TITLE: If $f$ is an entire function where every power series expansion has at least one 0 term, show it is a polynomial
QUESTION [11 upvotes]: Suppose $f$ is entire and that in every power series $f(z)=\sum_{n=0}^\infty c_n(z-a)^n$ at least one coefficient is $0$. Prove that $f$ is a polynomial.

REPLY [10 votes]: Hint: For each complex number $a$, there exists an $n$ for which $c_n(a) = 0$, where $c_n(a)$ is the $n$th coefficient for the expansion at $a$. There are uncountably many complex numbers, but only countably many naturals, so....