\displaystyle -\frac{\ln \left( \cos \frac{1}{n} \right)}{\frac{1}{n^2}} = \frac{\ln \left( 1 - \left(1 - \cos \frac{1}{n} \right) \right)}{-(1 - \cos \frac{1}{n})} \cdot \frac{1 - \cos \frac{1}{n}}{\frac{1}{n^2}} \to \frac{1}{2} \in \langle 0, +\infty \rangle