The series $\sum\limits_{n=0}^\infty {a_{n}}(x-c)^n $ is a polynomial in $x$.
(This is from this question.)
For $c=0$ this clearly means that, for some $n>k, \space a_n=0$, almost by definition.
Is it also true for all other values of $c$ that there exists a $k$ such that $n>k, \space a_n=0$?