I'm trying to figure out if the following series converges or diverges. I have spent hours on it and can't figure it out. Tried to use the comparison test, Dirichlet, and Abel, but none of it worked. Any hints?
$$ \sum_{n=1}^\infty \frac{\cos^2(n)}{\sqrt{n}} $$
On
Another approach is to observe that among $e^{in}, n=1,2,\dots, 7,$ at least one of these points lies in the arc $\{e^{it}: t\in (-\pi/4,\pi/4)\}.$ Thus
$$\sum_{n=1}^{7} \frac{\cos^2 n}{\sqrt n} \ge \frac{1}{ 2}\frac{1}{\sqrt 7}.$$
The same thing happens for $n=8,\dots,14.$ etc. So the series in question is at least
$$\sum_{m=1}^{\infty} \frac{1}{ 2}\frac{1}{\sqrt {7m}} = \infty.$$
Note that this idea will work if $\cos^2 n$ is replaced by $|\cos n|^p$ for any $p>0$.
HINT:
Recall that $\cos^2(x)=\frac{1+\cos(2x)}{2}$.