# Convergence is contagious

Suppose $g$ is function $g: \Bbb C\to \Bbb C$ and that at a point $z_0\in \Bbb C$ such function is analytic. If the infinite sum $g(z_0)+g'(z_0)+\cdots$ converges, then in fact $g$ is entire and such sum converges for any other choice of $w$.