Diverges by Divergence Test because $\cos(2n)$ does not approach 0

Converges by Abel's Test: $\sum \cos(2n)$ has bounded partial sums, and $\frac{1}{n^2+1}$ is monotone decreasing with limit 0

Diverges by Integral Test

Converges absolutely by Limit Comparison with $\frac{1}{n^2}$