# Power Series Equivalence Question

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?

## 2 Answers

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 $$g(x) = 2f(x)$$.

• 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. – user2661923 Nov 1 '18 at 2:46

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.

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