Discuss the convergence or divergence of the series .
We can see that this series converges since it is a geometric series with common ratio and also because it is an alternating series with terms decreasing in magnitude. But, to give another example for Dirichlet's test, we will show this series converges by it.
This is in just the form to use Dirichlet's test. The sequence of partial sums
is bounded since it alternates between -1 and 0. Also, decreases to 0 as .
Thus, by Dirichlet's test, this series converges.