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

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

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

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

I would appreciate any suggestions.