The series diverges because $\frac{2 + \cos(n)}{n^2} \geq \frac{1}{n^2}$

The series converges because $0 \leq 2 + \cos(n) \leq 3$ implies $0 \leq \frac{2 + \cos(n)}{n^2} \leq \frac{3}{n^2}$, and $\sum \frac{1}{n^2}$ converges

The series converges by the Ratio Test

Convergence cannot be determined without computing the exact sum