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>q<\/mi><mo class=\"MathClass-bin\">\u2212<\/mo><mn>1<\/mn><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-punc\">!<\/mo><\/mrow> <mrow><mi>q<\/mi><mo class=\"MathClass-punc\">!<\/mo><\/mrow><\/mfrac> <\/math> <span class=\"ecti-1095\">mit der Partialsumme <\/span><span class=\"maperiod\"><math display=\"inline\"><msubsup><mrow><mi class=\"MathClass-op\">\u2211<\/mi><mo> <\/mo> <\/mrow><mrow><mi>n<\/mi><mo class=\"MathClass-rel\">=<\/mo><mn>0<\/mn><\/mrow><mrow><mi>q<\/mi><\/mrow><\/msubsup><mfrac><mrow><msup><mrow><mo class=\"MathClass-open\">(<\/mo><mo class=\"MathClass-bin\">\u2212<\/mo><mn>1<\/mn><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msup><\/mrow> <mrow><mi>n<\/mi><mo class=\"MathClass-punc\">!<\/mo><\/mrow><\/mfrac> <\/math><\/span><span class=\"period\">.<\/span> <span class=\"ecti-1095\">Vergleichen Sie den Fehler im Leibniz-Kriterium und die Aussage, dass zwei rationale Zahlen<\/span> <span class=\"ecti-1095\">mit dem gleichem Nenner <\/span><math display=\"inline\"><mi>Q<\/mi><\/math> <span class=\"ecti-1095\">entweder gleich oder Mindestabstand <\/span><math display=\"inline\"> <mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mi>Q<\/mi><\/mrow><\/mfrac><\/math> <span class=\"ecti-1095\">haben.<\/span><\/p><\/details>  <\/div> <div class=\"me meexample\"> <p class=\"indent\"><\/p><h4 id=\"z1dfab298c823\"> <span class=\"ecbx-1095\">\u00dc<\/span><span class=\"ecbx-1095\">bung.<\/span><\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Wir wollen einen (spiels<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">chtigen) Kasinobesucher und ein un<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">bliches, den Spieler<\/span> <span class=\"ecti-1095\">bevorzugendes, Spiel betrachten. Das Spiel besteht aus einem einfachen M<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">nzwurf mit zwei<\/span> <span class=\"ecti-1095\">m<\/span><span class=\"ecti-1095\">\u00f6<\/span><span class=\"ecti-1095\">glichen gleich wahrscheinlichen Ergebnissen, n<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">mlich Kopf und Zahl. Bei Kopf gewinnt das<\/span> <span class=\"ecti-1095\">Kasino den Einsatz des Spielers und bei Zahl gewinnt der Spieler das Vierfache seines Einsatzes.<\/span> <span class=\"ecti-1095\">Wir wollen die verschiedenen Ergebnisse des iterierten Spieles anhand einer Funktion auf<\/span> <math display=\"inline\"><mo class=\"MathClass-open\">[<\/mo><mn>0<\/mn><mo class=\"MathClass-punc\">,<\/mo> <mn>1<\/mn><mo class=\"MathClass-close\">]<\/mo><\/math> <span class=\"ecti-1095\">beschreiben. Hierbei verwenden wir die bin<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">re Zifferndarstellung reeller Zahlen, wobei<\/span> <span class=\"ecti-1095\">wir die Nichteindeutigkeit einfach ignorieren, da diese f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r sich gesehen extrem<\/span> <span class=\"ecti-1095\">unwahrscheinlichen Ergebnissen des iterierten Spieles entsprechen.<\/span><button class=\"hover-trigger\" style=\"vertical-align: super;font: smaller\">\u2020<\/button><span class=\"hover-text\"><span class=\"marginpar\">\u2020 <span class=\"ecti-1095\">Dies genauer und<\/span> <span class=\"ecti-1095\">mathematisch exakt zu formulieren w<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">rde hier zu weit f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">hren.<\/span><\/span><\/span> <span class=\"ecti-1095\">Wir interpretieren die Ziffer<\/span> <math display=\"inline\"><mn>0<\/mn><\/math> <span class=\"ecti-1095\">als Kopf und die<\/span> <span class=\"ecti-1095\">Ziffer <\/span><math display=\"inline\"><mn>1<\/mn><\/math> <span class=\"ecti-1095\">als Zahl,<\/span> <span class=\"ecti-1095\">womit die reelle Zahl <\/span><math display=\"inline\"><mn>0<\/mn><mo class=\"MathClass-punc\">.<\/mo><mn>1<\/mn><mn>0<\/mn><mn>0<\/mn><mn>1<\/mn><mn>0<\/mn><mn>1<\/mn><mi class=\"MathClass-op\">\u2026<\/mi><mo> <\/mo><\/math> <span class=\"ecti-1095\">f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r die wiederholten W<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">rfe der M<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">nze mit den Ergebnissen Zahl-Kopf-Kopf-Zahl-Kopf-Zahl und so<\/span> <span class=\"ecti-1095\">weiter steht (und die Null vor dem Komma ignoriert wird).<\/span> <\/p><dl class=\"enumerate\"><dt class=\"enumerate\"> <span class=\"ecti-1095\">(i)<\/span><\/dt><dd class=\"enumerate\"><span class=\"ecti-1095\">Bestimmen Sie eine Funktion <\/span><span class=\"maperiod\"><math display=\"inline\"><msub><mrow><mi>f<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub> <mo class=\"MathClass-punc\">:<\/mo> <mo class=\"MathClass-open\">[<\/mo><mn>0<\/mn><mo class=\"MathClass-punc\">,<\/mo><mn>1<\/mn><mo class=\"MathClass-close\">]<\/mo> <mo class=\"MathClass-rel\">\u2192<\/mo> <mi>\u211d<\/mi><\/math><\/span><span class=\"period\">,<\/span> <span class=\"ecti-1095\">die den Gewinn nach <\/span><math display=\"inline\"><mi>n<\/mi><\/math> <span class=\"ecti-1095\">Wiederholungen des Spiels beschreibt, wobei der Spieler mit einem Franken das Spiel<\/span> <span class=\"ecti-1095\">beginnt und bei Gewinn jeweils sein Gesamtverm<\/span><span class=\"ecti-1095\">\u00f6<\/span><span class=\"ecti-1095\">gen im n<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">chsten Spiel wieder einsetzt.<\/span> <\/dd><dt class=\"enumerate\"> <span class=\"ecti-1095\">(ii)<\/span><\/dt><dd class=\"enumerate\"><span class=\"ecti-1095\">Bestimmen Sie den Erwartungswert f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r den Spieler, wenn dieser das Spiel <\/span><math display=\"inline\"><mi>n<\/mi><\/math> <span class=\"ecti-1095\">mal wiederholt (also das Integral von <\/span><math display=\"inline\"><msub><mrow><mi>f<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><\/math><span class=\"ecti-1095\">).<\/span> <\/dd><dt class=\"enumerate\"> <span class=\"ecti-1095\">(iii)<\/span><\/dt><dd class=\"enumerate\"><span class=\"ecti-1095\">Wir nehmen nun an, dass der Spieler spiels<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">chtig ist und auch bei Gewinn von wirklich<\/span> <span class=\"ecti-1095\">grossen Summen nicht aufh<\/span><span class=\"ecti-1095\">\u00f6<\/span><span class=\"ecti-1095\">ren kann zu spielen. Bestimmen Sie die Funktion, die den<\/span> <span class=\"ecti-1095\">Gewinn des Spielers beschreibt. Sie sollten bemerken, dass dies der punktweise Grenzwert<\/span> <span class=\"ecti-1095\">der Funktion <\/span><math display=\"inline\"><msub><mrow><mi>f<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><\/math> <span class=\"ecti-1095\">ist.<\/span> <\/dd><dt class=\"enumerate\"> <span class=\"ecti-1095\">(iv)<\/span><\/dt><dd class=\"enumerate\"><span class=\"ecti-1095\">Bestimmen  Sie  den  Erwartungswert  des  unbeschr<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">nkt  langen  Spiels  f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r  den<\/span> <span class=\"ecti-1095\">spiels<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">chtigen Spieler.<\/span><\/dd><\/dl> <\/div> <div class=\"me meexample\"> <p class=\"indent\"><\/p><h4 id=\"z370d7d57db5f\"> <span class=\"ecbx-1095\">\u00dc<\/span><span class=\"ecbx-1095\">bung <\/span>(Der Fall der absoluten Konvergenz am Rand)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Sei <\/span><math display=\"inline\"><msubsup><mrow><mi class=\"MathClass-op\"> \u2211<\/mi><mo> <\/mo> <\/mrow><mrow><mi>n<\/mi><mo class=\"MathClass-rel\">=<\/mo><mn>0<\/mn><\/mrow><mrow><mi>\u221e<\/mi><\/mrow><\/msubsup><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><msup><mrow><mi>z<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msup><\/math> <span class=\"ecti-1095\">eine Potenzreihe<\/span> <span class=\"ecti-1095\">mit Konvergenzradius <\/span><span class=\"maperiod\"><math display=\"inline\"><mi>R<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mo class=\"MathClass-open\">(<\/mo><mn>0<\/mn><mo class=\"MathClass-punc\">,<\/mo><mi>\u221e<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math><\/span><span class=\"period\">.<\/span> <span class=\"ecti-1095\">Sei des Weiteren <\/span><span class=\"maperiod\"><math display=\"inline\"><msubsup><mrow><mi class=\"MathClass-op\">\u2211<\/mi><mo> <\/mo> <\/mrow><mrow><mi>n<\/mi><mo class=\"MathClass-rel\">=<\/mo><mn>0<\/mn><\/mrow><mrow><mi>\u221e<\/mi><\/mrow><\/msubsup><mo class=\"MathClass-rel\">|<\/mo><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><mo class=\"MathClass-rel\">|<\/mo><msup><mrow><mi>R<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msup> <mo class=\"MathClass-rel\">&lt;<\/mo> <mi>\u221e<\/mi><\/math><\/span><span class=\"period\">.<\/span> <span class=\"ecti-1095\">Zeigen Sie, dass in diesem Fall die Funktion<\/span> <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mi>z<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo><mover accent=\"false\" class=\"mml-overline\"><mrow><msub><mrow><mi>B<\/mi><\/mrow><mrow><mi>R<\/mi><\/mrow><\/msub><mo class=\"MathClass-open\">(<\/mo><mn>0<\/mn><mo class=\"MathClass-close\">)<\/mo><\/mrow><mo accent=\"true\">\u00af<\/mo><\/mover> <mo class=\"MathClass-rel\">=<\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> {<\/mo><mrow><mi>z<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2102<\/mi><mo class=\"MathClass-rel\">\u2223<\/mo><mo class=\"MathClass-rel\">|<\/mo><mi>z<\/mi><mo class=\"MathClass-rel\">|<\/mo><mo class=\"MathClass-rel\">\u2264<\/mo> <mi>R<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">}<\/mo><\/mrow><mo class=\"MathClass-rel\">\u21a6<\/mo><munderover accent=\"false\" accentunder=\"false\"><mrow><mo>\u2211<\/mo> <\/mrow><mrow><mi>n<\/mi><mo class=\"MathClass-rel\">=<\/mo><mn>0<\/mn><\/mrow><mrow><mi>\u221e<\/mi><\/mrow><\/munderover><msub><mrow><mi>a<\/mi><\/mrow><mrow> <mi>n<\/mi><\/mrow><\/msub><msup><mrow><mi>z<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msup><\/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\">wohldefiniert und stetig ist.