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

Hello!

It is indeed odd to state that such a complex sequence is monotonic. If you are looking for help with the solution. Take the series of absolute values and show that it converges (for example with the ratio test) for show that the limit as is not true, where is the -th term of the sum.

Cheers!

The previous comment got concatenated so here is the rest.

For use Drichlet’s test to conclude that it converges for the case when the and Leibnitz’s test when . The in the previous comment is .

Cheers!