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.