Laurent Series at Infinity

Question

I thought that finding the Laurent series was something that was straightforward, however, I am having some difficulty of finding the Laurent series of

$$f(z) = \frac{1}{z(1-z)}$$

for $z= \infty$. Any suggestions?

Answer 1

Make the change of variable $w= 1/z$ and then find the Laurent series around $w=0$

Answer 2

We can write for $|z|>1$,

$$\frac{1}{z(1-z)}=-\frac1{z^2}\frac{1}{1-z^{-1}}=-\sum_{n=0}^{\infty}z^{-n-2}$$

Answer 3

Using generalized Laurentseries with two poles $$\sum _{k=0}^{n-1} x^k-\frac{x^n}{x-1}+\frac{1}{x}$$