Past exam of the mathematics course of the University of Cambridge / 2007 / ib / Paper 1 / 3F / Solution

The partial fraction decomposition is $f(z)=1/(z-i)+1/(z+i)$. Put $w=z-1$ and expand each reciprocal by a geometric series:Adding gives the Taylor seriesEquivalently, since $1\pm i=\sqrt{2}{e}^{\pm i\pi /4}$, its coefficient is $(-1{)}^n{2}^{(1-n)/2}\cos ((n+1)\pi /4)$. The first terms are $1-\frac{1}{2}{w}^2+\frac{1}{2}{w}^3-\frac{1}{4}{w}^4+\cdots$.

The nearest nonremovable poles are $i$ and $-i$, both at distance $\sqrt{2}$ from the center. The function is holomorphic throughout the open disk of that radius and cannot be holomorphic across either pole. Therefore the largest radius of convergence is $r=\sqrt{2}$.