
Convergence tests
When I'm testing $\displaystyle \sum_{n = 0}^\infty \cos \frac{1}{n} $ for convergence, how come I can apply the nth term test and show that it is divergent, i.e.
$\displaystyle \lim_{n \to \infty} \cos \frac{1}{n} = \cos 0 = 1, $
but when I'm testing $\displaystyle \sin \frac{1}{n} $, I can't apply the same reasoning, i.e.
$\displaystyle \lim_{n \to \infty} \sin \frac{1}{n} = \sin 0 = 0 \Rightarrow \sin \frac{1}{n} converges, $
but instead have to use the limit comparison test and using L'Hopital's Rule, ending up with an implication that $\displaystyle \sin \frac{1}{n}$ is divergent

Do you know the difference between necessary and sufficient conditions?

You can only show divergence that way. $\displaystyle \lim_{n\to \infty}a_n=0$ is necessary for convergence, but convergence does not follow from it.