Past exam of the mathematics course of the University of Cambridge / 2021 / ia / Paper 1 / 12F / b / Solution

Set $w=e^z$. ThenThe hypothesis that convergence and divergence both occur implies that this ordinary power series has a radius $R$ with $0<R<\infty$. Since $|w|=e^{\mathrm{Re}z}$, the half-plane of convergence of an exponential power series is determined bythe series converges for $\mathrm{Re}z<S$ and diverges for $\mathrm{Re}z>S$.

For the specified series,Writing $z=x+iy$ gives $|2e^{iz}|=2e^{-y}$. The series converges absolutely when $y>\log 2$ and diverges by the term test when $y<\log 2$. On $y=\log 2$, its absolute values are $(n+1{)}^{-2}$, so it still converges absolutely. Therefore the exact convergence set is