# Thread: Convergence of a Complex Power Series

1. ## Convergence of a Complex Power Series

Hello everyone. I am asked to show that the power series

$\displaystyle \sum_{n=1}^\infty \frac{z^n}{n}$

converges for $|z|\leq 1$, except for $z=1$.

It is easy to show that the series converges to $|z|<1$ by the root test. However, it is not clear to me why it should converge on the boundary. I tried putting $z=e^{i\theta}$, but it does not help.

I would appreciate any suggestions.

2. See...

Abel's test - Wikipedia, the free encyclopedia

Kind regards

$\chi$ $\sigma$