<\/span> <\/p> <\/div> <div class=\"me meexample\"> <p class=\"indent\"><\/p><h4 id=\"z084578002c20\"> <span class=\"ecbx-1095\">\u00dc<\/span><span class=\"ecbx-1095\">bung <\/span>(Bestimmte Divergenz am Rand)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Sei <\/span><math display=\"inline\"><msubsup><mrow><mi class=\"MathClass-op\"> \u2211<\/mi><mo> <\/mo> <\/mrow><mrow><mi>n<\/mi><mo class=\"MathClass-rel\">=<\/mo><mn>0<\/mn><\/mrow><mrow><mi>\u221e<\/mi><\/mrow><\/msubsup><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><msup><mrow><mi>x<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msup><\/math> <span class=\"ecti-1095\">eine Potenzreihe<\/span> <span class=\"ecti-1095\">mit Konvergenzradius <\/span><math display=\"inline\"><mi>R<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mo class=\"MathClass-open\">(<\/mo><mn>0<\/mn><mo class=\"MathClass-punc\">,<\/mo><mi>\u221e<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math> <span class=\"ecti-1095\">und<\/span> <span class=\"ecti-1095\">nicht-negativen Koeffizienten <\/span><math display=\"inline\"><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub> <mo class=\"MathClass-rel\">\u2265<\/mo> <mn>0<\/mn><\/math> <span class=\"ecti-1095\">f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r alle <\/span><math display=\"inline\"><mi>n<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2115<\/mi><\/math><span class=\"ecti-1095\">. Sei<\/span> <span class=\"ecti-1095\">des Weiteren <\/span><span class=\"maperiod\"><math display=\"inline\"><msubsup><mrow><mi class=\"MathClass-op\">\u2211<\/mi><mo> <\/mo> <\/mrow><mrow><mi>n<\/mi><mo class=\"MathClass-rel\">=<\/mo><mn>0<\/mn><\/mrow><mrow><mi>\u221e<\/mi><\/mrow><\/msubsup><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><msup><mrow><mi>R<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msup> <mo class=\"MathClass-rel\">=<\/mo> <mi>\u221e<\/mi><\/math><\/span><span class=\"period\">.<\/span> <span class=\"ecti-1095\">Zeigen Sie, dass in diesem Fall<\/span> <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><munder class=\"msub\"><mrow><mi class=\"qopname\">lim<\/mi><mo>  <\/mo><\/mrow><mrow><mi>x<\/mi><mo class=\"MathClass-rel\">\u2197<\/mo><mi>R<\/mi><\/mrow><\/munder><munderover accent=\"false\" accentunder=\"false\"><mrow><mo> \u2211<\/mo> <\/mrow><mrow><mi>n<\/mi><mo class=\"MathClass-rel\">=<\/mo><mn>0<\/mn><\/mrow><mrow><mi>\u221e<\/mi><\/mrow><\/munderover><msub><mrow><mi>a<\/mi><\/mrow><mrow> <mi>n<\/mi><\/mrow><\/msub><msup><mrow><mi>x<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msup> <mo class=\"MathClass-rel\">=<\/mo> <mi>\u221e<\/mi><mo class=\"MathClass-punc\">.<\/mo><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <\/div> <div class=\"me meexample\"> <p class=\"indent\"><\/p><h4 id=\"z04184e3c92b6\"> <span class=\"ecbx-1095\">\u00dc<\/span><span class=\"ecbx-1095\">bung <\/span>(Challenge)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Wir betrachten das Gitter <\/span><math display=\"inline\"><msup><mrow><mi>\u2115<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><\/math> <span class=\"ecti-1095\">in der Ebene <\/span><math display=\"inline\"><msup><mrow><mi>\u211d<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><\/math> <span class=\"ecti-1095\">und fixieren uns ein Quadrat darin, <\/span><span class=\"maperiod\"><math display=\"inline\"><mi>Q<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <msup><mrow><mo class=\"MathClass-open\">[<\/mo><mn>1<\/mn><mo class=\"MathClass-punc\">,<\/mo><mn>1<\/mn><mn>5<\/mn><mo class=\"MathClass-close\">]<\/mo><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup> <mo class=\"MathClass-bin\">\u2229<\/mo> <msup><mrow><mi>\u2115<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><\/math><\/span><span class=\"period\">.<\/span> <span class=\"ecti-1095\">Auf den Punkten <\/span><span class=\"maperiod\"><math display=\"inline\"><mo class=\"MathClass-open\">(<\/mo><mn>1<\/mn><mo class=\"MathClass-punc\">,<\/mo><mn>1<\/mn><mo class=\"MathClass-close\">)<\/mo><\/math><\/span><span class=\"period\">,<\/span> <span class=\"maperiod\"><math display=\"inline\"><mo class=\"MathClass-open\">(<\/mo><mn>2<\/mn><mo class=\"MathClass-punc\">,<\/mo> <mn>1<\/mn><mo class=\"MathClass-close\">)<\/mo><\/math><\/span><span class=\"period\">,<\/span> <span class=\"maperiod\"><math display=\"inline\"><mo class=\"MathClass-open\">(<\/mo><mn>3<\/mn><mo class=\"MathClass-punc\">,<\/mo> <mn>1<\/mn><mo class=\"MathClass-close\">)<\/mo><\/math><\/span><span class=\"period\">,<\/span> <span class=\"maperiod\"><math display=\"inline\"><mo class=\"MathClass-open\">(<\/mo><mn>1<\/mn><mo class=\"MathClass-punc\">,<\/mo> <mn>2<\/mn><mo class=\"MathClass-close\">)<\/mo><\/math><\/span><span class=\"period\">,<\/span> <span class=\"maperiod\"><math display=\"inline\"><mo class=\"MathClass-open\">(<\/mo><mn>2<\/mn><mo class=\"MathClass-punc\">,<\/mo> <mn>2<\/mn><mo class=\"MathClass-close\">)<\/mo><\/math><\/span><span class=\"period\">,<\/span> <math display=\"inline\"><mo class=\"MathClass-open\">(<\/mo><mn>1<\/mn><mo class=\"MathClass-punc\">,<\/mo> <mn>3<\/mn><mo class=\"MathClass-close\">)<\/mo><\/math> <span class=\"ecti-1095\">platzieren wir nun jeweils eine M<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">nze und beginnen dann folgendes Spiel. Sie als Spieler<\/span> <span class=\"ecti-1095\">d<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">rfen jeweils eine der M<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">nzen entfernen, worauf Sie eine M<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">nze oberhalb und eine M<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">nze<\/span> <span class=\"ecti-1095\">rechts von der entfernten M<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">nze platzieren. Dieser Zug ist aber nur dann erlaubt, wenn<\/span> <span class=\"ecti-1095\">die Pl<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">tze rechts und oberhalb der zu entfernenden M<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">nze noch frei sind. Ihre Aufgabe<\/span> <span class=\"ecti-1095\">besteht nun darin, nach endlich vielen Z<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">gen keine M<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">nzen mehr innerhalb des Quadrat<\/span> <math display=\"inline\"><mi>Q<\/mi><\/math> <span class=\"ecti-1095\">liegen zu haben. Ist das m<\/span><span class=\"ecti-1095\">\u00f6<\/span><span class=\"ecti-1095\">glich?<\/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\">Die Antwort ist nein. Um dies zu beweisen, suchen Sie nach einer Funktion <\/span><span class=\"maperiod\"><math display=\"inline\"><mi>f<\/mi> <mo class=\"MathClass-punc\">:<\/mo> <msup><mrow><mi>\u2115<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup> <mo class=\"MathClass-rel\">\u2192<\/mo> <mi>\u2115<\/mi><\/math><\/span><span class=\"period\">,<\/span> <span class=\"ecti-1095\">so dass die Summe von <\/span><math display=\"inline\"><mi>f<\/mi><\/math> <span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">ber die Positionen der M<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">nzen im <\/span><math display=\"inline\"><mi>n<\/mi><\/math><span class=\"ecti-1095\">-ten<\/span> <span class=\"ecti-1095\">Schritt nicht von <\/span><math display=\"inline\"><mi>n<\/mi><\/math> <span class=\"ecti-1095\">oder von der gew<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">hlten Strategie abh<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">ngt. In anderen Worten soll diese Summe unter der in<\/span> <span class=\"ecti-1095\">der Aufgabenstellung beschriebenen Operation erhalten bleiben.<\/span><\/p><\/details>  <\/div> \n","rendered":"\n<style scoped=\"scoped\">.cmr-5{font-size:50%;}\n.cmr-7{font-size:70%;}\n.cmmi-5{font-size:50%;font-style: italic;}\n.cmmi-7{font-size:70%;font-style: italic;}\n.cmmi-10{font-style: italic;}\n.cmsy-5{font-size:50%;}\n.cmsy-7{font-size:70%;}\n.cmbx-10{ font-weight: bold;}\n.cmbsy-10{font-weight: bold;}\n.cmbsy-10{font-weight: bold;}\n.cmbsy-10{font-weight: bold;}\n.cmbsy-7{font-size:70%;font-weight: bold;}\n.cmbsy-7{font-weight: bold;}\n.cmbsy-7{font-weight: bold;}\n.cmbsy-5{font-size:50%;font-weight: bold;}\n.cmbsy-5{font-weight: bold;}\n.cmbsy-5{font-weight: bold;}\n.cmex-7{font-size:70%;}\n.cmex-7x-x-71{font-size:49%;}\n.msam-7{font-size:70%;}\n.msam-5{font-size:50%;}\n.msbm-7{font-size:70%;}\n.msbm-5{font-size:50%;}\n.cmr-17{font-size:170%;}\n.cmr-12{font-size:120%;}\n.cmti-10{ font-style: italic;}\np{margin-top:0;margin-bottom:0}\np.indent{text-indent:0;}\np + p{margin-top:1em;}\np + div, p + pre {margin-top:1em;}\ndiv + p, pre + p {margin-top:1em;}\n@media print {div.crosslinks {visibility:hidden;}}\na img { border-top: 0; 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width:125%;}\ndt {text-align:right; font-weight:bold; clear:left; float:left;}\ndd {width:100%; padding-left:1em; padding-top: 0px; clear:right;}\ndd + dd {float:right; clear:both;}\ndd + dt {clear:both;}\ndt + dt {width: 100%; float: none; padding: 0 70% 0 0;}\ndt + dt + dd {margin-top: -2em;}\ndt + dt + dd + dt {margin-top: 2em;}\n<\/style>\n<style scoped=\"scoped\">\n\/* CSS Analysis-Skript D-Math ETHZ *\/\n\n\/* Uniform Font, also for headers *\/\nh3 {\n\tfont-family: \"Times New Roman\", serif;\n\tmargin-bottom: 35px;\n}\nh4 {\n\tfont-family: \"Times New Roman\", serif;\n}\nh5 {\n\tfont-family: \"Times New Roman\", serif;\n}\n\n\/* Bold font, e.g. for definitions *\/\n.ecbx-1095 {font-weight: 550 ;}\n\n\n\/* Uniform spacing, indent: larger, noindent, enumerate, itemize *\/\np.indent {\n\tmargin: 25px 0px 0px 0px;\n\ttext-indent: 0px; \n}\np.noindent {\n\tmargin: 15px 0px 0px 0px;\n\ttext-indent: 0px; \n}\ndl.enumerate {\n\tmargin: 0px 0px 0px 0px;\n}\ndl.enumerate dt, dl.enumerate dd {\n\tmargin-top: 15px;\n\tmargin-bottom: 0px;\n}\ndiv.custom-itemize {\n\tmargin: 0px 0px 0px 0px;\n}\ndiv.custom-itemize div.item-head {\n\tmargin-top: 15px;\n\tmargin-bottom: 0px;\n\ttext-align: center;\n}\ndiv.custom-itemize div.item-head:first-of-type {\n\tmargin-top: 0px;\n} \ndiv.custom-itemize div.item-content {\n\tmargin-top: 15px;\n\tmargin-bottom: 0px;\n}\n.MJXc-display {\n\tmargin: 15px 0px 0px 0px;\n}\n\n\n\n\/* green metheorem\/melemma CSS class for more\/medium important latex-theorem-environments *\/\n\/* metheorem box+header *\/\ndiv.metheorem {\n    margin-bottom: 40px;\n    margin-top: 40px;\n\tpadding: 0px 15px 15px 15px;\n    border: 1px solid #333;\n    border-color: #4eb79e;\n    background: #c7e4da;\n}\ndiv.metheorem h4 {\n    background: #4eb79e;\n    color: white;\n\tmargin-top: 12px;\n\tmargin-left: -15px;\n\tmargin-right: -15px;\n\tpadding: 0px 15px 0px 15px;\n}\n\/* melemma box+header *\/\ndiv.melemma {\n    margin-bottom: 40px;\n    margin-top: 40px;\n\tpadding: 0px 15px 15px 15px;\n    border: 1px solid #333;\n    border-color: #4eb79e;\n    background: #F2F2F2;\n}\ndiv.melemma h4 {\n    background: #4eb79e;\n    color: white;\n\tmargin-top: 12px;\n\tmargin-left: -15px;\n\tmargin-right: -15px;\n\tpadding: 0px 15px 0px 15px;\n}\n\/* meexample box+header *\/\ndiv.meexample {\n    margin-bottom: 30px;\n    margin-top: 30px;\n\tpadding: 0px 15px 15px 15px;\n\tborder-color: gainsboro;\n\tborder-style: solid;\n\tborder-width: thin;\n}\ndiv.meexample h4 {\n\tfont-size: inherit;\n\tfont-weight: bold;\n    padding: 15px 0px 0px 0px;\n\tmargin-top: 0px;\n\tmargin-bottom: 5px;\n}\ndiv.meexample h4+p.noindent, div.meexample h4+p.indent {\n\tmargin-top: 5px;\n\ttext-indent: 0px;\n}\n\/* padding and margins for stuff inside these boxes, CSS-selector &gt; doesn't work in WP *\/\ndiv.me details {\n\tmargin: 10px 0px 0px 0px;\n}\ndiv.me dd {\n    width: calc(100% - 30px);\n}\t\n\n\n\/* fixing background of pictures *\/\nimg {\n\tbackground: white;\n}\n\n\/* div-container for centered geoapplet *\/\ndiv.geoapplet {\n\tmargin-left: auto;\n\tmargin-right: auto;\n\tmargin-top: 15px;\n\tmax-width: 100%;\n}\ndiv.geoapplet iframe {\n\tborder-style: none;\n\tmax-height: 110vw;\n}\n\n\/* div-container for centered squeezed tables *\/\ndiv.websqueeze {\n\tmargin-left: auto;\n\tmargin-right: auto;\n}\n\n\/* two containers for squeezing text sizes *\/\ndiv.mesmalltext, div.mesmalltext * {\n\tfont-size: 15px;\n}\nspan.metinytext, span.metinytext * {\n\tfont-size: 12px;\n}\n\n\n\/* removing grid lines in equations *\/\n#content table.equation tr td, #content table.equation tr th {\n    border: none;\n}\n#content table.equation {\n    border: none;\n}\n\n\/* hover\/click-solution for short inline explanations and footnotes *\/\n.hover-text {    \/* hidden part *\/\n    display: none;\n}\n.marginpar {     \/* style for footnote as marginpar *\/\n\ttext-decoration: none;\n\tborder: solid;\n\tborder-width: 1pt;\n\tpadding: 3pt;\t\n\twidth: 30%;\n\tbackground: white;\n}\n.hover-trigger { \/* style for hover\/click-trigger text\/symbol *\/\n\tbackground: none;\n\tborder: none;\n\tpadding: 0;\n\toutline: inherit;\t\n\ttext-transform: none;\n\tfont: inherit;\n\tposition: inherit;\n\tvertical-align: baseline;\n    color: #FF7F00;\n\tcursor: help;\n}\n.hover-trigger:hover +.hover-text{\n    display: inline;\n}\n.hover-trigger:active +.hover-text{\n    display: inline;\n}\n\n\/* simplifying style of details\/summary, removing triangle *\/\ndetails summary {\n  background: none;\n  list-style: none;\n  outline: none;\n  cursor: pointer;\n}\ndetails summary::-webkit-details-marker { \n  display: inline;\n  display: none;\n}\n\n\/* MC-True\/False as inline details\/summary *\/\ndetails.mcquest, div.me details.mcquest {\n\tdisplay: inline;\n\tmargin-top: 0px;\n}\nsummary.mcquest {\n\tdisplay: inline;\n\tcolor: #FF7F00;\n\tcursor: help;\n}\n\n\/* proof style: simple black box with gray background \n                little black square at the end on the right *\/\ndiv.proof {\n\tborder-color: black;\n\tborder-style: solid;\n\tborder-width: thin;\n\tbackground-color: #F2F2F2;\n\tpadding: 15px;\n\tmargin-top: 1em; \n}\ndiv.proof p:first-of-type {\n\tmargin: 0px;\n}\ndiv.qed {\n\tmargin-top: -25px;\n\tmargin-bottom: -7px;\n\ttext-align: right;\n}\ntable.equation+div.qed {\n\tmargin-top: -65px;\n}\n\n\/* The following is making also math-formulas inside the headers of Lemmas, etc., white. *\/\ndiv.melemma h4 span {\n    color: white;\n}\ndiv.metheorem h4 span {\n    color: white;\n}\n\n\/* The following are used to avoid fullstop, period, colon, semicolon, and endquote (broader) to move by itself to the next line after a formula.\n   The math-environment before needs to be wrapped in span.maperiod and the fullstop etc. in a span.period --- together they achieve what we want.  *\/\nspan.maperiod {\n       margin-right: 5px;\n}\nspan.period {\n       display: inline-block;\n       width: 0px;\n       margin-left: -5px;\n       margin-right: 4.9px;\n\t   text-indent: 0px;\n}\nspan.maendquote {\n       margin-right: 8px;\n}\nspan.endquote {\n       display: inline-block;\n       width: 0px;\n       margin-left: -8px;\n       margin-right: 7.9px;\n}\n\n\n\/* The following is removing an extra space left of the equation side in aligned equations *\/\nspan.mjx-mtd {\n    padding-left: 0em !important;\n}\n\n\/* The following fixes the weird problem that math appears smaller if it was rendered while the details tag was closed. *\/\ndetails span.mjx-chtml, details span.MathJax_CHTML {\n font-size: 100% !important;\n}\n\n\/* trying to fix line breaks in verbatim, new lines are missing *\/\npre.verbatim {\n\twhite-space: pre-wrap;\n\tfont-size: small;\n}\n<\/style><h3 id=\"z6f8a0471ddc2\" class=\"sectionHead\"><span class=\"titlemark\">7.9 <\/span> <a id=\"x1-2220009\"><\/a>Weitere Lernmaterialien<\/h3> <a id=\"x1-222001r221\"><\/a> <h4 id=\"zbd77bd63a8cf\" class=\"subsectionHead\"><span class=\"titlemark\">7.9.1 <\/span> <a id=\"x1-2230001\"><\/a>Verwendung des Kapitels<\/h4> <p class=\"noindent\">Wie wir gesehen haben, sind Potenzreihen, deren Konvergenzradius und Konvergenzverhalten fundamentale Werkzeuge f\u00fcr die Definition von vielen Ihnen bereits bekannten Funktionen (und auch weiteren). Wir werden also ab nun sowohl die komplexe Exponentialfunktion, die trigonometrischen Funktionen auf <math display=\"inline\"><mi>\u211d<\/mi><\/math> und auf <span class=\"maperiod\"><math display=\"inline\"><mi>\u2102<\/mi><\/math><\/span><span class=\"period\">,<\/span> als auch die hyperbolischen Funktionen gemeinsam mit den wichtigsten Eigenschaften dieser Funktionen (meist ohne Verweise) verwenden. (Die Umkehrfunktionen der trigonometrischen Funktionen werden wir erst im n\u00e4chsten Kapitel einf\u00fchren.) <\/p><p class=\"indent\">F\u00fcr den Begriff der Potenzreihe ben\u00f6tigten wir die grundlegenden Definitionen der Reihe und der Funktionenfolgen. F\u00fcr Reihen ist die Unterscheidung der bedingten und absoluten Konvergenz fundamental, da gewisse Operationen (Umordnen, Cauchy-Produkt) nur f\u00fcr den letzteren Konvergenzbegriff erlaubt sind. Dabei ist es sehr hilfreich, dass f\u00fcr Potenzreihen im Inneren des Konvergenzbereichs absolute Konvergenz vorliegt und damit alle Operationen erlaubt sind. Die folgenden Konvergenzkriterien sind f\u00fcr Beispiele aber auch f\u00fcr die Theorie unabdingbar: <\/p> <div class=\"custom-itemize\"><div class=\"item-head\"> <span class=\"tcrm-1095\">\u2022<\/span><\/div><div class=\"item-content\">die geometrische Reihe in Beispiel <a href=\"..\/..\/chapter\/reihen#x1-187003r3\">7.3<\/a>, <\/div><div class=\"item-head\"> <span class=\"tcrm-1095\">\u2022<\/span><\/div><div class=\"item-content\">Majoranten-   und   Minorantenkriterium   f\u00fcr   Reihen   mit   positiven   Gliedern   in Korollar&nbsp;<a href=\"..\/..\/chapter\/reihen#x1-188002r12\">7.12<\/a> und Korollar <a href=\"..\/..\/chapter\/absolute-konvergenz#x1-193001r29\">7.29<\/a>, <\/div><div class=\"item-head\"> <span class=\"tcrm-1095\">\u2022<\/span><\/div><div class=\"item-content\">Verdichtung in Proposition <a href=\"..\/..\/chapter\/reihen#x1-188006r16\">7.16<\/a>, <\/div><div class=\"item-head\"> <span class=\"tcrm-1095\">\u2022<\/span><\/div><div class=\"item-content\"><math display=\"inline\"><mi>p<\/mi><\/math>-Test in Beispiel <a href=\"..\/..\/chapter\/reihen#x1-188007r17\">7.17<\/a>, <\/div><div class=\"item-head\"> <span class=\"tcrm-1095\">\u2022<\/span><\/div><div class=\"item-content\">Leibniz-Kriterium in Proposition <a href=\"..\/..\/chapter\/reihen#x1-190001r25\">7.25<\/a> (welches vor allem f\u00fcr bedingt konvergente aber wegen der  Fehlerabsch\u00e4tzung  auch  f\u00fcr  absolut  konvergente  Reihen  n\u00fctzlich  sein kann), <\/div><div class=\"item-head\"> <span class=\"tcrm-1095\">\u2022<\/span><\/div><div class=\"item-content\">Cauchy-Kriterium in Satz <a href=\"..\/..\/chapter\/reihen#x1-191001r26\">7.26<\/a> (meist als theoretisches Hilfsmittel),                                                                                                                                                                           <\/div><div class=\"item-head\"> <span class=\"tcrm-1095\">\u2022<\/span><\/div><div class=\"item-content\">Wurzelkriterium in Korollar <a href=\"..\/..\/chapter\/absolute-konvergenz#x1-193002r30\">7.30<\/a> (als theoretisches und praktisches Hilfsmittel), <\/div><div class=\"item-head\"> <span class=\"tcrm-1095\">\u2022<\/span><\/div><div class=\"item-content\">Quotientenkriterium  in  Korollar  <a href=\"..\/..\/chapter\/absolute-konvergenz#x1-193004r32\">7.32<\/a>  (meist  als  praktisches  Hilfsmittel,  da  es  oft einfacher anwendbar ist, aber im Gegensatz zu dem Wurzelkriterium zum Beispiel f\u00fcr Potenzreihen weniger allgemein einsetzbar ist), <\/div><div class=\"item-head\"> <span class=\"tcrm-1095\">\u2022<\/span><\/div><div class=\"item-content\">aber wenn sonst nichts zum Erfolg f\u00fchrt, sollte man nicht vergessen, dass auf Grund von Proposition <a href=\"..