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I am studying a differential equation $$ y'(x)=g(x,y), $$

which has no analytic solution, however I have found that $y(x)$ is asymptotic to a series

$$ f(x)=\sum_{k=0}^\infty a_kx^{-k} $$

as $|x|\to\infty$. Clearly, such expansion is not well defined for $x=0$. Furtheremore, it is likely that the series diverges for $x\sim 0$.

Let $U$ be a neighborhood of $x=0$.

Is it possible to construct a (analytic?) function $F(x)$ such that $f(x)=F(x)\, \forall x\notin U$?

My first idea was just to construct another series around $x=0$ and then try to match them together. But, I do not have any analytic information on $y(0), f(0)$.

What would it be a good way to proceed?

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You may be interested in the section "How to 'Sum' an Asymptotic Series" in Miller's Applied Asymptotic Analysis. Google Books link. –  Antonio Vargas Oct 15 '13 at 17:42
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