Let $p_n$ be the nth prime number, $p_1=2,p_2=3,p_3=5,\ldots$
How to prove this series converges/diverges?
If it converges, then this disproves the twin prime conjecture, I believe.
If $\lim\ \cos p_n = 0$ and the twin prime conjecture were true, then we would have that
as $p_n$ runs through the lower twin prime (i.e. both $p_n$ and $p_n + 2$ are primes),
$0 = \lim\ \cos (p_n + 2) = \lim\ (\cos p_n \cos 2 + \sin p_n \sin 2) = \pm \sin 2$
If $\lim\ \cos p_n = 0$, then for any odd integer $M$, we must have that $\lim\ \cos (M\times p_n) = 0$ (as $\cos Mx$ can be written as an odd polynomial in $\cos x$), which I guess, implies that $ \lim\ \cos (2n+1) = 0$
If I remember correctly there was a previous question which disproved this.