Expanding on Convergence of a sequence of holomorphic functions -- derivative relations:
Let $f,f_n,n\in\mathbb N$ be holomorphic functions given by $$f(x)=\sum_{k=0}^\infty \alpha_kx^k,\quad f_n(x)=\sum_{k=0}^\infty\alpha^{(n)}_kx^k$$ on a compact domain $D$. Moreover we know that $$\lim_{n\to\infty}\alpha^{(n)}_k=\alpha_k$$ for all $k\in\mathbb N_0$.
Under which conditions do we have pointwise convergence of $f_n\to f$?
The simple counter example $f_n(x)=x^n$ shows that this is certainly not always the case.
I found one possible condition: Assume that $|\alpha_k^{(n)}|\leq|\alpha_k|$. Let $\rho_n$ and $\rho$ be the radii of convergence for the power series for $f_n$ and $f$ respectively and $x\in\mathbb C$ such that $|x|<\min\{\rho,\rho_n,n\in\mathbb N\}$. Then we have $$|\alpha_k^{(n)}x^k|\leq |\alpha_kx^k|$$ for all $k$ and since power series converge absolutely inside of their radius of convergence we can apply the dominated convergence theorem to $$\lim_{n\to\infty}f_n(x)=\lim_{n\to\infty}\sum_{k=0}^\infty\alpha_k^{(n)}x^k=\sum_{k=0}^\infty\lim_{n\to\infty}\alpha_k^{(n)}x^k=\sum_{k=0}^\infty\alpha_kx^k=f(x)$$