Solution
= Solution
Partial fractions give
$$
f=\frac1{2z^2}-\frac3{4z}+\frac1{z+1}-\frac1{4(z+2)}.
$$
For $1\lt |z|\lt 2$ this becomes
$$
\boxed{\frac1{2z^2}-\frac3{4z}+\sum_{n\ge0}(-1)^nz^{-n-1}-\frac18\sum_{n\ge0}(-z/2)^n.}
$$
= Solution
Partial fractions give
$$
f=\frac1{2z^2}-\frac3{4z}+\frac1{z+1}-\frac1{4(z+2)}.
$$
For $1\lt |z|\lt 2$ this becomes
$$
\boxed{\frac1{2z^2}-\frac3{4z}+\sum_{n\ge0}(-1)^nz^{-n-1}-\frac18\sum_{n\ge0}(-z/2)^n.}
$$