\/..\/chapter\/reihen#x1-187002r2\">7.2<\/a> die Folgenglieder einer konvergenten Reihe eine Nullfolge bilden.<\/div><\/div> <p class=\"noindent\">Wir bemerken noch, dass diese Kriterien sehr hilfreich sind f\u00fcr die Entscheidung ob Konvergenz oder Divergenz bei einer Reihe vorliegt, doch haben wir sehr wenige allgemeine Gesetze um den Grenzwert von Reihen zu bestimmen. <\/p><p class=\"indent\">Wie bereits erw\u00e4hnt war der Begriff der Funktionenfolge auch f\u00fcr die Besprechung der Potenzreihen notwendig. F\u00fcr Funktionenfolgen haben wir zwei unterschiedliche Konvergenzbegriffe besprochen. Der Begriff der punktweisen Konvergenz mag zwar als der nat\u00fcrliche Konvergenzbegriff f\u00fcr Funktionen betrachtet werden, doch hat dieser keine guten Eigenschaften (weder f\u00fcr Stetigkeit noch f\u00fcr das Riemann-Integral). Sie sollten die entsprechenden Gegenbeispiele im Ged\u00e4chtnis behalten. Dies motivierte die Definition der gleichm\u00e4ssigen Konvergenz, welche wegen den guten Eigenschaften f\u00fcr Stetigkeit und das Riemann-Integral f\u00fcr uns immer wieder wichtig sein wird. Die Unterscheidung dieser Konvergenzbegriffe ist wohlgemerkt keine Spitzfindigkeit. <a id=\"x1-223001r223\"><\/a> <\/p> <h4 id=\"zcbbfaabbe411\" class=\"subsectionHead\"><span class=\"titlemark\">7.9.2 <\/span> <a id=\"x1-2240002\"><\/a>\u00dcbungen<\/h4> <div class=\"me meexample\"> <div class=\"wp-nocaption \"><\/div><h4 id=\"z2a25b2aa8160\"> <span class=\"ecbx-1095\">\u00dc<\/span><span class=\"ecbx-1095\">bung.<\/span><\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Sei <\/span><math display=\"inline\"><msubsup><mrow><mi class=\"MathClass-op\"> \u2211<\/mi><mo> <\/mo> <\/mrow><mrow><mi>k<\/mi><mo class=\"MathClass-rel\">=<\/mo><mn>1<\/mn><\/mrow><mrow><mi>\u221e<\/mi><\/mrow><\/msubsup><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>k<\/mi><\/mrow><\/msub><\/math> <span class=\"ecti-1095\">eine konvergente Reihe. Falls <\/span><math display=\"inline\"><msub><mrow><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>k<\/mi><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mi>k<\/mi><\/mrow><\/msub><\/math> <span class=\"ecti-1095\">eine monoton fallende Folge ist, so ist nicht nur <\/span><span class=\"maperiod\"><math display=\"inline\"><msub><mrow><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>k<\/mi><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mi>k<\/mi><\/mrow><\/msub><\/math><\/span><span class=\"period\">,<\/span> <span class=\"ecti-1095\">sondern auch <\/span><math display=\"inline\"><msub><mrow><mo class=\"MathClass-open\">(<\/mo><mi>k<\/mi><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>k<\/mi><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mi>k<\/mi><\/mrow><\/msub><\/math> <span class=\"ecti-1095\">eine Nullfolge. Beweisen Sie dies.<\/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\">\u00dc<\/span><span class=\"ecti-1095\">berzeugen Sie sich von der Ungleichung<\/span> <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mo class=\"MathClass-open\">(<\/mo><mi>n<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>m<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mn>1<\/mn><mo class=\"MathClass-close\">)<\/mo><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub> <mo class=\"MathClass-rel\">\u2264<\/mo><munderover accent=\"false\" accentunder=\"false\"><mrow><mo>\u2211<\/mo> <\/mrow><mrow><mi>k<\/mi><mo class=\"MathClass-rel\">=<\/mo><mi>m<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/munderover><msub><mrow><mi>a<\/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\"><span class=\"ecti-1095\">f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r <\/span><span class=\"maperiod\"><math display=\"inline\"><mi>n<\/mi> <mo class=\"MathClass-rel\">&gt;<\/mo> <mi>m<\/mi><\/math><\/span><span class=\"period\">.<\/span> <\/p><\/details>  <\/div> <div class=\"me meexample\"> <div class=\"wp-nocaption \"><\/div><h4 id=\"z926c9b97e4d5\"> <span class=\"ecbx-1095\">\u00dc<\/span><span class=\"ecbx-1095\">bung.<\/span><\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Sei <\/span><math display=\"inline\"><mi>f<\/mi> <mo class=\"MathClass-punc\">:<\/mo> <mi>\u2115<\/mi> <mo class=\"MathClass-rel\">\u2192<\/mo> <mi>\u2115<\/mi><\/math> <span class=\"ecti-1095\">eine bijektive Abbildung. Konvergiert die Reihe <\/span><span class=\"maendquote\"><math display=\"inline\"><msubsup><mrow><mi class=\"MathClass-op\">\u2211<\/mi><mo> <\/mo> <\/mrow><mrow><mi>n<\/mi><mo class=\"MathClass-rel\">=<\/mo><mn>1<\/mn><\/mrow><mrow><mi>\u221e<\/mi><\/mrow><\/msubsup><mfrac><mrow><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>n<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow> <mrow><msup><mrow><mi>n<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><\/mrow><\/mfrac> <\/math><\/span><span class=\"endquote\">?<\/span> <\/p><div class=\"wp-nocaption \"><\/div><details><summary style=\"color:#FF7F00\"><span class=\"ecti-1095\">Hinweis.<\/span><\/summary><p class=\"indent\" style=\"margin-top: 0\"><span class=\"ecti-1095\">F<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r jedes <\/span><math display=\"inline\"><mi>n<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2115<\/mi><\/math> <span class=\"ecti-1095\">gilt <\/span><span class=\"maperiod\"><math display=\"inline\"><mo class=\"MathClass-rel\">|<\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> {<\/mo><mrow><mi>k<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2115<\/mi><mo class=\"MathClass-rel\">\u2223<\/mo><msup><mrow><mn>2<\/mn><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msup> <mo class=\"MathClass-rel\">\u2264<\/mo> <mi>k<\/mi> <mo class=\"MathClass-rel\">&lt;<\/mo> <msup><mrow><mn>2<\/mn><\/mrow><mrow><mi>n<\/mi><mo class=\"MathClass-bin\">+<\/mo><mn>1<\/mn><\/mrow><\/msup><mstyle class=\"text\"><mtext>&nbsp;und&nbsp;<\/mtext><\/mstyle><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>k<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">\u2265<\/mo> <msup><mrow><mn>2<\/mn><\/mrow><mrow><mi>n<\/mi><mo class=\"MathClass-bin\">\u2212<\/mo><mn>1<\/mn><\/mrow><\/msup><\/mrow><mo fence=\"true\" form=\"postfix\">}<\/mo><\/mrow> <mo class=\"MathClass-rel\">|<\/mo><mo class=\"MathClass-rel\">\u2265<\/mo> <msup><mrow><mn>2<\/mn><\/mrow><mrow><mi>n<\/mi><mo class=\"MathClass-bin\">\u2212<\/mo><mn>1<\/mn><\/mrow><\/msup><\/math><\/span><span class=\"period\">.<\/span><\/p><\/details>  <\/div> <div class=\"me meexample\"> <div class=\"wp-nocaption \"><\/div><h4 id=\"zeb59e7c92ab8\"> <span class=\"ecbx-1095\">\u00dc<\/span><span class=\"ecbx-1095\">bung <\/span>(Raabes Quotientenkriterium)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Sei <\/span><math display=\"inline\"><msub><mrow><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi> <\/mrow> <\/msub> <mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><\/math> <span class=\"ecti-1095\">eine Folge<\/span> <span class=\"ecti-1095\">komplexer Zahlen mit <\/span><math display=\"inline\"><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><mo class=\"MathClass-rel\">\u2260<\/mo><mn>0<\/mn><\/math> <span class=\"ecti-1095\">f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r alle <\/span><span class=\"maperiod\"><math display=\"inline\"><mi>n<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2115<\/mi><\/math><\/span><span class=\"period\">,<\/span> <span class=\"ecti-1095\">so dass <\/span><math display=\"inline\"><munder class=\"msub\"><mrow><mi class=\"qopname\"> lim<\/mi><mo>  <\/mo> <\/mrow><mrow><mi>n<\/mi><mo class=\"MathClass-rel\">\u2192<\/mo><mi>\u221e<\/mi><\/mrow><\/munder><mfrac><mrow><mo class=\"MathClass-rel\">|<\/mo><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi><mo class=\"MathClass-bin\">+<\/mo><mn>1<\/mn><\/mrow><\/msub><mo class=\"MathClass-rel\">|<\/mo><\/mrow> <mrow><mo class=\"MathClass-rel\">|<\/mo><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><mo class=\"MathClass-rel\">|<\/mo><\/mrow><\/mfrac> <mo class=\"MathClass-rel\">=<\/mo> <mn>1<\/mn><\/math> <span class=\"ecti-1095\">und<\/span> <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mi>Q<\/mi> <mo class=\"MathClass-rel\">=<\/mo><munder class=\"msub\"><mrow><mi class=\"qopname\"> lim<\/mi><mo>  <\/mo><\/mrow><mrow><mi>n<\/mi><mo class=\"MathClass-rel\">\u2192<\/mo><mi>\u221e<\/mi><\/mrow><\/munder><mi>n<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mn>1<\/mn> <mo class=\"MathClass-bin\">\u2212<\/mo><mfrac><mrow><mo class=\"MathClass-rel\">|<\/mo><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi><mo class=\"MathClass-bin\">+<\/mo><mn>1<\/mn><\/mrow><\/msub><mo class=\"MathClass-rel\">|<\/mo><\/mrow> <mrow><mo class=\"MathClass-rel\">|<\/mo><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><mo class=\"MathClass-rel\">|<\/mo><\/mrow><\/mfrac> <\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\"><span class=\"ecti-1095\">existiert. Zeigen Sie, dass die Reihe <\/span><math display=\"inline\"><msubsup><mrow><mi class=\"MathClass-op\">\u2211<\/mi><mo> <\/mo> <\/mrow><mrow><mi>n<\/mi><mo class=\"MathClass-rel\">=<\/mo><mn>1<\/mn><\/mrow><mrow><mi>\u221e<\/mi><\/mrow><\/msubsup><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><\/math> <span class=\"ecti-1095\">konvergiert, falls <\/span><span class=\"maperiod\"><math display=\"inline\"><mi>Q<\/mi> <mo class=\"MathClass-rel\">&gt;<\/mo> <mn>1<\/mn><\/math><\/span><span class=\"period\">.<\/span> <span class=\"ecti-1095\">(J.<\/span><span class=\"ecti-1095\">&nbsp;Raabe war einer der ersten Mathematikprofessoren an der ETH Z<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">rich.)