Find all complex numbers such that the series
First, we simplify the expression as follows,
Then, we consider the first series,
Let and use the ratio test,
This series converges if and diverges if . If and then the series also converges by Dirichlet’s test. Finally, if , then the series is which diverges.
Next, we know
Hence, this series converges anywhere converges. Therefore, the sum
converges for with .