For what complex values of $z$ does the series $\sum_{n=0}^\infty \frac{z^n}{\log(n)}$ converge or diverge?

Question

I used the ratio test to find that it converges when $|z| < 1$ and diverges when $|z| > 1$, but I'm not sure how to proceed with the $|z| = 1$ case. Because $z^n$ is function that traces out the unit circle for $z \notin \{1,-1\}$, I'm fairly certain that it only diverges when z = 1 and converges on all other points of the unit circle.

Answer 1

Hint: Take $z\neq 1$ such that $|z|=1$ and apply Dirichlet's test.