I suspect that the series
$$\sum\_{n=1}^\infty\left(\frac{\cos(2\pi n/3)}{\ln(1+n)}\right)^3$$
diverges, but having used all the criterion for divergence I can think of, I haven't come up with anything yet. Note that for $n=1,2$ the numerator is $-1/8$, and for $n=3$ it is 1. I suspect it diverges because we don't even have the conditions to apply the Dirichlet criterion, since the partial sums of $\cos(2\pi n/3)$ are not bounded.

Does anyone have an idea?