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Mathematik-Online problems:

Problem 100: Convergence 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

a)
Given the series $ {\displaystyle{\sum_{n=1}^\infty a_n \ = \ \sum_{n=1}^\infty \frac{1}{n\,(n+1)} \ = \ \frac{1}{2}+\frac{1}{6}+\frac{1}{12}+\frac{1}{20}+\ldots}}$

Give the general element $ a_n$ as a difference of two fractions. Does the series converge, and if so, what's the limit?

b)
Test the series $ {\displaystyle{\sum_{n=1}^\infty \frac{1}{n^2} \ = \ 1+\frac{1}{4}+\frac{1}{9}+\frac{1}{16}+\ldots}}$ for convergence.

Hint: Use a).

(Authors: Kimmerle/Roggenkamp/Höfert)

see also:

[Solutions]
  automatisch erstellt am 14. 10. 2004