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Hey folks! I was trying to find the Laurent series of f=1/(z^2+z) in the region 1<|z-1|<2.
f can be expressed as 1/(w+1) - 1/(w+2), where w=z -1
f=1/(1-(-w)) - (1/2)(1/(1-(-w/2))
Anyway, at first I tried making 1/(1-(-w))= the sum from 0 to infinity of (-w)^n. However, that's only valid when |w|<1, which is not the case here, so I got a wrong answer.
Then I looked for some examples on the internet and found the following modified geometric series
1/(1-z)= -(1/z^0 + 1/z^1 + 1/z^2+...), for |z|>1.
Using this formula I got the right answer. My question now is, how is that formula derived?