<\/span> <\/p><p class=\"indent\"><span class=\"ecti-1095\">Hinweis: Finden Sie ein <\/span><math display=\"inline\"><mi>p<\/mi> <mo class=\"MathClass-rel\">&gt;<\/mo> <mn>1<\/mn><\/math><span class=\"ecti-1095\">, so<\/span> <span class=\"ecti-1095\">dass f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r alle bis auf endlich viele <\/span><math display=\"inline\"><mi>n<\/mi><\/math> <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mfrac><mrow><mo class=\"MathClass-rel\">|<\/mo><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi><mo class=\"MathClass-bin\">+<\/mo><mn>1<\/mn><\/mrow><\/msub><mo class=\"MathClass-rel\">|<\/mo><\/mrow> <mrow><mo class=\"MathClass-rel\">|<\/mo><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><mo class=\"MathClass-rel\">|<\/mo><\/mrow><\/mfrac> <mo class=\"MathClass-rel\">&lt;<\/mo> <mn>1<\/mn> <mo class=\"MathClass-bin\">\u2212<\/mo><mfrac><mrow><mi>p<\/mi><\/mrow> <mrow><mi>n<\/mi><\/mrow><\/mfrac><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\"><span class=\"ecti-1095\">gilt. Verwenden Sie die kontinuierliche Bernoulli-Ungleichung aus <\/span><span class=\"ecti-1095\">\u00dc<\/span><span class=\"ecti-1095\">bung <\/span><a href=\"..\/..\/chapter\/die-exponentialfunktion#x1-173006r36\"><span class=\"ecti-1095\">6.36<\/span><\/a><span class=\"ecti-1095\">, um zu zeigen,<\/span> <span class=\"ecti-1095\">dass<\/span> <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mfrac><mrow><mo class=\"MathClass-rel\">|<\/mo><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi><mo class=\"MathClass-bin\">+<\/mo><mn>1<\/mn><\/mrow><\/msub><mo class=\"MathClass-rel\">|<\/mo><\/mrow> <mrow><mo class=\"MathClass-rel\">|<\/mo><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><mo class=\"MathClass-rel\">|<\/mo><\/mrow><\/mfrac> <mo class=\"MathClass-rel\">\u2264<\/mo><msup><mrow> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mn>1<\/mn> <mo class=\"MathClass-bin\">\u2212<\/mo> <mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mi>n<\/mi><\/mrow><\/mfrac> <\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><\/mrow><mrow><mi>p<\/mi><\/mrow><\/msup> <mo class=\"MathClass-rel\">=<\/mo> <mfrac><mrow><msup><mrow><mo class=\"MathClass-open\">(<\/mo><mi>n<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <mn>1<\/mn><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mi>p<\/mi><\/mrow><\/msup><\/mrow> <mrow><msup><mrow><mi>n<\/mi><\/mrow><mrow><mi>p<\/mi><\/mrow><\/msup><\/mrow><\/mfrac> <mo class=\"MathClass-punc\">.<\/mo><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\"><span class=\"ecti-1095\">Schliessen Sie nun auf die Aussage unter Verwendung von Korollar <\/span><a href=\"..\/..\/chapter\/absolute-konvergenz#x1-193001r29\"><span class=\"ecti-1095\">7.29<\/span><\/a> <span class=\"ecti-1095\">und Beispiel<\/span> <a href=\"..\/..\/chapter\/reihen#x1-188007r17\"><span class=\"ecti-1095\">7.17<\/span><\/a><span class=\"ecti-1095\">.<\/span> <\/p> <\/div> <div class=\"me meexample\"> <div class=\"wp-nocaption \"><\/div><h4 id=\"z1b0d38f7ef06\"> <span class=\"ecbx-1095\">\u00dc<\/span><span class=\"ecbx-1095\">bung <\/span>(Kronecker\u2019s Lemma)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Sei <\/span><math display=\"inline\"><msubsup><mrow><mi class=\"MathClass-op\"> \u2211<\/mi><mo> <\/mo> <\/mrow><mrow><mi>k<\/mi><mo class=\"MathClass-rel\">=<\/mo><mn>1<\/mn><\/mrow><mrow><mi>\u221e<\/mi><\/mrow><\/msubsup><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>k<\/mi><\/mrow><\/msub><\/math> <span class=\"ecti-1095\">eine konvergente<\/span> <span class=\"ecti-1095\">Reihe und sei <\/span><math display=\"inline\"><msub><mrow><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>b<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><\/math> <span class=\"ecti-1095\">eine divergente monoton wachsende Folge positiver Zahlen. Dann gilt<\/span> <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><munder class=\"msub\"><mrow><mi class=\"qopname\">lim<\/mi><mo>  <\/mo><\/mrow><mrow><mi>n<\/mi><mo class=\"MathClass-rel\">\u2192<\/mo><mi>\u221e<\/mi><\/mrow><\/munder> <mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><msub><mrow><mi>b<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><\/mrow><\/mfrac><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> <mn>0<\/mn><mo class=\"MathClass-punc\">.<\/mo><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\"><span class=\"ecti-1095\">Beweisen Sie Kronecker\u2019s Lemma unter Verwendung von Abel-Summation (<\/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#x1-78001r3\"><span class=\"ecti-1095\">3.3<\/span><\/a><span class=\"ecti-1095\">).<\/span> <\/p> <\/div> <div class=\"me meexample\"> <div class=\"wp-nocaption \"><\/div><h4 id=\"z8c69610e1c6f\"> <span class=\"ecbx-1095\">\u00dc<\/span><span class=\"ecbx-1095\">bung <\/span>(Vertauschung der Summationsreihenfolge)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Wie schon vor dem Beweis des Produktsatzes (Satz <\/span><a href=\"..\/..\/chapter\/absolute-konvergenz#x1-195001r36\"><span class=\"ecti-1095\">7.36<\/span><\/a><span class=\"ecti-1095\">) angedeutet, ist der<\/span> <span class=\"ecti-1095\">Produktsatz stark mit Vertauschbarkeit von Summationsreihenfolge verwandt. Wir<\/span> <span class=\"ecti-1095\">wollen dies hier genauer formulieren. Dazu betrachten wir eine doppelt indizierte Folge<\/span> <span class=\"maperiod\"><math display=\"inline\"><msub><mrow><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>a<\/mi><\/mrow><mrow><mo class=\"MathClass-open\">(<\/mo><mi>m<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>n<\/mi><mo class=\"MathClass-close\">)<\/mo> <\/mrow> <\/msub> <mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mo class=\"MathClass-open\">(<\/mo><mi>m<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>n<\/mi><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-rel\">\u2208<\/mo><msup><mrow><mi>\u2115<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><\/mrow><\/msub><\/math><\/span><span class=\"period\">.<\/span> <\/p><dl class=\"enumerate\"><dt class=\"enumerate\"> <span class=\"ecti-1095\">a)<\/span><\/dt><dd class=\"enumerate\"><span class=\"ecti-1095\">Um zu sehen, dass die Vertauschbarkeit der Summationsreihenfolge f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r Reihen nicht immer gilt,<\/span> <span class=\"ecti-1095\">definieren wir<\/span> <math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><msub><mrow><mi>a<\/mi><\/mrow><mrow><mo class=\"MathClass-open\">(<\/mo><mi>m<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>n<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow><\/msub> <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=\"left\"><mn>1<\/mn> <\/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> <\/mtd> <\/mtr> <mtr><mtd class=\"array\" columnalign=\"left\"> <mo class=\"MathClass-bin\">\u2212<\/mo> <mn>1<\/mn><\/mtd><mtd class=\"array\" columnalign=\"left\"><mstyle class=\"text\"><mtext>falls&nbsp;<\/mtext><\/mstyle><mi>m<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mn>1<\/mn> <mo class=\"MathClass-rel\">=<\/mo> <mi>n<\/mi><\/mtd> <\/mtr> <mtr><mtd class=\"array\" columnalign=\"left\"><mn>0<\/mn> <\/mtd><mtd class=\"array\" columnalign=\"left\"><mstyle class=\"text\"><mtext>sonst&nbsp;<\/mtext><\/mstyle><mo class=\"MathClass-punc\">.<\/mo> <\/mtd><\/mtr><\/mtable> <\/mrow><mo fence=\"true\" form=\"postfix\" \/><\/mrow><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\"><span class=\"ecti-1095\">f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r alle <\/span><span class=\"maperiod\"><math display=\"inline\"><mo class=\"MathClass-open\">(<\/mo><mi>m<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>n<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">\u2208<\/mo> <msup><mrow><mi>\u2115<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><\/math><\/span><span class=\"period\">.<\/span> <span class=\"ecti-1095\">Zeigen Sie, dass die Doppelreihen<\/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>m<\/mi><mo class=\"MathClass-rel\">=<\/mo><mn>1<\/mn><\/mrow><mrow><mi>\u221e<\/mi><\/mrow><\/munderover><mstyle><mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow \/><mo fence=\"true\" form=\"postfix\" \/><\/mrow><\/mstyle><munderover accent=\"false\" accentunder=\"false\"><mrow><mo>\u2211<\/mo> <\/mrow><mrow><mi>n<\/mi><mo class=\"MathClass-rel\">=<\/mo><mn>1<\/mn><\/mrow><mrow><mi>\u221e<\/mi><\/mrow><\/munderover><msub><mrow><mi>a<\/mi><\/mrow><mrow> <mo class=\"MathClass-open\">(<\/mo><mi>m<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>n<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow><\/msub><mstyle><mrow><mo fence=\"true\" form=\"prefix\"> )<\/mo><mrow \/><mo fence=\"true\" form=\"postfix\" \/><\/mrow><\/mstyle><mo class=\"MathClass-punc\">,<\/mo><mspace class=\"quad\" width=\"1em\" \/><munderover accent=\"false\" accentunder=\"false\"><mrow><mo>\u2211<\/mo> <\/mrow><mrow><mi>n<\/mi><mo class=\"MathClass-rel\">=<\/mo><mn>1<\/mn><\/mrow><mrow><mi>\u221e<\/mi><\/mrow><\/munderover><mstyle><mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow \/><mo fence=\"true\" form=\"postfix\" \/><\/mrow><\/mstyle><munderover accent=\"false\" accentunder=\"false\"><mrow><mo>\u2211<\/mo> <\/mrow><mrow><mi>m<\/mi><mo class=\"MathClass-rel\">=<\/mo><mn>1<\/mn><\/mrow><mrow><mi>\u221e<\/mi><\/mrow><\/munderover><msub><mrow><mi>a<\/mi><\/mrow><mrow> <mo class=\"MathClass-open\">(<\/mo><mi>m<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>n<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow><\/msub><mstyle><mrow><mo fence=\"true\" form=\"prefix\"> )<\/mo><mrow \/><mo fence=\"true\" form=\"postfix\" \/><\/mrow><\/mstyle><\/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\">konvergieren, aber verschieden sind.