Suppose that $f$ is analytic on the open unit disk such that there exists a constant $M$ with $|f^k(0)| \leq k^4M^k$ for all $k \geq 0$. Show that $f$ can be extended to be analytic on $\mathbb{C}$.
I'm not sure how to apply the basic facts about analytic continuation that I know to this problem. Are there any results that might be helpful here? A hint would be appreciated.
Context: I'm studying for a qual, so just a hint at this point would be most helpful.
Hint: Look at the Maclaurin series for $f$. What can you say about its radius of convergence?