In case of (ii),

As $n \rightarrow \infty$, $\cos \frac {1}{n^2} \rightarrow 1 \neq 0$. Hence this series diverges.

In case of (i),

$\cos \frac 1n = 1 - 2\sin^2 \frac {1}{2n}$. But since $\sum \sin \frac {1}{k}$ diverges, Hence $\sum\_{n=1}^{\infty} 1 - 2\sin^2 \frac{1}{2n}$ also. Hence $\sum \cos \frac 1n$ diverges. Am I correct here? If not,then what should I do?