TITLE: Taylor expansion at infinity
QUESTION [29 upvotes]: Are there case where does make sense to speak about the "Taylor expansion of a function ad infinity"?  
By inversion, sending $x \to \frac{1}{x}$ one could exchange $0\leftrightarrows\infty$; then if the values of the derivatives of a function are finite at infinity I was wondering if it is possible to give some sense to $(x-\infty)^n$ in order to define the "Taylor expansion of a function ad infinity".

REPLY [14 votes]: While studying complex analysis concepts, we come across Laurent series at infinity.
There, if after substituting $z=\frac{1}{x}$, the function $f$ has Taylor expansion $c_0+c_1 z+c_2 z^2 + \dots$ for $z$ near $0$, then, substituting back $x=\frac{1}{z}$, the series $c_0+c_1 x^{-1} + c_2 x^{-2}+\dots$ could be called as Taylor series at $x_0=\infty$