0
$\begingroup$

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!

$\endgroup$
2
$\begingroup$

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

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.