= 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 hence
$$
g(z)=-\frac{4\pi^2}{L(z)^2}+F\left(\frac{L(z)}{2\pi i}\right)
=-4\pi^2h^{-2}-4\pi^2h^{-1}+O(1).
$$
Therefore \b[$z=1$ is a double pole], with leading coefficient $\lim_{z\to1}(z-1)^2g(z)=-4\pi^2$. Its <residue>, if desired, is $-4\pi^2$.
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