I am interested in theorems that allow me to differentiate a divergent asymptotic power series for $x \to \infty$. I have a function $f:\mathbb{R}\to\mathbb{R}$ that is differentiable for large enough inputs and has a power series expansion of the form
$$f(x) \sim \sum_{n\geq 1}^\infty \frac{a_n}{x^n}$$
Under what conditions can I now obtain a power series expansion for $f'(x)$ by term-wise differentiation?
The two conditions that I know are:
if $f'$ has a power series expansion, it must be the one obtained from $f$ by term-wise differentiation
if the power series expansion of $f$ holds uniformly on a suitably shaped subset of the complex plane, then its differentiation can be justified somehow using Cauchy's integral theorem (although I have not quite understood how – I only found the proof in A. Erdelyi's Asymptotic Expansions and that is not very detailed)
Are there other simple conditions like that? Perhaps something involving monotonicity of $f'$?
UPDATE: Seeing as my function is holomorphic, applying Cauchy's integral theorem in a similar fashion as Erdelyi worked fine, so I my problem is solved. Still, I think the question is of general interest, so I'll leave it open.