Find all complex numbers such that the series

converges.

If then we apply the Leibniz rule, where is monotonically decreasing and has limit 0 as . If with , then we apply Dirichlet’s test with to conclude that the series converges. If then and the series diverges since it equals the harmonic series.

Regarding Leibniz’ rule, I don’t understand how a complex sequence is defined as monotonic. I can see how a power series has a radius of convergence < 1 though. Perhaps I'm just tired…

I do not think these answers are correct, it should be solved somehow else. The complex sequence cannot indeed be decreasing