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Mathematik-Online problems:

Problem 101: Limits of Series


A B C D E F G H I J K L M N O P Q R S T U V W X Y Z

Show that the following series have the given limits:

a) $ {\displaystyle \sum_{n=1}^{\infty}\: \frac{(-3)^n}{4^n}
\;=\; -\frac{3}{7} }$ b) $ {\displaystyle \sum_{n=0}^{\infty}\: \frac{3^n + (-2)^n}{6^n}
\;=\; \frac{11}{4} }$
c) $ {\displaystyle \sum_{n=1}^{\infty}\: \frac{1}{4n^2 - 1}
\;=\; \frac{1}{2} }$ d) $ {\displaystyle \sum_{n=1}^{\infty}\: \frac{1}{n\,(n+1)\,(n+2)}
\;=\; \frac{1}{4} }$
e) $ {\displaystyle \sum_{n=0}^{\infty}\:
\left(\frac{1+{\rm {i}}}{2}\right)^n \;=\; 1+{\rm {i}} }$ f) $ {\displaystyle \sum_{n=1}^{\infty}\: \frac{\cos n\pi}{n}
\;=\; -{\rm {ln}}\,2 }$

(Authors: Werner/Höfert)

see also:


[Solutions]

  automatisch erstellt am 14. 10. 2004