I am looking at the convergence of $$\sum_n (-1)^n\frac{|\cos(n)|}{n}.$$ If I look at the convergence of the series with absolute value of the general term, that is $$\sum_n \frac{|\cos(n)|}{n},$$ I can show that it diverges since $$\sum_n \frac{|\cos(n)|}{n}\geq \sum_n \frac{|\cos(n)|^2}{n}=\sum_n \frac{1+\cos(2n)}{2n}$$ which is the sum of the divergent series $\sum_n \frac{1}{2n}$ and the convergent series $\sum_n\frac{\cos(2n)}{n}$ (It can be obtained using Abel's theorem).
Now with the $(-1)^n$, I have no idea if it converges or not. We can't use the alternate series theorem since $(|\cos(n)|/n)_n$ is not decreasing. I try to use Abel's theorem and thus I try to show that $$\sum_n\left|\frac{|\cos(n)|}{2n+1}-\frac{|\cos(n+1)|}{2(n+1)+1}\right|$$ converges but I can't manage to do it.
