Does this tricky trigonometric series converge? My question was to know if the following series converges
$$
\sum_{n \geq 0}^{ }\cos\left(\frac{\pi}{4}\left(7+4\sqrt{3}\right)^n\right)
$$
I may have found a ( weird ) way to do it, but I would like to know how you, you would be doing this. Is there specific theorem that could conclude directly about this ?
Is it possible that
$$
\lim\limits_{n \rightarrow +\infty}\cos\left(\frac{\pi}{4}\left(7+4\sqrt{3}\right)^n\right)=0
$$ ?

$$ (7 + \sqrt {48})^n + (7 - \sqrt {48})^n =  (7 + \sqrt {48})^n + \left( \frac{1}{7 + \sqrt {48}}\right)^n $$
is always an integer. Indeed, always an integer $m$ such that $m \equiv 2 \pmod 4.$ Including negative $n,$ the values are
$$ ..., 2702, 194, 14, 2, 14, 194, 2702,...    $$
such that
$$ a_{n+1} + a_{n-1} = 14 a_n $$ 
