In interval q, $0\notin q$, ( e.g.[1,2] ), $ \sum_{n=0}^\infty a_nx^{n} $ converge to $0$. Can there be an $a_i$ that $a_i \ne 0$?
p.s I know the case when $0 \in q$, where you can prove it by derivative.
p.p.s I asked the question in mathoverflow(now I know it is for mathematicians to ask each other questions about their research.).
it seems it
follows from the uniqueness theorem for holomorphic functions
?