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?