example of using dirichlet's test

**fqqs** · May 28th 2012, 12:27 AM

I need an exmaple of example of using dirichlet's test for assessing if functions series are convergent. please help

**Siron** · May 29th 2012, 06:43 AM

Dirichlet's test says:
Let $\sum a_n$ be a complex serie with bounded partial sums and let $(v_n)_n$ be a decreasing sequence of real numbers then the serie $\sum v_na_n$ converges.
Take for example the decreasing sequence $(v_n)_n = \frac{1}{n}$ and the serie $\sum a_n=\sum(-1)^n = 1-1+1-1+1-1+\ldots$ , this serie doesn't converge but the partial sums are bounded therefore by Dirichlet's criterium the serie $\sum \frac{(-1)^n}{n}$ converges.

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