Nested radicals and iterated roots

Using the theorem on the existence of the limit of a monotone and bounded sequence, prove the convergence of the sequence$x_{1}=\sqrt{2}$, $x_{2}=\sqrt{2+\sqrt{2}}$, $\ldots$,

The sequence $x_{n}$ is given by the following equalities:

Find $\lim \limits_{n\to \infty }x_{n}$.

A sequence of values $u_{n}$ is given by the formulas: $u_{1} = \sqrt{2}$, $u_{2} = \sqrt{2+\sqrt{2}}$, $u_{3} = \sqrt{2+\sqrt{2+\sqrt{2}}}$ and in general $u_{n} = \sqrt{2+u_{n-1}}$. Replacing the last 2 under the radical by 4, verify that $u_{n}$ is bounded. Having proved that $\lim_{n \to \infty } u_{n}$ exists, find it from the equality $u_{n}^{2} = 2+u_{n-1}$.

A sequence $v_{n}$ is given by the equalities:

Replacing the number $a$ under the last radical by the quantity $a+x$, where $x$ is the positive root of the equation $x^{2} = a+x$, verify that $v_{n}$ is bounded, and prove that $v_{n} \to x$ as $n \to \infty$.

Prove that the sequence $\left\{ x_{n}\right\}$ has a limit, and find it, if:

Let $a > 0, x_{1} = \sqrt{a}, x_{n+1} = \sqrt{a+x_{n}}, n \in \mathbb {N}$. Prove that $\lim_{n \rightarrow \infty } x_{n}$ exists, and find it.

Let $x_{1} = \sqrt{2}, x_{n+1} = \sqrt{2+\sqrt{x_{n}}}, n \in \mathbb {N}$. Prove that the sequence $\left\{ x_{n}\right\}$ converges.

Prove the convergence of the sequence and find its limit, if:

See also [D]. Let $a>0$ and $x_{n}=\sqrt{a+\sqrt{a+\ldots +\sqrt{a}}}$ ($n$ radicals). Prove that

Let $x_{n}=\sqrt[3]{6+\sqrt[3]{6+\ldots +\sqrt[3]{6}}}$ ($n$ roots). Prove that $2-x_{n} \sim \frac{C}{(12)^{n}}$ for some $C>0$.

Let $p>1$ and $x_{n}=\sqrt[p]{1+\sqrt[p]{1+\ldots +\sqrt[p]{1}}}$ ($n$ roots). Prove that the sequence $\left\{ x_{n}\right\}$ converges to the positive root of the equation $x^{p}-x-1=0$.

Let $\alpha_{n}=\sqrt[n]{1+\sqrt[n+1]{1+\sqrt[n+2]{1+\ldots }}}$ (an infinite number of roots). Are these quantities finite? What is $\lim \alpha_{n}$?

Prove that

Let $x_{n}=\sqrt{1+\sqrt{2+\ldots +\sqrt{n}}}$. Prove that $x_{n} \rightarrow a \in \mathbb {R}$ and $\sqrt[n]{a-x_{n}} \sim \frac{1}{2} \sqrt{\frac{e}{n}}$.

Considering positive square roots only we find

Let $a_1, a_2, \ldots , a_n, \ldots$ be positive numbers and put

where only positive square roots are considered. The convergence of the sequence

is related to

in the following way:

(continued). The sequence $(t)$ is certainly convergent if the series

Let the numbers $\varepsilon_0, \varepsilon_1, \varepsilon_2, \ldots , \varepsilon_n, \ldots$ assume one of the three values $-1, 0, 1$. Then we have

Every value $x$ of the interval $[-2,2]$ can be written in the form

where $\varepsilon_k$, $k=0,1,2,\ldots$, is either $-1$ or $+1$. The representation is unique if $x$ is not of the form $2 \cos \dfrac {p}{2^q} \pi$, $p, q$ integers, $0<p<2^q$. These are the only numbers that may be written in the finite form

The finite representation can be extended to an infinite one in two ways: by putting either

or

(continued). The number $x$ is of the form $x=2\cos k\pi$, $k$ rational, if and only if the sequence $\varepsilon_0, \varepsilon_1, \varepsilon_2, \ldots$ is periodical after a certain term.

If $s_{1} = \sqrt{2}$, and

prove that $\left\{ s_{n}\right\}$ converges, and that $s_{n} < 2$ for $n = 1,2,3, \ldots$.