Recurrent Sequences

Let $x_{0}, x_{1} \in \mathbb {R}, x_{n+1}=\frac{1}{3}\left(2 x_{n}+x_{n-1}\right)(n \in \mathbb {N})$. Prove that the limit $\lim x_{n}$ exists and find it.

Let $x_{0}, x_{1} \in \mathbb {R}$. Prove that the limit $\lim x_{n}$ exists and compute it in the following cases:

Let $x_{1} \in \mathbb {R}$ and $x_{n+1}=\left(2+\frac{2}{n}\right) x_{n}-1 \quad (n \in \mathbb {N})$.

Let $b \in \mathbb {R}, a_{n} \rightarrow a \in \mathbb {R}$ and $x_{n+1}=a_{n}+b x_{n}$. Prove that for any $x_{1} \in \mathbb {R}$

Let the sequence $\left\{ x_{n}\right\}$ and the number $p \in \mathbb {R}$ be such that $x_{n+1} \leqslant p x_{n}+(1-p) x_{n-1}$ for all $n=2,3, \ldots$ Prove that if $p>0$, then the limit $\lim x_{n} \in \overline{\mathbb {R}}$ exists, while for $p \leqslant 0$ this statement is false for some sequences.

Let the positive sequence $\left\{ x_{n}\right\}$ and the number $p \in \mathbb {R}$ be such that $x_{n+1} \leqslant x_{n}^{p} x_{n-1}^{1-p}$ for $n=2,3, \ldots$ Prove that if $p \in (0,2)$, then the sequence $\left\{ x_{n}\right\}$ converges, while for $p \notin (0,2)$ this statement is false for some sequences.

Let $x_{1}>0, x_{n+1}=\frac{1}{n^{\pi } x_{n}}(n \in \mathbb {N})$. For which value of $x_{1}$ is the sequence monotonic? What is the character of the monotonicity?

Let $p>0$ and $L_{p}=\lim x_{n}$, where $\left\{ x_{n}\right\}$ is a sequence such that $x_{1} \in (0,1)$ and $x_{n+1}=x_{n}+\left(\frac{x_{n}}{n}\right)^{p}$ for all $n \in \mathbb {N}$. Prove that $L_{p}<+\infty$ only for $p>1$. How fast does the limit $L_{2}$ grow if the first term of the sequence $x_{1}$ approaches one?

In solving the equation $x=\frac{1}{x}+a$ ($a$ a positive parameter) by the iteration method, a recurrent sequence $x_{n}=\frac{1}{x_{n-1}}+a, x_{0}>0$ arises. Prove that the limit $l=\lim x_{n}$ exists, and find the asymptotics of the differences $x_{n}-l$.

In solving the equation $x^{2}-a=0$ ($a$ a positive parameter) by Newton's method, a recurrent sequence $x_{n}=\frac{1}{2}\left(x_{n-1}+\frac{a}{x_{n-1}}\right), x_{0}>0$ arises. Prove that $x_{n} \rightarrow \sqrt{a}$ and that "superconvergence" occurs here: $x_{n}-\sqrt{a} \sim C q^{2^{n}}$ for some $q \in (0,1)$ and $C>0$. What are $C$ and $q$?

Let $f \in C(\left[a, b\right])$, with $f(x) \in [a, b]$ for all $x \in [a, b]$. Consider a sequence $\left\{ x_{n}\right\}$ such that $x_{1} \in [a, b]$ and $x_{n+1}= \left(1-c_{n}\right) x_{n}+c_{n} f\left(x_{n}\right), c_{n} \in (0,1)$ for all $n \in \mathbb {N}$.

Let a sequence $\left\{ x_{n}\right\}$ of nonnegative numbers be such that $x_{n} \leqslant \frac{1}{n^{2}}\left(x_{n-1}+x_{n-2}\right)$ for $n \geqslant 3$. Prove that $x_{n}=O\left(\frac{1}{n!}\right)$.

Let $k \in \mathbb {N}, A_{n} \uparrow +\infty$ and let $\left\{ x_{n}\right\}$ be a nonnegative sequence such that

for $n>k$. Set $S_{n}=\frac{1}{A_{1}}+\ldots +\frac{1}{A_{n}}$. Prove that

Let $k \in \mathbb {N}, A_{n} \uparrow +\infty$ and let $\left\{ x_{n}\right\}$ be a nonnegative sequence such that

for $n>k$. Assume that $A_{n+1}-A_{n} \geqslant \delta >0$. Prove that

Let $k \in \mathbb {N}, C>0$ and let $\left\{ x_{n}\right\}$ be a nonnegative sequence such that

for $n>k$. Prove that $x_{n}=O\left(\frac{C^{n}}{n!} n^{p}\right)$, where $p=\frac{1}{k}\left(C-\frac{k(k-1)}{2}\right)$. Verify that this estimate cannot be improved.

Let $k \in \mathbb {N}, A_{n} \uparrow +\infty , n\left(1-\frac{\ln A_{n-1}}{\ln A_{n}}\right) \rightarrow \theta \geqslant 0$ and let $\left\{ x_{n}\right\}$ be a nonnegative sequence such that

Prove that $x_{n}=O\left(A_{n}^{-\beta n}\right)$, if $0<\beta <\frac{1}{1+\theta }$.