Practical Use of Series Expansion at $x=\infty$

Asking WolframAlpha on certain functions, it happens that you get a series expansion at $\infty$. Thinking of the expansion as an approximation of the function in the vincinity of a point $a$, like in Taylor Series $$ f(x)= f(a)+\frac {f'(a)}{1!} (x-a)+ \frac{f''(a)}{2!} (x-a)^2+ \cdots. \tag{$*$} $$ I wonder for which values, other than $\infty$, this expansion is a good one. Put $a=\infty$ in $(*)$ doesn't seem to make sense. I can image that something like this is used, when it comes to limit calculations, but are they of any other use?

One can apply a linear fractional transformation to move a point at $\infty$ to any finite point, e.g. changing variables by $z = 1/x$ transforms the series expansion from around $x= \infty$ to an ordinary Taylor (or Laurent) series around $z = 0$, e.g. see the discussion of singular points at $\infty$ in the Wiki page on regular singular points of ODEs, via Möbius transformations on the Riemann sphere.

This enables one to translate notions such as "converges in a neighborhood of $\infty$" to more standard notions such as "converges in a neighborhood of $0$". However, one will gain greater insight by instead learning about how to extend such such notions to such "completions" having ideal point(s) at infinity, since their theory is often simpler, due to having more uniform structure (so eliminating exceptional cases). For a nice introduction see the reference here on points at infinity, projective closure, compactifications, and modifications.

Your expression is a good expansion around any finite point $a$. If you look at the Alpha approximation at $\infty$, the terms are $x^{-n}$ instead of $x^{n}$ and you can often think of it informally as a standard approximation in the variable $y=\frac 1x$ around $y=0$. You could write an expansion in $y=\frac 1x$ around some non-zero value and get powers of $\frac 1x -a$, but I haven't seen it. I have used (but forget when) expansions around $\infty$ of the form $f(x)=a+\frac bx$ to approximate values.