Cauchy convergence criterion for sequences

Using the Cauchy criterion, prove the convergence of the sequence$x_{n}=a_{0}+a_{1}q+\cdots +a_{n}q^{n}$, where $\left|a_{k}\right|<M (k=0,1,2,\ldots )$ and $\left|q\right|<1$.

Using the Cauchy criterion, prove the convergence of the sequence$x_{n}=\dfrac {\sin 1}{2}+\dfrac {\sin 2}{2^{2}}+\cdots +\dfrac {\sin n}{2^{n}}$.

Using the Cauchy criterion, prove the convergence of the sequence$x_{n}=\dfrac {\cos 1!}{1 \cdot 2}-\dfrac {\cos 2!}{2 \cdot 3}+\cdots +\dfrac {\cos n!}{n(n+1)}$.

Using the Cauchy criterion, prove the convergence of the sequence$x_{n}=1+\dfrac {1}{2^{2}}+\cdots +\dfrac {1}{n^{2}}$.

If there exists a number $c$ such that

then the sequence $x_{n}(n=1,2,\ldots )$ is said to have bounded variation. Prove that any sequence of bounded variation is convergent. Give an example of a convergent sequence that does not have bounded variation.

State the meaning of "a certain sequence does not satisfy the Cauchy criterion."

Using the Cauchy criterion, prove the divergence of the sequence

Prove that the sequence $\left\{ x_{n}\right\}$ is fundamental (Cauchy), if

Let $a_{0}$ be a nonnegative integer, and let $\left\{ a_{n}\right\}$ be a sequence whose terms are digits. Consider the sequence of finite decimal fractions $x_{n} = a_{0}, a_{1} a_{2} \ldots a_{n}, n \in \mathbb {N}$. Prove that this sequence is fundamental (Cauchy)

Prove that the sequence $\left\{ x_{n}\right\}$ converges, if $x_{n}$ equals

Prove that a fundamental (Cauchy) sequence is bounded

Prove that every subsequence of a fundamental (Cauchy) sequence is fundamental

Prove that for the sequence $\left\{ x_{n}\right\}$ to be fundamental, it is necessary and sufficient that the condition hold

Using the negation of the Cauchy condition, prove that the sequence $\left\{ x_{n}\right\}$ diverges, if $x_{n}$ equals

Let $\left\{ x_{n}\right\}$ and $\left\{ y_{n}\right\}$ be fundamental sequences. Prove that:

Prove that the sequence $\left\{ x_{n}\right\}$ converges, if $x_{n}$ equals:

The sequence $\left\{ x_{n}\right\}$ is such that the sequence $\left\{ \sum_{k = 1}^{n} x_{n}\right\}$ converges. Prove that $\lim_{n \rightarrow \infty } x_{n} = 0$.

The sequence $\left\{ x_{n}\right\}$ is monotone and $\lim_{n \rightarrow \infty } x_{n} = 0$. Prove that the sequence

converges.

Prove that a monotone bounded sequence is fundamental (Cauchy).

The sequence $\left\{ x_{n}\right\}$ is such that $\left|x_{n+1}-x_{n}\right| \leqslant C \alpha^{n}$, where $0 < \alpha < 1, \quad n \in \mathbb {N}$. Prove that the sequence $\left\{ x_{n}\right\}$ converges.

Prove that for any $x \in \mathbb {R}$ the sequence converges.

Give an example of a divergent sequence $\left\{ x_{n}\right\}$ such that for any $p \in \mathbb {N}$

The sequence $\left\{ x_{n}\right\}$ does not converge. Prove that there exists a sequence of natural numbers $\left\{ p_{n}\right\}$ such that the sequence $\left\{ x_{n+p_{n}}-x_{n}\right\}$ does not tend to 0.

For the sequence $\left\{ x_{n}\right\}$, denote

Prove: for the sequence $\left\{ x_{n}\right\}$ to satisfy the Cauchy condition, it is necessary and sufficient that $\lim_{n \rightarrow \infty } M_{n} = 0$.

Prove that the sequence

where $\alpha > 0$, converges.

Prove the convergence of a sequence bounded above and satisfying the condition:

(Leibniz's test.) The sequence $\left\{ a_{n}\right\}$ is such that

Prove that the sequence $\left\{ a_{n}\right\}$ converges.

A sequence $\left\{ x_{n}\right\}$ has bounded variation if the sequence is bounded

Prove that:

Establish Proposition A.2.3.