I have to find Laurent representation of a function $\displaystyle f(x)= \frac{1}{(z-1)(z-2)}$ , $\displaystyle |z|<1$

So

$\displaystyle f(x)= \frac{1}{(z-1)(z-2)} = \frac{1}{z-2}- \frac{1}{z-1}$

$\displaystyle \frac{1}{z-2} = \frac{1}{z} \cdot \frac{1}{1- \frac{2}{z} } = \sum_{0}^{n} \frac{2^{n}}{z^{n+1}}$

$\displaystyle \frac{1}{z-1}= - \frac{1}{1-z} = \sum_{0}^{n} - z^{n}$

Finally

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

Good?