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Suppose that the power series $p(x) = \sum b_nx^n$ converges for $|x| \le 1$. Suppose that for some $\delta \gt 0 , p(x) = 0$ for $|x| \lt \delta$. Show that $b_n = 0$ for all $n \ge 1$.

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For $|x|<1$ $p(x)$ is analytic. But also $p(x)=0$ for $|x|<\delta $. But then $p(x)$ must vanish for all $|x|<1$. Hence $b_{n}=0$.

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