Find the Laurent Series for f(z)=1/(z(z-1)) Valid on 1<|z-1|<infinity

[g1990]

Homework Statement

Find the Laurent series for f(z)=1/(z(z-1)) valid on 1<|z-1|<infinity

Homework Equations

1/(1+a)=1-a+a^2-a^3... where |a|<1
we are not supposed to use integrals for this problem

The Attempt at a Solution

I want 1/(z-1) to be in my final answer, so I have 1/(z(z-1))=(1/(z-1))(1/(z-1))(1/(1+1/(z-1))=(*)
I can then expand the last of the three terms in (*) as 1/(1-1/(z-1))=1-(z-1)^-1+(z-1)^-2 etc.
Is this right? can I then multiply it by the first two (multiplicative) terms in (*) to get an extra (z-1)^-2 in each term of the series

 

[vela]

Yes, that's right.