Solution
= Solution
Use $\sin z=z+O(z^3)$ and
$$
(1-e^{-z})^{-3}=z^{-3}(1-z/2+O(z^2))^{-3}=z^{-3}(1+3z/2+O(z^2)).
$$
Their product is $z^{-2}+(3/2)z^{-1}+O(1)$. Thus the singularity is a \b[<pole> of order two], with $\boxed{\operatorname{Res}_0 f=3/2}$.