Note that,

n/(n+2) >= n/(n+n) = n/(2n) = 1/2 for sufficiently large n.

And Σ(1,inf)1/2 = +oo thus, Σ(1,inf)n/(n+2) = +oo

Note, cos(nΠ) = (-1)^{n} because it alternates between 1 and -1.Σ(1,inf) cos(nΠ) / n

Thus, instead you can consider,

Σ(1,inf) (-1)^n/n which converges by alternating series test.