Find all complex numbers such that the series

converges.

First, the series is not absolutely convergent for any since

By limit comparison with this always diverges.

Then, if for any positive integer the series is convergent since is monotonically decreasing and . If for some positive integer , then the series is not defined since it is undefined for that term.

Let

, and the real and imaginary parts of the original series are alternating series(es), which converge by Lebniz’s rule.

I think that is a bit strange you say 1/(z+n) is decreasing, since complex field isn’t ordered.