Qual season...
Let $\Omega$ be a region in which $g$ is analytic and suppose that for a $z_0\in\Omega$, $\sum_{n=0}^\infty g^{(n)}(z_0)<\infty.$
Prove that $g$ is entire and that the series converges uniformly on discs of the form $|z|
The function $g$ is going to have a convergent Taylor series expansion at all points in $\Omega$, but I'm guessing that the stronger convergence given in the problem statement imply that the Taylor series expanded at a point in our region converges across the plane. Does the M-test play a role here?