Let $D$ be an open connected set. How can I show that the series $$f(z) = \sum_{k=1}^{\infty}kz^k,$$ where $|z|<1$ for all $z\in D$ is uniformly convergent on $D$?
I wanted to use Weierstrass-M test but I can't find a convergent series $\sum_{k=1}^{\infty}a_k$ such that $|kz^k|\leq a_k$ for all $k\in \mathbb{N}$ and all $z\in D$.
It is not uniformly convergent in $D$. If it is, there would be an integer $m$ such that $|kz^{k}| <1$ for all $z \in D$ for all $k \geq m$. Letting $z\to 1$ we get $k \leq 1$ for all $k \geq m $ which is absurd.
[If $s_n$ is the $n-$th partial sum of the series the $kz^{k}=s_k-s_{k-1}$. This proves the existence of $m$].