If a sequence converges to $\pi/2$ can it be determined whether $\sum_{n=0}^\infty \cos(a_n)$ converges or diverges?

Question

If $\lim_{n\to \infty}a_n=\pi/2$ then can you determine if $\sum_{n=0}^\infty \cos(a_n)$ converges or diverges? Would more information be required?

Answer 1

Yes, more information is required. For example, if $a_n = \arccos \frac{1}{n}$ then $$\lim_{n \to \infty} a_n = \arccos 0 = \frac{\pi}{2} \tag{*}$$ and $\sum \cos a_n$ diverges; while if $a_n = \arccos \frac{1}{n^2}$ then (*) holds and the cosine series converges.

Answer 2

Note that $a_n\to 0$ is just a necessary condition for convergence .

Thus for $\sum_{n=0}^\infty = cos(a_n)$ you can't conclude anything without other information to determine whether it converges or not.