Past exam of the mathematics course of the University of Cambridge / 2012 / ia / Paper 1 / 12F / a / ii / Solution

The radius of convergence of a power series is the extended real number $R\in [0,\infty ]$ characterized by absolute convergence for $|x|<R$ and divergence for $|x|>R$. Equivalently,For $R=0$, only the central point is guaranteed; for $R=\infty$, every real point is inside. At finite endpoints $x=\pm R$, convergence must be investigated separately and is not determined by $R$.