Past exam of the mathematics course of the year 2 of the natural sciences course of the University of Cambridge / 2018 / Paper 1 / 7 / b / Solution

Put $w=z-4i$. Factor out the nonzero constant in the denominator and use the geometric series:This Taylor series converges when $|z-4i|<|3-4i|$. The simple pole at $z=3$ is the only finite obstruction to analyticity, whose distance from the expansion centre is five. Hence the radius of convergence is