Past exam of the mathematics course of the University of Cambridge / 2023 / ia / Paper 4 / 5F / d

Every term is nonnegative. Since $1-\cos x\le x^2/2$,Comparison with $\sum k^{-2}$ shows that the partial sums are bounded. Being increasing, they converge to the finite limit

Taking real parts of the geometric sum givesso $(a_n)$ is bounded when $\cos y\ne 1$. It cannot converge: convergence would imply $\cos (ny)=a_n-a_{n-1}\to 0$, but thenwhereas the same necessary condition applied to the subsequence $2n$ would give a limit of zero.