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

Dirichlet's test says:
Let $\displaystyle \sum a_n$ be a complex serie with bounded partial sums and let $\displaystyle (v_n)_n$ be a decreasing sequence of real numbers then the serie $\displaystyle \sum v_na_n$ converges.
Take for example the decreasing sequence $\displaystyle (v_n)_n = \frac{1}{n}$ and the serie $\displaystyle \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 $\displaystyle \sum \frac{(-1)^n}{n}$ converges.