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