How to compute $\lim\limits_{n\rightarrow \infty} \{(2+\sqrt{3})^{n}\}$, where $\{x\}$ is the fractional part of $x$?

Question

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?

Answer 1

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 $0 \lt 2 - \sqrt{3} \lt 1$.