I have a function $f(x)$ that has convergent Taylor series expansion around $x=0$ in the following form:
where $(-1)^ig_i(x)$ denotes the $i$-th derivative of $f(x)$ with respect to $x$. I know that $g_i(x)>0$ for all $i$ at $x=0$. I am interested in bounding $f(x)$ around small positive $x$, say $0<x\leq\epsilon$.
I would like to make sure that, given all the facts that I described, I can make a claim that the terms $0$ through $i$ of Taylor series above form an upper bound on $f(x)$ for small positive $x$ if $i$ is even, and lower bound if $i$ is odd, or whether these facts are insufficient to make such a claim.
(I haven't worked with Taylor series in a while and I would rather make a fool of myself in front of the experts here than at work.)