# Are these two power series equal?

Let $f(x)=\sum_{n=0}^\infty a_n x^n$ and $g(x)=\sum_{n=0}^\infty b_n x^n$ where $a_n,b_n\in[0,1]$ for all $n\geq 0$. Hence we know these two power series are convergent on $(-1,1)$.

Now assume there exists $0<a<b<1$ such that $f(x)=g(x)$ for all $x\in[a,b]$. Can we conclude that $a_n=b_n$ for all $n\geq 0$?

Thanks!

• Have you considered analytic continuation? mathworld.wolfram.com/AnalyticContinuation.html Aug 8 '15 at 8:55
• @GeorgSaliba, yes thanks. Theorem 8.5 in baby Rudin is exactly this result. Aug 8 '15 at 9:35

HINT If the power series of $0$ is $$0 = c_0 + \sum_{k=1}^{\infty}c_kx^k$$ then $c_k=0$ for all $k$.