TITLE: How to compute $\lim\limits_{n\rightarrow \infty} \{(2+\sqrt{3})^{n}\}$, where $\{x\}$ is the fractional part of $x$?
QUESTION [6 upvotes]: I've stumbled upon the following problem:
$$\lim_{n\rightarrow \infty} \{(2+\sqrt{3})^{n}\}$$
where "{}" notates the fractional part.
I've never studied this kind of problem, there exists any reference that I could read about
this type of problem?

REPLY [13 votes]: Hint: Note, for all integers $n \ge 0$, that
$$f(n) = (2 + \sqrt{3})^n + (2 - \sqrt{3})^n \tag{1}\label{eq1A}$$
is an integer, which you can fairly easily show by expanding both terms and collecting the powers of $\sqrt{3}$ using the Binomial theorem. Also, note $0 \lt 2 - \sqrt{3} \lt 1$.