I would like to know if there is a systematic way of writing down an asymptotic series representation of a function? (like one can use Taylor series expansion for doing a power series).

Conversely given a series and a function is there a way to decide whether it is an asymptotic series of the function?

Most examples I see in books are kind of very special where one could develop an asymptotic series because of some special properties of the integral representation of the function like for the $Ei(x)$ etc. You basically integrate by parts and and in those special cases it works out.

Or if there is some simple test that one can do on a function to test whether it has an asymptotic expansion about that point?