Here is an old exam problem that I still can't figure out: Let $M > 0$ and consider the set $\mathcal{F}$ of all functions $f$ such that $$ f(z)=\sum_{n=0}^\infty a_{n}z^n $$ with $a_n \in \mathbb{C}$ and $\sum_{n=0}^\infty |a_n|\leq M $. Give an example of a sequence $\{f_k\}\subset \mathcal{F}$ that converges uniformly on each compact subset of $\mathbb{D}$, but does not converge uniformly on $\overline{\mathbb{D}}$.
Any insights will be appreciated. It will be especially helpful if the thinking process of forming an example is explained. Thanks.