Prove if a power series is zero on an interval then all coefficients are zero Suppose the power series $P(x) =  \sum_{n=1}^\infty b_n x^n$  converges for $|x| \leq 1$ and that for some $0<c\le1$ it is given that $$P(x)=0 \quad \forall x \;\text{such that}\;|x| < c$$ Show that $b_n = 0$ for all $n \geq 1$.
What I was able to do was:
$$\lim_{n\to\infty} |b_n x^n|^{1/n} < 1$$ (by the root theorem for convergence) 
and as $P(x)=0$ for $|x| < c$, we have:
$$\sum_{n=1}^\infty b_n x^n = 0$$ 
Now how to get what is asked in the question from these known facts? I am not able to proceed beyond this. Thanks.

A power series has the same radius of convergence as the series of its derivatives. In particular, we can take the derivative term by term for $|x|<c$. We have $P^{(k)}(0)=0$, and translating this in term of $\sum_k a_kx^k$, this means that $a_k=0$.
