\mnb150ÿ{\rtf1\ansi\deff0\deftab720{\fonttbl{\f0\fswiss MS Sans Serif;}{\f1\froman\fcharset2 Symbol;}{\f2\fswiss\fprq2 System;}{\f3\fswiss\fprq2 Arial;}{\f4\fmodern\fprq1 Courier New;}{\f5\fswiss\fprq2 Helvetica;}} {\colortbl\red0\green0\blue0;\red0\green0\blue255;\red0\green128\blue0;\red255\green0\blue0;} \deflang1031\pard\ri4\plain\f4\fs20\cf0\b ________________________________________________________________________________ \par \par \plain\f4\fs20\cf0 Inhalt....: Berechnung von Summen \par Kategorie.: Handwerkskasten \par Mathematik: Analysis \par MuPAD.....: 3.0.0 \par Datum.....: 2002-02-06 \par Autoren...: Kai Gehrs \par Funktionen: sum, infinity, nops \par \plain\f4\fs20\cf0\b ________________________________________________________________________________ \par \plain\f3\fs36\cf0\b \par \plain\f3\fs40\cf0\b Elementare MuPAD-Funktionen: \par Berechnung von Summen \plain\f3\fs24\cf2 \par \par Summation ist per Hand oder mit dem Taschenrechner oft eine leidige Angelegenheit - vor \par allem dann, wenn man nicht nur \'fcber zwei, sondern vielleicht \'fcber 10 oder 2000 Summanden \par addieren m\'f6chte. \par \plain\f3\fs28\cf0 \par \plain\f3\fs28 Summation erledigt man in MuPAD mit Hilfe der Funktion \plain\f3\fs28\cf3 sum\plain\f3\fs28 . \plain\f3\fs28\cf0 Sie erh\'e4lt \par stets \plain\f3\fs28\cf2 zwei Argumente\plain\f3\fs28\cf0 : \par \par \plain\f3\fs28\cf3 1. Argument: \plain\f3\fs28\cf0 Die \plain\f3\fs28\cf2 Folge oder mathematische Vorschrift\plain\f3\fs28\cf0 , \'fcber deren Werte \par aufsummiert werden soll \par \par \plain\f3\fs28\cf3 2. Argument: \plain\f3\fs28\cf0 Der \plain\f3\fs28\cf2 Laufindex der Summe\plain\f3\fs28\cf0 , in Form einer Gleichung \par \plain\f3\fs28\cf2 Laufindex = untere_Grenze..obere_Grenze\plain\f3\fs28\cf0 . \par \par Ein Aufruf ist also im allgemeinen stets von der Form \par \plain\f3\fs28\cf3 sum( \plain\f3\fs28\cf2 Folge, Laufindex = untere_Grenze..obere_Grenze\plain\f3\fs28\cf0 \plain\f3\fs28\cf3 )\plain\f3\fs28\cf0 . \par \par \plain\f3\fs28 Betrachten wir einige Beispiele. Wir wollen die Summe der Zahlen von 1 bis \par 100 berechnen. Schon Carl Friedrich Gau\'df wu\'dfte: Das Ergebnis ist gegeben \par durch \par \par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f4\fs28\cf3 {\pntext\f1\'b7\tab}(100*101) / 2 \par \pard\li600\ri1\fi-300\plain\f3\fs28\cf3 \par \pard\ri4\plain\f3\fs28 Wir k\'f6nnen \plain\f3\fs28\cf0 uns\plain\f3\fs28 nun die Summe auch explizit berechnen lassen: \par \par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f4\fs28\cf3 {\pntext\f1\'b7\tab}sum(i, i = 1..100) \par \pard\ri4\plain\f3\fs28\cf0 \par F\'fcr die Summe der ersten 100 Quadratzahlen gibt es ebenfalls \par entsprechende Formeln. Wir verzichten hier jedoch \plain\f3\fs28 auf diese \par Formeln und berechnen die Summe explizit mit Hilfe von \plain\f3\fs28\cf3 sum\plain\f3\fs28\cf0 : \par \par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f4\fs28\cf3 {\pntext\f1\'b7\tab}sum(i^2, i = 1..100) \par \pard\ri4\plain\f3\fs28 \par N\'fctzlich ist die Funktion \plain\f3\fs28\cf3 sum\plain\f3\fs28\cf0 dar\'fcber hinaus auch dann, wenn \'fcber die Werte \par einer Liste L aufsummiert werden soll. Solche Anwendungen tauchen recht \par h\'e4ufig in der Stochastik, bei der Berechnung von Erwartungswerten oder \par Varianzen, auf. \par \par L sei ein Liste mit den folgenden Elementen \par \par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f4\fs28\cf3 {\pntext\f1\'b7\tab}L:= [31,11,73,35,343,6,5,323,21,6,9,23,21,45,76,23, \par \pard\li600\ri1\fi-300\plain\f4\fs28\cf3 122,456,67,88,89] \par \pard\ri4\plain\f3\fs28 \par Wir erfragen die L\'e4nge der Liste \'fcber den Befehl \plain\f3\fs28\cf3 nops\plain\f3\fs28 (number of oprands): \par \par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f4\fs28\cf3 {\pntext\f1\'b7\tab}nops(L) \par \pard\ri4\plain\f3\fs28 \par Jetzt l\'e4sst sich die Summe \'fcber all ihre Elemente spielend leicht und vor allem \par schnell berechnen. Mit L[ i ] erhalten wir das i-te Element der Liste. Wir \par summieren f\'fcr i = 1 bis 21 auf. \par \par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f4\fs28\cf3 {\pntext\f1\'b7\tab}sum(L[i], i = 1..21) \par \pard\ri4\plain\f3\fs28 \par Ebenso kann man \'fcber die Elemente von Mengen aufsummieren. \par \par Neben konkreten Summ\plain\f3\fs28\cf0 ationen bietet MuPAD zudem die M\'f6glichkeit mit \par allgemeinen Laufindizes zu a\plain\f3\fs28 rbeiten: So liefert uns MuPAD z.B. die \par Formel von Gau\'df oder die Formel zur Berechnung der ersten n \par Quadratzahlen: \par \par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f4\fs28\cf3 {\pntext\f1\'b7\tab}sum(i, i = 1..n) \par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f4\fs28\cf3 {\pntext\f1\'b7\tab}sum(i^2, i = 1..n) \par \pard\ri4\plain\f3\fs28 \par Auch unendliche Summen k\'f6nnen wir auf diese Weise berechnen: Die \par harmonische Reihe divergiert und MuPAD liefert uns das korrekte Ergebnis \par \par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f4\fs28\cf3 {\pntext\f1\'b7\tab}sum(1/i, i = 1..infinity) \par \pard\ri4\plain\f3\fs28 \par Im Gegensatz dazu wurde die Konvergenz der Reihe \par {\pict\wmetafile8\picw1482\pich1550\picscalex99\picscaley99\picwgoal843\pichgoal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Wir erhalten das Ergebnis \par \par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f4\fs28\cf3 {\pntext\f1\'b7\tab}sum(1/i^2, i = 1..infinity) \par \pard\ri4\plain\f3\fs28 \par \plain\f4\fs20\cf0\b ________________________________________________________________________________ \par \plain\f3\fs22\cf0 \par \plain\f3\fs22\cf1\b Aufgaben:\plain\f3\fs22\cf1 \par \plain\f3\fs20\cf1\b 1\plain\f3\fs20\cf1 . 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