$\sum_{n=1}^\infty\frac{z^n}{n}$ does not converge uniformly on $\mathbb{D}$.
Solution 1:
Use the uniform Cauchy test. Given $n\in\mathbb{N}$ $$ \Bigl|\sum_{k=n+1}^{2n}\frac1n\,\Bigl(1-\frac1n\Bigr)^k\Bigr|\ge\Bigl(1-\frac1n\Bigr)^{2n}\to e^{-2}>0. $$