Uniform convergence of the series $\sum_{n=1}^{\infty} \frac{z^n}{n}.$

Question

How to check uniform convergence of the series $\sum_{n=1}^{\infty} \frac{z^n}{n} $ on $|z|<1.$ Clearly it is power series with radius of convergence as $1$ and power series converges in any compact subset of its domain of convergence on $|z|<1$. How to check its uniform convergence? $M_{n}$ test is not working here. Thanks.

Answer 1

If there was uniform convergence on the open unit disk, then we would have $$\lim_{N\to +\infty}a_N\mbox{ where }a_N :=\sup_{|z|\lt 1}\left|\sum_{n=N +1}^{2N}z^n/n\right|. $$ We have the bounds $$a_N\geqslant \sup_{0\lt t\lt 1}\sum_{n=N+1}^{2N}t^n/n\geqslant \sup_{0\lt t\lt 1}t^{2N}/2 =1/2 $$
since the sequence $\left(t^n/n\right)_{n\geqslant 1}$ is decreasing.