Past exam of the mathematics course of the University of Cambridge / 2003 / ia / Paper 1 / 4C (Analysis)

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.

The absolute ratio of successive terms is $|z|n^2/(n+1{)}^2\to |z|$. The ratio test gives convergence for $|z|<1$ and divergence for $|z|>1$, so the radius of convergence is $\boxed{R=1}$. On the boundary, $|n^{-2}{z}^n|=n^{-2}$ andThe positive partial sums are bounded, and thus converge. By the comparison test for series, the original power series converges absolutely at every point of $|z|=1$.

The coefficient ratio isThus the ratio test again gives radius of convergence $\boxed{R=1}$. On $|z|=1$, however, the terms have absolute value $n+2^{-n}$, which does not tend to zero. The necessary term condition for a convergent series fails at every boundary point. Therefore this power series diverges everywhere on $|z|=1$.