<\/span> <\/p><\/dd><dt class=\"enumerate\"> <span class=\"ecti-1095\">b)<\/span><\/dt><dd class=\"enumerate\"><span class=\"ecti-1095\">Angenommen <\/span><math display=\"inline\"><msub><mrow><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>a<\/mi><\/mrow><mrow><mo class=\"MathClass-open\">(<\/mo><mi>m<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>n<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mo class=\"MathClass-open\">(<\/mo><mi>m<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>n<\/mi><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-rel\">\u2208<\/mo><msup><mrow><mi>\u2115<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><\/mrow><\/msub><\/math> <span class=\"ecti-1095\">erf<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">llt<\/span> <math display=\"inline\"><msubsup><mrow><mi class=\"MathClass-op\">\u2211<\/mi><mo> <\/mo> <\/mrow><mrow><mi>m<\/mi><mo class=\"MathClass-rel\">=<\/mo><mn>1<\/mn><\/mrow><mrow><mi>\u221e<\/mi><\/mrow><\/msubsup><mstyle><mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow \/><mo fence=\"true\" form=\"postfix\" \/><\/mrow><\/mstyle><msubsup><mrow><mi class=\"MathClass-op\">\u2211<\/mi><mo> <\/mo> <\/mrow><mrow><mi>n<\/mi><mo class=\"MathClass-rel\">=<\/mo><mn>1<\/mn><\/mrow><mrow><mi>\u221e<\/mi><\/mrow><\/msubsup><mo class=\"MathClass-rel\">|<\/mo><msub><mrow><mi>a<\/mi><\/mrow><mrow><mo class=\"MathClass-open\">(<\/mo><mi>m<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>n<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow><\/msub><mo class=\"MathClass-rel\">|<\/mo><mstyle><mrow><mo fence=\"true\" form=\"prefix\"> )<\/mo><mrow \/><mo fence=\"true\" form=\"postfix\" \/><\/mrow><\/mstyle> <mo class=\"MathClass-rel\">&lt;<\/mo> <mi>\u221e<\/mi><\/math><span class=\"ecti-1095\">. Zeigen Sie, dass<\/span> <span class=\"ecti-1095\">die beiden Reihen <\/span><math display=\"inline\"><msubsup><mrow><mi class=\"MathClass-op\">\u2211<\/mi><mo> <\/mo> <\/mrow><mrow><mi>m<\/mi><mo class=\"MathClass-rel\">=<\/mo><mn>1<\/mn><\/mrow><mrow><mi>\u221e<\/mi><\/mrow><\/msubsup><mstyle><mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow \/><mo fence=\"true\" form=\"postfix\" \/><\/mrow><\/mstyle><msubsup><mrow><mi class=\"MathClass-op\">\u2211<\/mi><mo> <\/mo> <\/mrow><mrow><mi>n<\/mi><mo class=\"MathClass-rel\">=<\/mo><mn>1<\/mn><\/mrow><mrow><mi>\u221e<\/mi><\/mrow><\/msubsup><msub><mrow><mi>a<\/mi><\/mrow><mrow><mo class=\"MathClass-open\">(<\/mo><mi>m<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>n<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow><\/msub><mstyle><mrow><mo fence=\"true\" form=\"prefix\"> )<\/mo><mrow \/><mo fence=\"true\" form=\"postfix\" \/><\/mrow><\/mstyle><\/math> <span class=\"ecti-1095\">und <\/span><math display=\"inline\"><msubsup><mrow><mi class=\"MathClass-op\"> \u2211<\/mi><mo> <\/mo> <\/mrow><mrow><mi>n<\/mi><mo class=\"MathClass-rel\">=<\/mo><mn>1<\/mn><\/mrow><mrow><mi>\u221e<\/mi><\/mrow><\/msubsup><mstyle><mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow \/><mo fence=\"true\" form=\"postfix\" \/><\/mrow><\/mstyle><msubsup><mrow><mi class=\"MathClass-op\">\u2211<\/mi><mo> <\/mo> <\/mrow><mrow><mi>m<\/mi><mo class=\"MathClass-rel\">=<\/mo><mn>1<\/mn><\/mrow><mrow><mi>\u221e<\/mi><\/mrow><\/msubsup><msub><mrow><mi>a<\/mi><\/mrow><mrow><mo class=\"MathClass-open\">(<\/mo><mi>m<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>n<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow><\/msub><mstyle><mrow><mo fence=\"true\" form=\"prefix\"> )<\/mo><mrow \/><mo fence=\"true\" form=\"postfix\" \/><\/mrow><\/mstyle><\/math> <span class=\"ecti-1095\">konvergent sind und den gleichen Wert haben.<\/span><\/dd><\/dl> <\/div> <div class=\"me meexample\"> <div class=\"wp-nocaption \"><\/div><h4 id=\"z003dfb34fab8\"> <span class=\"ecbx-1095\">\u00dc<\/span><span class=\"ecbx-1095\">bung <\/span>(Zerlegung in gerade und ungerade Funktionen)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Sei<\/span> <math display=\"inline\"><mi>f<\/mi> <mo class=\"MathClass-punc\">:<\/mo> <mi>\u2102<\/mi> <mo class=\"MathClass-rel\">\u2192<\/mo> <mi>\u2102<\/mi><\/math> <span class=\"ecti-1095\">eine               Funktion.               Zeigen               Sie,               dass               sich<\/span> <math display=\"inline\"><mi>f<\/mi><\/math> <span class=\"ecti-1095\">als Summe einer geraden und einer ungeraden Funktion schreiben l<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">sst.<\/span> <\/p> <\/div> <div class=\"me meexample\"> <div class=\"wp-nocaption \"><\/div><h4 id=\"z3f56df5a7d19\"> <span class=\"ecbx-1095\">\u00dc<\/span><span class=\"ecbx-1095\">bung.<\/span><\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Seien <\/span><math display=\"inline\"><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo> <mi>b<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u211d<\/mi><\/math> <span class=\"ecti-1095\">. Zeigen<\/span> <span class=\"ecti-1095\">Sie, dass es ein <\/span><math display=\"inline\"><mi>\ud835\udf03<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u211d<\/mi><\/math> <span class=\"ecti-1095\">gibt, so dass<\/span> <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mi>a<\/mi><mi class=\"qopname\">sin<\/mi><mo>  <\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>\u03c6<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-bin\">+<\/mo> <mi>b<\/mi><mi class=\"qopname\">cos<\/mi><mo>  <\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>\u03c6<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-rel\">=<\/mo> <msqrt><mrow><msup><mrow><mi>a<\/mi><\/mrow><mrow><mn>2<\/mn> <\/mrow> <\/msup> <mo class=\"MathClass-bin\">+<\/mo> <msup><mrow><mi>b<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><\/mrow><\/msqrt><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\"> sin<\/mi><mo>  <\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>\u03c6<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mi>\ud835\udf03<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\"><span class=\"ecti-1095\">f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r alle <\/span><span class=\"maperiod\"><math display=\"inline\"><mi>\u03c6<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u211d<\/mi><\/math><\/span><span class=\"period\">.<\/span> <\/p><div class=\"geoapplet\" style=\"width: 688px\"><iframe height=\"695px\" scrolling=\"no\" src=\"https:\/\/www.geogebra.org\/material\/iframe\/id\/SvkmUmkp\/width\/688\/height\/695\/border\/888888\/rc\/false\/ai\/false\/sdz\/false\/smb\/false\/stb\/false\/stbh\/false\/ld\/false\/sri\/false\" style=\"border:0px\"><\/iframe><\/div><div class=\"wp-nocaption \"><\/div><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\">Betrachten Sie <\/span><math display=\"inline\"><msqrt><mrow><msup><mrow><mi>a<\/mi><\/mrow><mrow><mn>2<\/mn> <\/mrow> <\/msup> <mo class=\"MathClass-bin\">+<\/mo> <msup><mrow><mi>b<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><\/mrow><\/msqrt> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow> <mfrac><mrow><mi>a<\/mi><\/mrow> <mrow><msqrt><mrow><msup><mrow><mi>a<\/mi><\/mrow><mrow><mn>2<\/mn> <\/mrow> <\/msup> <mo class=\"MathClass-bin\">+<\/mo><msup><mrow><mi>b<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><\/mrow><\/msqrt><\/mrow><\/mfrac><mi class=\"qopname\"> sin<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>\u03c6<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">+<\/mo> <mfrac><mrow><mi>b<\/mi><\/mrow> <mrow><msqrt><mrow><msup><mrow><mi>a<\/mi><\/mrow><mrow><mn>2<\/mn> <\/mrow> <\/msup> <mo class=\"MathClass-bin\">+<\/mo><msup><mrow><mi>b<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><\/mrow><\/msqrt><\/mrow><\/mfrac><mi class=\"qopname\"> cos<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>\u03c6<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><\/math> <span class=\"ecti-1095\">und finden Sie eine reelle Zahl<\/span><span class=\"ecti-1095\">&nbsp;<\/span><math display=\"inline\"><mi>\ud835\udf03<\/mi><\/math> <span class=\"ecti-1095\">mit<\/span><span class=\"ecti-1095\">&nbsp;<\/span><math display=\"inline\"><mi class=\"qopname\"> cos<\/mi><mo>  <\/mo> <mo class=\"MathClass-open\">(<\/mo><mi>\ud835\udf03<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <mfrac><mrow><mi>a<\/mi><\/mrow> <mrow><msqrt><mrow><msup><mrow><mi>a<\/mi><\/mrow><mrow><mn>2<\/mn> <\/mrow> <\/msup> <mo class=\"MathClass-bin\">+<\/mo><msup><mrow><mi>b<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><\/mrow><\/msqrt><\/mrow><\/mfrac><\/math> <span class=\"ecti-1095\">und<\/span><span class=\"ecti-1095\">&nbsp;<\/span><span class=\"maperiod\"><math display=\"inline\"><mi class=\"qopname\"> sin<\/mi><mo>  <\/mo> <mo class=\"MathClass-open\">(<\/mo><mi>\ud835\udf03<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <mfrac><mrow><mi>b<\/mi><\/mrow> <mrow><msqrt><mrow><msup><mrow><mi>a<\/mi><\/mrow><mrow><mn>2<\/mn> <\/mrow> <\/msup> <mo class=\"MathClass-bin\">+<\/mo><msup><mrow><mi>b<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><\/mrow><\/msqrt><\/mrow><\/mfrac><\/math><\/span><span class=\"period\">.<\/span><\/p><\/details>  <\/div> <div class=\"me meexample\"> <div class=\"wp-nocaption \"><\/div><h4 id=\"z70c53ef216c1\"> <span class=\"ecbx-1095\">\u00dc<\/span><span class=\"ecbx-1095\">bung <\/span>(Irrationalit\u00e4t der Eulerschen Zahl)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Zeigen                                              Sie,                                              dass<\/span> <math display=\"inline\"><mi class=\"qopname\">e<\/mi><mo>  <\/mo><mo class=\"MathClass-rel\">=<\/mo><msubsup><mrow><mi class=\"qopname\"> \u2211<\/mi><mo>  <\/mo> <\/mrow><mrow><mi>k<\/mi><mo class=\"MathClass-rel\">=<\/mo><mn>0<\/mn><\/mrow><mrow><mi>\u221e<\/mi><\/mrow><\/msubsup><mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mi>k<\/mi><mo class=\"MathClass-punc\">!<\/mo><\/mrow><\/mfrac><\/math> <span class=\"ecti-1095\">irrational.              Nehmen              Sie              indirekt              an,              dass<\/span> <math display=\"inline\"><mi class=\"qopname\">e<\/mi><mo>  <\/mo><\/math> <span class=\"ecti-1095\">rational ist und verwenden Sie die Exponentialreihe.