Q:

Does $\sum_{n=1}^{\infty} \sin(\pi(2+\sqrt{3})^n)$ converge? Converge absolutely?

Per the title, does $\displaystyle\sum_{n=1}^{\infty} \sin(\pi(2+\sqrt{3})^n)$ converge? Converge absolutely? 
I'm stuck on this question, not sure how to approach it.
Thank you!

A:

Hint: Let 
$$A_n=(2+\sqrt{3})^n+(2-\sqrt{3})^n.$$
Show first that $A_n$ is an integer.  There are various ways of seeing this, including expanding both powers  using the Binomial Theorem, and noting the cancellations.  Or else, if you are familiar with recurrences, the $A_n$ satisfy a nice Fibonacci-like recurrence relation. 
But $0<2-\sqrt{3}<0.27$, so $(2-\sqrt{3})^n$ goes to $0$ rapidly as $n$ gets large. Thus, for large $n$, $(2+\sqrt{3})^n$ is only a tiny bit below an integer.  That gives us a good handle on the size of $\sin(\pi(2+\sqrt{3})^n)$. For
$$\sin(\pi(2+\sqrt{3})^n)=\sin(\pi A_n -\pi(2-\sqrt{3})^n).$$
Now use the formula $\sin(x+y)=\sin x\cos y +\cos x\sin y$, and standard estimates for $|\sin t|$ when $t$ is close to $0$.

