Past exam of the mathematics course of the University of Cambridge / 2003 / ia / Paper 1 / 4C / Solution

The radius of convergence of a power series ${\sum }_{n\ge 0}{a}_n{z}^n$ is the unique $R\in [0,\infty ]$ for which the series converges absolutely when $|z|<R$ and diverges when $|z|>R$. When $R=\infty$ it converges for every complex $z$; when $R=0$ it converges only at the centre. The definition makes no assertion about convergence on $|z|=R$, so boundary points must be checked separately.