Nested radical

Example

The infinite nested radical $$\sqrt{1 + \sqrt{1 + \sqrt{1 + \cdots}}}$$ is defined as the limit of the sequence $x_1 = 1$, $x_{n+1} = \sqrt{1 + x_n}$. This is an infinite composition of the map $f(x) = \sqrt{1 + x}$, since $x_n = f^{\circ n}(0)$. Its value is the golden ratio $\varphi = \frac{1 + \sqrt5}{2}$.

Check: the sequence is increasing, since $x_2 = \sqrt2 > x_1$ and $f$ is increasing. It is bounded above by $2$, since $x_n < 2$ gives $x_{n+1} < \sqrt3 < 2$. By the monotone convergence theorem it has a limit $L \ge 1$, and continuity of $f$ gives $L = \sqrt{1 + L}$. So $L^2 - L - 1 = 0$, and the positive root is $L = \varphi$.

The same method gives $\sqrt{2 + \sqrt{2 + \cdots}} = 2$, from $L = \sqrt{2 + L}$.