$\sum\_{n=1}^{\infty} (-1)^n\cos (1/n^2)$

For this sum, if I use the divergence test, am I allowed to say that:

$\lim\_{x \to \infty} (-1)^n\cos (1/n^2)$
= $\lim\_{x \to \infty} (-1)^n\*\lim\_{x \to \infty}\cos (1/n^2)$

Which diverges because first limit doesn't exist and the second one is equal to 1?

This means $\lim\_{x \to \infty} (-1)^n\cos (1/n^2) \neq 0$

So it diverges.

If I can't do that, what convergence test should I should?