Taylor expansion at infinity

Solution 1:

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 $x_0=\infty$

Solution 2:

Think of Taylor expansion as an approximation formula, with main term $f(x_0)$ and $\epsilon = x-x_0$ being a small parameter. When expanding around $x_0 = \infty$, $x-x_0$ is no longer small, but $x^{-1}$ is.