$\sum_{n=1}^\infty\frac{z^n}{n}$ does not converge uniformly on $\mathbb{D}$.

Question

It is required to prove that the series $\sum_{n=1}^\infty\frac{z^n}{n}$ does not converge uniformly on the open unit disc centered at $0$, i.e. $\mathbb{D}$. Clearly by virtue of the ratio test the series converges pointwise on $\mathbb{D}$. However I find it difficult to see why the convergence is not uniform. Could someone please give me a hint?

Answer 1

Use the uniform Cauchy test. Given $n\in\mathbb{N}$ $$ \Bigl|\sum_{k=n+1}^{2n}\frac1n\,\Bigl(1-\frac1n\Bigr)^k\Bigr|\ge\Bigl(1-\frac1n\Bigr)^{2n}\to e^{-2}>0. $$