Does this alternating series converge or diverge $\sum _{n=1}^{\infty } (-1)^n \frac{n + \cos(n)}{n + \cos(\cos(n))}$

Question

I'm trying to figure out whether this series converges or diverges but I am not sure which test to use as I always get a super messy formula. Am I missing something obvious that tells me if it converges or not ?.

$$ \sum _{n=1}^{\infty }\:\left(-1\right)^{n}\, \frac{n + \cos\left(n\right)}{n + \cos\left(\cos\left(n\right)\right)} $$

Answer 1

Since$$\lim_{n\to\infty}\left|(-1)^n\frac{n+\cos n}{n+\cos(\cos n)}\right|=1,$$your series diverges.