In an answer to this question the problem was raised of the convergence or divergence of $$\sum_{k=1}^{\infty}\frac{1}{k^{2+\cos k}}\ .$$ The problem was (quite properly) dismissed as being significantly different from the original question. But I would like to know the answer, so here it is as a separate question.
Clearly the difficulty is that while $2+\cos k$ is always greater than $1$, there is no $\varepsilon>0$ such that $2+\cos k$ is always greater than $1+\varepsilon$.
Any ideas?
