Finding Laurent Series for a Rational Function on an Annulus

[QuantumLuck]

Homework Statement

find the Laurent series for f(z) = 1/(z(z-1)(z-2)) on the annulus between 1 and 2. with the origin as center.

Homework Equations

The Attempt at a Solution

so i found the partial fraction decomposition of this function and it turns out to be f(z) = 1/2z + -1/(z-1) + 1/(2(z-2)). In order to find the Laurent series do I just Taylor expand each of my 3 different decomposition around their singularities? However, their singularities are not contained on the annulus so this doesn't seem to make much sense. I am unsure how to proceed from this spot.

 

[yyat]

Note that 1/2z and -1/(z-1) have their singularity "inside" the annulus, while 1/(2(z-2)) has a singularity "outside" the annulus. Try expressing the first two as series in 1/z and the third one as a series in z (i.e. Taylor series).