roots of series $\sum_{n=1}^{\infty}a_nx^n$

Question

I think if we have series $\sum_{n=1}^{\infty}a_nx^n$ and exist $a_n\neq0$ then series can have not more then countable number of roots, is it right? What theorem can proof it?

Answer 1

If the series converges for all $x$ the the sum is an entire function. This implies that its zeros do not have any limit points. Hence the set of zeros is at most countable.