The problem is to prove that $$ \sum_{n=0}^\infty a_n(z-z_0)^n $$ uniformly converges on a connected open set $D$ if and only if it uniformly converges on $\overline{D}$.
The sufficiency is trivial.Since it uniformly converges on $\overline{D}$, it must uniformly converges on $D$. But the necessity seems hard to prove.
Please show me how to prove the problem.
Given $\varepsilon>0$, there is some $N$ so that any two partial sums with at least $N$ terms differ by at most $\varepsilon$ on $D$. Note that partial sums are polynomials, and so continuous on all of $\mathbb{C}$. I'll let you take it from there.