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Mathematik-Online problems:

Problem 124: Convergence Criterion for 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
Test the following series for convergence and if so, calculate the limit.

a) $ {\displaystyle{\sum_{n=1}^\infty \frac{(-{\rm {e}})^{n-1}}{\pi^{n+1}}}}$ b) $ {\displaystyle{\sum_{n=1}^\infty \frac{n}{(n+1)(n+2)}}}$ c) $ {\displaystyle{\sum_{n=2}^\infty\, \ln\left(1-\frac{1}{n^2}\right)}}$
For which $ x\in\mathbb{R}$ the following series are convergent?
d) $ {\displaystyle{\sum_{n=0}^\infty\, {\rm {e}}^{-n}\,(x-1)^n}}$ e) $ {\displaystyle{\sum_{n=1}^\infty\, (1+2+\ldots +n)\, x^n}}$    

(Authors: /Höfert)

see also:

[Solutions]
  automatisch erstellt am 12.  4. 2005