To prove convergence, use Dirichlet's test.

If you want to know the sum of the series, notice that . If then . Take the real part to see that . In particular, if then . That argument is not rigorous, because the series for has radius of convergence 1, so you cannot assume that it behaves well when |z| = 1. However, once you have used the Dirichlet test to prove convergence, the rest of the argument works to give the correct result.