Past exam of the mathematics course of the year 2 of the natural sciences course of the University of Cambridge / 2019 / Paper 1 / 7 / e / Solution

Put $\zeta =z-3$ and apply the geometric series:ThusThe radius is the distance from the expansion centre $3$ to the nearest singularity $2i$, not the distance of that pole from the origin. On the boundary circle the series terms all have magnitude $1/\sqrt{13}$, so they do not approach zero: this particular series diverges at every boundary point.