Does $\sum_{n\ge0} \cos (\pi \sqrt{n^2+n+1}) $ converge/diverge? How would you prove convergence/divergence of the following series?
$$\sum_{n\ge0} \cos (\pi \sqrt{n^2+n+1}) $$
I'm interested in more ways of proving convergence/divergence for this series. 
My thoughts
Let
$$u_{n}= \cos (\pi \sqrt{n^2+n+1})$$
Let's first check A necessary condition for the convergence of that seires  is that the limit $$\lim\limits_{n \to \infty} u_n$$ should exist and be equal $0$
but i have answer in my old book but i can't remember how they got that
$$u_n=\cos \left(n\pi+\frac{\pi}{2}+\frac{3\pi}{8n}+o(\frac{1}{n^2})\right)=\frac{(-1)^{n+1}.3\pi}{8n}+o(\frac{1}{n^2})$$
since $|\frac{(-1)^{n+1}.3\pi}{8n}|$ decreasing. and converge $0$
then by compraison $u_n$ is convergent
any help would be appreciated

A hint:
$$\sqrt{n^2+n+1}-\left(n+{\textstyle{1\over2}}\right)={3/4 \over 
\sqrt{n^2+n+1}+\left(n+{1\over2}\right)}=:\ \alpha_n\searrow\ 0\qquad(n\to\infty)\ .$$
