Past exam of the mathematics course of the University of Oxford / 2015 / Prelims / Mathematics II / 3 / b / i / Solution

The radius of convergence of a power series ${\sum }_{n=0}^{\infty }{a}_n{z}^n$ is the number $R\in [0,\infty ]$ such that the series has absolute convergence for $|z|<R$ and diverges for $|z|>R$. Equivalently,Use $r^0=1$, including at $r=0$. The supremum may be $+\infty$. There is no general conclusion at $|z|=R$: boundary convergence depends on the coefficients. If $R=0$, only the centre is guaranteed; if $R=\infty$, every finite $z$ lies inside the disk of convergence.