Obtaining all possible series expansion of $f(z) = \frac{1}{z(z-1)^2}$ about $z=1$

Question

The function is $f(z) = \frac{1}{z(z-1)^2}$ and we have to obtain all possible expansions about $z=1$.

I tried using Binomial Expansion but failed.

Answer 1

HINT:

Using partial fraction expansion, we can write

$$\frac{1}{z(z-1)^2}=\frac{1}{z}-\frac{1}{z-1}+\frac{1}{(z-1)^2}$$

Then, write $\frac{1}{z}=\frac{1}{1+(z-1)}$ and expand the latter expression as a geometric series.