 Solution

ID: 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 - 4 i

. Factor out the nonzero constant in the denominator and use the geometric series:

\frac{1}{3 - z} = \frac{1}{3 - 4 i - w} = \frac{1}{3 - 4 i} \frac{1}{1 - w / ( 3 - 4 i )} = n = 0 \sum \infty \frac{( z - 4 i ) ^{n}}{( 3 - 4 i ) ^{n + 1}} .

This Taylor series converges when

∣ z - 4 i ∣ < ∣3 - 4 i ∣

z = 3

is the only finite obstruction to analyticity, whose distance from the expansion centre is five. Hence the radius of convergence is

R = 5 .

 New to topics? Read the docs here!