Why does the series $\sum\limits_{n=2}^\infty\frac{\cos(n\pi/3)}{n}$ converge? Why does this series
$$\sum\limits_{n=2}^\infty\frac{\cos(n\pi/3)}{n}$$
converge? Can't you use a limit comparison with $1/n$? 

Note that $$\cos(n\pi/3) = 1/2, \  -1/2, \  -1, \  -1/2, \  1/2, \  1, \  1/2, \  -1/2, \  -1, \  \cdots $$ so your series is just 3 alternating (and convergent) series inter-weaved. Exercise: Prove that if $\sum a_n, \sum b_n$ are both convergent, then the sequence $$a_1, a_1+b_1, a_1+b_1+a_2, a_1+b_1+a_2+b_2, \cdots $$ is convergent. Applying that twice proves your series converges. 
