10.8.7 Series with complex terms

Problem: Expand \( f(z) \) into the Laurant series (Taylor) in the given ring or the neighbourhood of the given point (in this case indicate the domain of convergence of the obtained series). \[ f(z)=\frac{z}{z+1}+e \frac{z}{z-1}, \quad z_{0}=1 . \]