If $f$ is an entire function where every power series expansion has at least one 0 term, show it is a polynomial
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....