# Comparison of terms of power series and the functions they represent

Hopefully someone better versed in real analysis than I can help with the following:

If $f(x)$ and $g(x)$ are functions on the real line, and $f_n$ and $g_n$ are the coefficients of their series expansions over the same indices, and the terms of both expansions are monotone decreasing, does

$f(x) \leq g(x) \forall x \geq 0 \in \mathbb R, \implies f_n \leq g_n \forall n?$

With the implication the other way it seems intuitively true, but I'm not sure with the implication in this direction. Is there a counterexample?

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Technically, their series expansions are only true locally. So the coefficients could change depending on the point you expand around. It is also unclear if you require coefficients on every term. If not an idea to approach is this: If you think about this on say $(0,1)$ let $g(x)=x$ and $f(x)=x^2$. Then these two functions would be a counter example. – toypajme Dec 19 '12 at 5:10
@toypajme Thanks for the comment, sorry I was unclear. I understand that the coefficients can change depending on where the expansion takes place; I suppose what I'm interested in is if the expression holds wherever the expansion is made. I am interested in the case where there are equivalent coefficients on every term in both expansions, so for example comparing an expansion that has only even powers to one with odd powers would be invalid. – Bitrex Dec 19 '12 at 5:36

I doubt it. I bet you can find two quadratics, $f(x)=ax^2+bx+c$, $g(x)=a'x^2+b'x+c'$, with $f\le g$ for $x\ge0$, $c\gt b\gt a$, $c'\gt b'\gt a'$, and $b\gt b'$. Try it!