Past exam of the mathematics course of the University of Cambridge / 2004 / ib / Paper 3 / 17E / iii / Solution

Near zero the $n=0$ summand supplies the pole and the remaining series is holomorphic, so $f(w)=w^{-2}+F(w)$ with $F$ holomorphic near zero. Choose the local logarithm with $L(1)=0$. Writing $h=z-1$ gives $L(z)=h-h^2/2+O(h^3)$ and henceTherefore $z=1$ is a double pole, with leading coefficient ${\lim }_{z\to 1}(z-1{)}^{2}g(z)=-4{\pi }^2$. Its residue, if desired, is $-4{\pi }^2$.