Take the function
$$ f(z) = \frac{z+1}{z-1} $$
defined on the complex plane.
We can write a Taylor expansion about $z = 0$ for $f$, which turns out to be:
$$ f(z) = -1 - 2\sum_{n=1}^{\infty}z^n $$
with region of convergence $|z| < 1$.
Now, let's say I wanted an Laurent series for this function, still around $z = 0$, but this time for the region $|z| > 1$. How would I go about this?
I have tried applying the definition of the Laurent series, but as far as I can tell, it only applies for an expansion around a pole, and $z = 0$ is not a pole of $f$.
The result for the Laurent expansion about $z = 0$ in this region, which I am trying to reach, is
$$ f(z) = 1 + 2\sum_{n=1}^{\infty}z^{-n} $$
but I have absolutely no clue how to get there.