<s>[INST] Simplifying $\sqrt{5+2\sqrt{5+2\sqrt{5+2\sqrt {5 +\cdots}}}}$ How to simplify the expression:
$$\sqrt{5+2\sqrt{5+2\sqrt{5+2\sqrt{\cdots}}}}.$$
If I could at least know what kind of reference there is that would explain these type of expressions that would be very helpful.
Thank you.
 [/INST] Put
$$x_0:=0,\quad x_1:=\sqrt{5},\quad x_2:=\sqrt{5+2\sqrt{5}},\quad x_3:=\sqrt{5+2\sqrt{5+2\sqrt{5}}}\ ,$$
and so on, which amounts to
$$x_0:=0,\qquad x_{n+1}:=\sqrt{5+2x_n}\quad(n\geq0)\ .$$
Then
$$x_{n+1}-x_n=\sqrt{5+2x_n}-\sqrt{5+2x_{n-1}}={2(x_n-x_{n-1}) \over \sqrt{5+2x_n}+\sqrt{5+2x_{n-1}}}\ .$$
As $x_1-x_0>0$ this shows that the sequence $(x_n)_{n\geq0}$ is momotonically increasing. 
Furthermore $0\leq x_0<4$, and for any $n\geq0$ the statement $0\leq x_n< 4$  implies $$0\leq x_{n+1}<\sqrt{5+2\cdot 4}<4\ .$$ This shows that our sequence is as well bounded, so it has a limit $\xi\in[0,4]\ $. This limit satisfies the equation $x=\sqrt{5+2x}$ and  is therefore given by $\xi=1+\sqrt{6}\doteq 3.45$.
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