<\/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\">Nehmen Sie indirekt an, dass <\/span><math display=\"inline\"><msup><mrow><mi class=\"qopname\">e<\/mi><mo>  <\/mo><\/mrow><mrow><mo class=\"MathClass-bin\">\u2212<\/mo><mn>1<\/mn><\/mrow><\/msup> <mo class=\"MathClass-rel\">=<\/mo> <mfrac><mrow><mi>p<\/mi><\/mrow> <mrow><mi>q<\/mi><\/mrow><\/mfrac><\/math> <span class=\"ecti-1095\">f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r <\/span><span class=\"maperiod\"><math display=\"inline\"><mi>p<\/mi><mo class=\"MathClass-punc\">,<\/mo> <mi>q<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2115<\/mi><\/math><\/span><span class=\"period\">.<\/span> <span class=\"ecti-1095\">Nun vergleichen Sie, <\/span><math display=\"inline\"><mi class=\"qopname\">e<\/mi><mo>  <\/mo> <mo class=\"MathClass-rel\">=<\/mo> <mfrac><mrow><mi>p<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>q<\/mi><mo class=\"MathClass-bin\">\u2212<\/mo><mn>1<\/mn><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-punc\">!<\/mo><\/mrow> <mrow><mi>q<\/mi><mo class=\"MathClass-punc\">!<\/mo><\/mrow><\/mfrac> <\/math> <span class=\"ecti-1095\">mit der Partialsumme <\/span><span class=\"maperiod\"><math display=\"inline\"><msubsup><mrow><mi class=\"MathClass-op\">\u2211<\/mi><mo> <\/mo> <\/mrow><mrow><mi>n<\/mi><mo class=\"MathClass-rel\">=<\/mo><mn>0<\/mn><\/mrow><mrow><mi>q<\/mi><\/mrow><\/msubsup><mfrac><mrow><msup><mrow><mo class=\"MathClass-open\">(<\/mo><mo class=\"MathClass-bin\">\u2212<\/mo><mn>1<\/mn><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msup><\/mrow> <mrow><mi>n<\/mi><mo class=\"MathClass-punc\">!<\/mo><\/mrow><\/mfrac> <\/math><\/span><span class=\"period\">.<\/span> <span class=\"ecti-1095\">Vergleichen Sie den Fehler im Leibniz-Kriterium und die Aussage, dass zwei rationale Zahlen<\/span> <span class=\"ecti-1095\">mit dem gleichem Nenner <\/span><math display=\"inline\"><mi>Q<\/mi><\/math> <span class=\"ecti-1095\">entweder gleich oder Mindestabstand <\/span><math display=\"inline\"> <mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mi>Q<\/mi><\/mrow><\/mfrac><\/math> <span class=\"ecti-1095\">haben.<\/span><\/p><\/details>  <\/div> <div class=\"me meexample\"> <div class=\"wp-nocaption \"><\/div><h4 id=\"z1dfab298c823\"> <span class=\"ecbx-1095\">\u00dc<\/span><span class=\"ecbx-1095\">bung.<\/span><\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Wir wollen einen (spiels<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">chtigen) Kasinobesucher und ein un<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">bliches, den Spieler<\/span> <span class=\"ecti-1095\">bevorzugendes, Spiel betrachten. Das Spiel besteht aus einem einfachen M<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">nzwurf mit zwei<\/span> <span class=\"ecti-1095\">m<\/span><span class=\"ecti-1095\">\u00f6<\/span><span class=\"ecti-1095\">glichen gleich wahrscheinlichen Ergebnissen, n<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">mlich Kopf und Zahl. Bei Kopf gewinnt das<\/span> <span class=\"ecti-1095\">Kasino den Einsatz des Spielers und bei Zahl gewinnt der Spieler das Vierfache seines Einsatzes.<\/span> <span class=\"ecti-1095\">Wir wollen die verschiedenen Ergebnisse des iterierten Spieles anhand einer Funktion auf<\/span> <math display=\"inline\"><mo class=\"MathClass-open\">[<\/mo><mn>0<\/mn><mo class=\"MathClass-punc\">,<\/mo> <mn>1<\/mn><mo class=\"MathClass-close\">]<\/mo><\/math> <span class=\"ecti-1095\">beschreiben. Hierbei verwenden wir die bin<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">re Zifferndarstellung reeller Zahlen, wobei<\/span> <span class=\"ecti-1095\">wir die Nichteindeutigkeit einfach ignorieren, da diese f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r sich gesehen extrem<\/span> <span class=\"ecti-1095\">unwahrscheinlichen Ergebnissen des iterierten Spieles entsprechen.<\/span><button class=\"hover-trigger\" style=\"vertical-align: super;font: smaller\">\u2020<\/button><span class=\"hover-text\"><span class=\"marginpar\">\u2020 <span class=\"ecti-1095\">Dies genauer und<\/span> <span class=\"ecti-1095\">mathematisch exakt zu formulieren w<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">rde hier zu weit f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">hren.<\/span><\/span><\/span> <span class=\"ecti-1095\">Wir interpretieren die Ziffer<\/span> <math display=\"inline\"><mn>0<\/mn><\/math> <span class=\"ecti-1095\">als Kopf und die<\/span> <span class=\"ecti-1095\">Ziffer <\/span><math display=\"inline\"><mn>1<\/mn><\/math> <span class=\"ecti-1095\">als Zahl,<\/span> <span class=\"ecti-1095\">womit die reelle Zahl <\/span><math display=\"inline\"><mn>0<\/mn><mo class=\"MathClass-punc\">.<\/mo><mn>1<\/mn><mn>0<\/mn><mn>0<\/mn><mn>1<\/mn><mn>0<\/mn><mn>1<\/mn><mi class=\"MathClass-op\">\u2026<\/mi><mo> <\/mo><\/math> <span class=\"ecti-1095\">f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r die wiederholten W<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">rfe der M<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">nze mit den Ergebnissen Zahl-Kopf-Kopf-Zahl-Kopf-Zahl und so<\/span> <span class=\"ecti-1095\">weiter steht (und die Null vor dem Komma ignoriert wird).<\/span> <\/p><dl class=\"enumerate\"><dt class=\"enumerate\"> <span class=\"ecti-1095\">(i)<\/span><\/dt><dd class=\"enumerate\"><span class=\"ecti-1095\">Bestimmen Sie eine Funktion <\/span><span class=\"maperiod\"><math display=\"inline\"><msub><mrow><mi>f<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub> <mo class=\"MathClass-punc\">:<\/mo> <mo class=\"MathClass-open\">[<\/mo><mn>0<\/mn><mo class=\"MathClass-punc\">,<\/mo><mn>1<\/mn><mo class=\"MathClass-close\">]<\/mo> <mo class=\"MathClass-rel\">\u2192<\/mo> <mi>\u211d<\/mi><\/math><\/span><span class=\"period\">,<\/span> <span class=\"ecti-1095\">die den Gewinn nach <\/span><math display=\"inline\"><mi>n<\/mi><\/math> <span class=\"ecti-1095\">Wiederholungen des Spiels beschreibt, wobei der Spieler mit einem Franken das Spiel<\/span> <span class=\"ecti-1095\">beginnt und bei Gewinn jeweils sein Gesamtverm<\/span><span class=\"ecti-1095\">\u00f6<\/span><span class=\"ecti-1095\">gen im n<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">chsten Spiel wieder einsetzt.<\/span> <\/dd><dt class=\"enumerate\"> <span class=\"ecti-1095\">(ii)<\/span><\/dt><dd class=\"enumerate\"><span class=\"ecti-1095\">Bestimmen Sie den Erwartungswert f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r den Spieler, wenn dieser das Spiel <\/span><math display=\"inline\"><mi>n<\/mi><\/math> <span class=\"ecti-1095\">mal wiederholt (also das Integral von <\/span><math display=\"inline\"><msub><mrow><mi>f<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><\/math><span class=\"ecti-1095\">).<\/span> <\/dd><dt class=\"enumerate\"> <span class=\"ecti-1095\">(iii)<\/span><\/dt><dd class=\"enumerate\"><span class=\"ecti-1095\">Wir nehmen nun an, dass der Spieler spiels<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">chtig ist und auch bei Gewinn von wirklich<\/span> <span class=\"ecti-1095\">grossen Summen nicht aufh<\/span><span class=\"ecti-1095\">\u00f6<\/span><span class=\"ecti-1095\">ren kann zu spielen. Bestimmen Sie die Funktion, die den<\/span> <span class=\"ecti-1095\">Gewinn des Spielers beschreibt. Sie sollten bemerken, dass dies der punktweise Grenzwert<\/span> <span class=\"ecti-1095\">der Funktion <\/span><math display=\"inline\"><msub><mrow><mi>f<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><\/math> <span class=\"ecti-1095\">ist.<\/span> <\/dd><dt class=\"enumerate\"> <span class=\"ecti-1095\">(iv)<\/span><\/dt><dd class=\"enumerate\"><span class=\"ecti-1095\">Bestimmen  Sie  den  Erwartungswert  des  unbeschr<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">nkt  langen  Spiels  f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r  den<\/span> <span class=\"ecti-1095\">spiels<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">chtigen Spieler.<\/span><\/dd><\/dl> <\/div> <div class=\"me meexample\"> <div class=\"wp-nocaption \"><\/div><h4 id=\"z370d7d57db5f\"> <span class=\"ecbx-1095\">\u00dc<\/span><span class=\"ecbx-1095\">bung <\/span>(Der Fall der absoluten Konvergenz am Rand)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Sei <\/span><math display=\"inline\"><msubsup><mrow><mi class=\"MathClass-op\"> \u2211<\/mi><mo> <\/mo> <\/mrow><mrow><mi>n<\/mi><mo class=\"MathClass-rel\">=<\/mo><mn>0<\/mn><\/mrow><mrow><mi>\u221e<\/mi><\/mrow><\/msubsup><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><msup><mrow><mi>z<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msup><\/math> <span class=\"ecti-1095\">eine Potenzreihe<\/span> <span class=\"ecti-1095\">mit Konvergenzradius <\/span><span class=\"maperiod\"><math display=\"inline\"><mi>R<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mo class=\"MathClass-open\">(<\/mo><mn>0<\/mn><mo class=\"MathClass-punc\">,<\/mo><mi>\u221e<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math><\/span><span class=\"period\">.<\/span> <span class=\"ecti-1095\">Sei des Weiteren <\/span><span class=\"maperiod\"><math display=\"inline\"><msubsup><mrow><mi class=\"MathClass-op\">\u2211<\/mi><mo> <\/mo> <\/mrow><mrow><mi>n<\/mi><mo class=\"MathClass-rel\">=<\/mo><mn>0<\/mn><\/mrow><mrow><mi>\u221e<\/mi><\/mrow><\/msubsup><mo class=\"MathClass-rel\">|<\/mo><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><mo class=\"MathClass-rel\">|<\/mo><msup><mrow><mi>R<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msup> <mo class=\"MathClass-rel\">&lt;<\/mo> <mi>\u221e<\/mi><\/math><\/span><span class=\"period\">.