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

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 $|z|=1$.