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Given:
1. $f(x) = a_0 + a_1(x) + a_2(x^2) + ... \;=\;$ an infinite power series with real coefficients
2. $g(x) = b_0 + b_1(x) + b_2(x^2) + ... \;=\;$ an infinite power series with real coefficients
3. There exist an infinite number of discrete real numbers $\;x_1, x_2, x_3, ...\;$ such that $\;f(x_i)=g(x_i)\;$ for each of these numbers.
4. Note: $\;0\;$ is not necessarily in $\;\{x_1, x_2, x_3, ...\}.$

To Prove: $\;\forall i\in\{0,1,2,...\},\;a_i=b_i.$

My Research:

The closest that I've come is theorem 11.10, p434, in "Calculus Volume 1, 2nd Edition", 1966, by Apostol: if two power series $\;\sum a_n(x-a)^n\;$ and $\;\sum b_n(x-a)^n\;$ have the same function $f$ in some neighborhood of the point $a,$ then the two series are equal term by term.

I don't see how I can use that theorem for my problem. I am aware that if $\;f(x)\;$ and $\;g(x)\;$ were each of finite degree, then the problem would be solved because $\;[f(x)-g(x)]\;$ has an infinite number of roots. However, I don't see how to use this insight.

Is there a relevant theorem? If not, how do I attack this problem?

If I'm understanding my own question correctly, it can be re-stated as: can a non-trivial (infinite) power series have an infinite number of roots?

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It's not true. For example, let $f(x)$ be any power series with a discrete infinite set of zeros (e.g., the power series for $\sin(x)$, centered at $x=0$), and let $g(x) = 2f(x)$.

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  • $\begingroup$ interesting, thanks. I was reading a paper where my (false) conjecture was apparently assumed. Based on your answer, I'll have to make a (separate) follow-up posting. $\endgroup$ – user2661923 Nov 1 '18 at 2:46
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The two functions $$ f(x)=\sin x$$ and $$ g(x)=-\sin x$$ have the same values at infinitely many points and the power series are not the same at all.

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  • $\begingroup$ Thanks. If I hadn't already accepted quasi's answer I would have accepted yours. $\endgroup$ – user2661923 Nov 1 '18 at 2:48
  • $\begingroup$ @user2661923 thanks for your attention $\endgroup$ – Mohammad Riazi-Kermani Nov 1 '18 at 3:09

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