<\/span> <span class=\"ecti-1095\">Zeigen Sie, dass in diesem Fall die Funktion<\/span> <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mi>z<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo><mover accent=\"false\" class=\"mml-overline\"><mrow><msub><mrow><mi>B<\/mi><\/mrow><mrow><mi>R<\/mi><\/mrow><\/msub><mo class=\"MathClass-open\">(<\/mo><mn>0<\/mn><mo class=\"MathClass-close\">)<\/mo><\/mrow><mo accent=\"true\">\u00af<\/mo><\/mover> <mo class=\"MathClass-rel\">=<\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> {<\/mo><mrow><mi>z<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2102<\/mi><mo class=\"MathClass-rel\">\u2223<\/mo><mo class=\"MathClass-rel\">|<\/mo><mi>z<\/mi><mo class=\"MathClass-rel\">|<\/mo><mo class=\"MathClass-rel\">\u2264<\/mo> <mi>R<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">}<\/mo><\/mrow><mo class=\"MathClass-rel\">\u21a6<\/mo><munderover accent=\"false\" accentunder=\"false\"><mrow><mo>\u2211<\/mo> <\/mrow><mrow><mi>n<\/mi><mo class=\"MathClass-rel\">=<\/mo><mn>0<\/mn><\/mrow><mrow><mi>\u221e<\/mi><\/mrow><\/munderover><msub><mrow><mi>a<\/mi><\/mrow><mrow> <mi>n<\/mi><\/mrow><\/msub><msup><mrow><mi>z<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msup><\/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\">wohldefiniert und stetig ist.<\/span> <\/p> <\/div> <div class=\"me meexample\"> <div class=\"wp-nocaption \"><\/div><h4 id=\"z084578002c20\"> <span class=\"ecbx-1095\">\u00dc<\/span><span class=\"ecbx-1095\">bung <\/span>(Bestimmte Divergenz am Rand)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Sei <\/span><math display=\"inline\"><msubsup><mrow><mi class=\"MathClass-op\"> \u2211<\/mi><mo> <\/mo> <\/mrow><mrow><mi>n<\/mi><mo class=\"MathClass-rel\">=<\/mo><mn>0<\/mn><\/mrow><mrow><mi>\u221e<\/mi><\/mrow><\/msubsup><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><msup><mrow><mi>x<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msup><\/math> <span class=\"ecti-1095\">eine Potenzreihe<\/span> <span class=\"ecti-1095\">mit Konvergenzradius <\/span><math display=\"inline\"><mi>R<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mo class=\"MathClass-open\">(<\/mo><mn>0<\/mn><mo class=\"MathClass-punc\">,<\/mo><mi>\u221e<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math> <span class=\"ecti-1095\">und<\/span> <span class=\"ecti-1095\">nicht-negativen Koeffizienten <\/span><math display=\"inline\"><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub> <mo class=\"MathClass-rel\">\u2265<\/mo> <mn>0<\/mn><\/math> <span class=\"ecti-1095\">f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r alle <\/span><math display=\"inline\"><mi>n<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2115<\/mi><\/math><span class=\"ecti-1095\">. Sei<\/span> <span class=\"ecti-1095\">des Weiteren <\/span><span class=\"maperiod\"><math display=\"inline\"><msubsup><mrow><mi class=\"MathClass-op\">\u2211<\/mi><mo> <\/mo> <\/mrow><mrow><mi>n<\/mi><mo class=\"MathClass-rel\">=<\/mo><mn>0<\/mn><\/mrow><mrow><mi>\u221e<\/mi><\/mrow><\/msubsup><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><msup><mrow><mi>R<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msup> <mo class=\"MathClass-rel\">=<\/mo> <mi>\u221e<\/mi><\/math><\/span><span class=\"period\">.<\/span> <span class=\"ecti-1095\">Zeigen Sie, dass in diesem Fall<\/span> <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><munder class=\"msub\"><mrow><mi class=\"qopname\">lim<\/mi><mo>  <\/mo><\/mrow><mrow><mi>x<\/mi><mo class=\"MathClass-rel\">\u2197<\/mo><mi>R<\/mi><\/mrow><\/munder><munderover accent=\"false\" accentunder=\"false\"><mrow><mo> \u2211<\/mo> <\/mrow><mrow><mi>n<\/mi><mo class=\"MathClass-rel\">=<\/mo><mn>0<\/mn><\/mrow><mrow><mi>\u221e<\/mi><\/mrow><\/munderover><msub><mrow><mi>a<\/mi><\/mrow><mrow> <mi>n<\/mi><\/mrow><\/msub><msup><mrow><mi>x<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msup> <mo class=\"MathClass-rel\">=<\/mo> <mi>\u221e<\/mi><mo class=\"MathClass-punc\">.<\/mo><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <\/div> <div class=\"me meexample\"> <div class=\"wp-nocaption \"><\/div><h4 id=\"z04184e3c92b6\"> <span class=\"ecbx-1095\">\u00dc<\/span><span class=\"ecbx-1095\">bung <\/span>(Challenge)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Wir betrachten das Gitter <\/span><math display=\"inline\"><msup><mrow><mi>\u2115<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><\/math> <span class=\"ecti-1095\">in der Ebene <\/span><math display=\"inline\"><msup><mrow><mi>\u211d<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><\/math> <span class=\"ecti-1095\">und fixieren uns ein Quadrat darin, <\/span><span class=\"maperiod\"><math display=\"inline\"><mi>Q<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <msup><mrow><mo class=\"MathClass-open\">[<\/mo><mn>1<\/mn><mo class=\"MathClass-punc\">,<\/mo><mn>1<\/mn><mn>5<\/mn><mo class=\"MathClass-close\">]<\/mo><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup> <mo class=\"MathClass-bin\">\u2229<\/mo> <msup><mrow><mi>\u2115<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><\/math><\/span><span class=\"period\">.<\/span> <span class=\"ecti-1095\">Auf den Punkten <\/span><span class=\"maperiod\"><math display=\"inline\"><mo class=\"MathClass-open\">(<\/mo><mn>1<\/mn><mo class=\"MathClass-punc\">,<\/mo><mn>1<\/mn><mo class=\"MathClass-close\">)<\/mo><\/math><\/span><span class=\"period\">,<\/span> <span class=\"maperiod\"><math display=\"inline\"><mo class=\"MathClass-open\">(<\/mo><mn>2<\/mn><mo class=\"MathClass-punc\">,<\/mo> <mn>1<\/mn><mo class=\"MathClass-close\">)<\/mo><\/math><\/span><span class=\"period\">,<\/span> <span class=\"maperiod\"><math display=\"inline\"><mo class=\"MathClass-open\">(<\/mo><mn>3<\/mn><mo class=\"MathClass-punc\">,<\/mo> <mn>1<\/mn><mo class=\"MathClass-close\">)<\/mo><\/math><\/span><span class=\"period\">,<\/span> <span class=\"maperiod\"><math display=\"inline\"><mo class=\"MathClass-open\">(<\/mo><mn>1<\/mn><mo class=\"MathClass-punc\">,<\/mo> <mn>2<\/mn><mo class=\"MathClass-close\">)<\/mo><\/math><\/span><span class=\"period\">,<\/span> <span class=\"maperiod\"><math display=\"inline\"><mo class=\"MathClass-open\">(<\/mo><mn>2<\/mn><mo class=\"MathClass-punc\">,<\/mo> <mn>2<\/mn><mo class=\"MathClass-close\">)<\/mo><\/math><\/span><span class=\"period\">,<\/span> <math display=\"inline\"><mo class=\"MathClass-open\">(<\/mo><mn>1<\/mn><mo class=\"MathClass-punc\">,<\/mo> <mn>3<\/mn><mo class=\"MathClass-close\">)<\/mo><\/math> <span class=\"ecti-1095\">platzieren wir nun jeweils eine M<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">nze und beginnen dann folgendes Spiel. Sie als Spieler<\/span> <span class=\"ecti-1095\">d<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">rfen jeweils eine der M<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">nzen entfernen, worauf Sie eine M<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">nze oberhalb und eine M<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">nze<\/span> <span class=\"ecti-1095\">rechts von der entfernten M<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">nze platzieren. Dieser Zug ist aber nur dann erlaubt, wenn<\/span> <span class=\"ecti-1095\">die Pl<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">tze rechts und oberhalb der zu entfernenden M<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">nze noch frei sind. Ihre Aufgabe<\/span> <span class=\"ecti-1095\">besteht nun darin, nach endlich vielen Z<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">gen keine M<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">nzen mehr innerhalb des Quadrat<\/span> <math display=\"inline\"><mi>Q<\/mi><\/math> <span class=\"ecti-1095\">liegen zu haben. Ist das m<\/span><span class=\"ecti-1095\">\u00f6<\/span><span class=\"ecti-1095\">glich?<\/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\">Die Antwort ist nein. Um dies zu beweisen, suchen Sie nach einer Funktion <\/span><span class=\"maperiod\"><math display=\"inline\"><mi>f<\/mi> <mo class=\"MathClass-punc\">:<\/mo> <msup><mrow><mi>\u2115<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup> <mo class=\"MathClass-rel\">\u2192<\/mo> <mi>\u2115<\/mi><\/math><\/span><span class=\"period\">,<\/span> <span class=\"ecti-1095\">so dass die Summe von <\/span><math display=\"inline\"><mi>f<\/mi><\/math> <span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">ber die Positionen der M<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">nzen im <\/span><math display=\"inline\"><mi>n<\/mi><\/math><span class=\"ecti-1095\">-ten<\/span> <span class=\"ecti-1095\">Schritt nicht von <\/span><math display=\"inline\"><mi>n<\/mi><\/math> <span class=\"ecti-1095\">oder von der gew<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">hlten Strategie abh<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">ngt. In anderen Worten soll diese Summe unter der in<\/span> <span class=\"ecti-1095\">der Aufgabenstellung beschriebenen Operation erhalten bleiben.<\/span><\/p><\/details>  <\/div> \n","protected":false},"author":1089,"menu_order":9,"template":"","meta":{"pb_show_title":"","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-84","chapter","type-chapter","status-publish","hentry"],"part":75,"_links":{"self":[{"href":"https:\/\/wp-prd.let.ethz.ch\/analysis19\/wp-json\/pressbooks\/v2\/chapters\/84","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\/84\/revisions"}],"part":[{"href":"https:\/\/wp-prd.let.ethz.ch\/analysis19\/wp-json\/pressbooks\/v2\/parts\/75"}],"metadata":[{"href":"https:\/\/wp-prd.let.ethz.ch\/analysis19\/wp-json\/pressbooks\/v2\/chapters\/84\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/wp-prd.let.ethz.ch\/analysis19\/wp-json\/wp\/v2\/media?parent=84"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/wp-prd.let.ethz.ch\/analysis19\/wp-json\/pressbooks\/v2\/chapter-type?post=84"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/wp-prd.let.ethz.ch\/analysis19\/wp-json\/wp\/v2\/contributor?post=84"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/wp-prd.let.ethz.ch\/analysis19\/wp-json\/wp\/v2\/license?post=84"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}