Find all complex numbers such that the series

converges.

First, we have

We know from the previous exercise (Section 10.20, Exercise #44) that the series

converges if and only if

Therefore, the series converges if since both terms in the sum converge. The series diverges if since diverges and converges. Finally, the series diverges if since .

Very nice, but here we could just use asymptotic equality to argue, that the sum behaves like in the previous exercise.