convergence or divergence of $\sum^{\infty}_{n=1} {\cos n\over \sqrt n + \cos n}$

Question

Does

$$\sum^{\infty}_{n=1} {\cos n\over \sqrt n + \cos n}$$

converge or diverge, and why?

It's not positive so I can't apply comparison tests, and it isn't monotone (right?) so Abel and Dirichlet tests hasn't helped neither.

Answer 1

Considering that the following sums converge:

\begin{align*} \sum_n \frac{\cos(2n)}{n} &\quad& \sum_n \frac{\cos(n)}{\sqrt n} &\quad&\sum_n \frac{1}{n^{3/2}} \end{align*}

(where the last one converges absolutely) and that $\sum_n \frac1n$ doesn't, rewrite the summand:

$$\frac{\cos(n)}{\sqrt n+\cos(n) }=1- \frac{1}{1+\frac{\cos(n)}{\sqrt n}} = \frac{\cos(n)}{\sqrt n}- \frac{\cos^2(n)}{n}+\frac{1}{n^{3/2}} f(n)$$

where $f(n)$ is a bounded function, which follows from the power series of $\frac{1}{1+z}$ for $z$ near to zero.

The first and the last term converge in sum, for the second term $\cos^2(n)=\frac{1+\cos(2n)}{2}$, and while $\sum_n \frac{\cos(2n)}{n}$ converges, $\sum_n \frac{1}{n}$ doesn't. This means that the original sum doesn't converge, being a sum of convergent sums and a divergent one.