Find all complex numbers such that the series

converges.

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

Thus,

Hence, this series converges anywhere converges. Therefore, the sum

converges for with .

You should probably split complex a_n into the real and imaginary parts, prove they converge individually, and apply linearity to arrive to the final result. Ratio test requires positive sequence terms.