Past exam of the algebra, analysis and differential equations II course of the University of Oxford / 2010 / Analysis / 5 / c

For $|z|<1$, differentiate the geometric series to obtain $(1-t{)}^{-2}={\sum }_{k\ge 0}(k+1)t^k$. Since $(z-i{)}^2=-(1-z/i{)}^2$,The convergence region is precisely $0<|z|<1$, the specified punctured disk; the other pole is on its boundary at $i$.