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By Dr. Ludwig Bieberbach (auth.)

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Additional info for Differential- und Integralrechnung: Differentialrechnung

Example text

V,. I< Iul II vl I+ · · · + Iu,. II v" I Also

1 Aber alle diese Hilfsmittel versagen, wenn wir sie auf die Reihen (von welchen wir oben schon einen Spezialfall hatten) anwenden. Wenn wir diese also noch besonders untersuchen, erhalten wir Ili. Unendliche Reihen 38 einen weiteren Vorrat an Vergleichsreihen. Wir beginnen mit der sog. harmonischen R&ihe: 00 1 1 "-=1 n 1 1 ... +-+-+ 3 2 Wir betrachten die folgenden feilsummen: s2 , s4 , s8 , •••. , ••• Falls wir zeigen können, daß lim s2" = co, so wissen wir damit, n-+oo daß die harmonische Reibe divergiert.

Trotzdem werden wir gleich sehr nützliche Anwendungen derselben sehen. Ich behaupte nämlich folgenden Satz: Eine Reihe ist sicher dann konvergent, wenn die Reihe ihrer absoluten Beträge konvergiert. Denn wenn Iu1 I Iull I konvergiert, so gibt es für jedes E ein N( E), so daß für alle positiven p die Summe uu+tl uN+lll ttu+P E ist. Nun ist aber + I +I + ··· + I + · ·· I< + uu+l! + ···+ ttN+p IS:: lttu+l! + luu+21 + ···+luu+p I· Daher ist auch für n > N( <) die Differenz I sn + s,. I < Die J uu+l P- E.

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