Theory of Sequences

Let

Prove that

that is: for any given $\varepsilon >0$, find a number $N=N(\varepsilon )$ such that when $n>N$, $\left|x_{n}-1\right|<\varepsilon$. Fill in the table below:

Suppose:

Prove that the sequences

Prove that

is unbounded, but as $n \to \infty$ it does not become infinite.

Express the following statements using inequalities:

Let $n$ run through the sequence of natural numbers; find the value of the following expression:$\lim \limits_{n \to \infty } \dfrac {10000n}{n^{2}+1}$.

Let $n$ run through the sequence of natural numbers; find the value of the following expression:$\lim \limits_{n \to \infty } (\sqrt{n+1}-\sqrt{n})$.

Let $n$ run through the sequence of natural numbers; find the value of the following expression:$\lim \limits_{n \to \infty } \dfrac {\sqrt[3]{n^{2}}\sin n!}{n+1}$.

Let $n$ run through the sequence of natural numbers; find the value of the following expression:$\lim \limits_{n \to \infty } \dfrac {(-2)^{n}+3^{n}}{(-2)^{n+1}+3^{n+1}}$.

Let $n$ run through the sequence of natural numbers; find the value of the following expression:$\lim \limits_{n \to \infty } \dfrac {1+a+a^{2}+\cdots +a^{n}}{1+b+b^{2}+\cdots +b^{n}} \quad (\left|a\right|<1, \left|b\right|<1)$.

Let $n$ run through the sequence of natural numbers; find the value of the following expression:$\lim \limits_{n \to \infty } \left(\dfrac {1}{n^{2}}+\dfrac {2}{n^{2}}+\cdots +\dfrac {n-1}{n^{2}}\right)$.

Let $n$ run through the sequence of natural numbers; find the value of the following expression:$\lim \limits_{n \to \infty } \left[\dfrac {1}{n}-\dfrac {2}{n}+\dfrac {3}{n}-\cdots +\dfrac {(-1)^{n-1}n}{n}\right]$.

Let $n$ run through the sequence of natural numbers; find the value of the following expression:$\lim \limits_{n \to \infty } \left[\dfrac {1^{2}}{n^{3}}+\dfrac {2^{2}}{n^{3}}+\cdots +\dfrac {(n-1)^{2}}{n^{3}}\right]$.

Let $n$ run through the sequence of natural numbers; find the value of the following expression:$\lim \limits_{n \to \infty } \left[\dfrac {1^{2}}{n^{3}}+\dfrac {3^{2}}{n^{3}}+\cdots +\dfrac {(2n-1)^{2}}{n^{3}}\right]$.

Let $n$ run through the sequence of natural numbers; find the value of the following expression:$\lim \limits_{n \to \infty } \left(\dfrac {1}{2}+\dfrac {3}{2^{2}}+\dfrac {5}{2^{3}}+\cdots +\dfrac {2n-1}{2^{n}}\right)$.

Let $n$ run through the sequence of natural numbers; find the value of the following expression:$\lim \limits_{n \to \infty } \left[\dfrac {1}{1 \cdot 2}+\dfrac {1}{2 \cdot 3}+\cdots +\dfrac {1}{n(n+1)}\right]$.

Let $n$ run through the sequence of natural numbers; find the value of the following expression:$\lim \limits_{n \to \infty } \left(\sqrt{2} \cdot \sqrt[4]{2} \cdot \sqrt[8]{2} \cdots \sqrt[2^{n}]{2}\right)$.

Prove the following identity:$\lim \limits_{n \to \infty } \dfrac {n}{2^{n}}=0$.

Prove the following identity:$\lim \limits_{n \to \infty } \dfrac {2^{n}}{n!}=0$.

Prove the following identity:$\lim \limits_{n \to \infty } \dfrac {n^{k}}{a^{n}}=0 \quad (a>1)$.

Prove the following identity:$\lim \limits_{n \to \infty } \dfrac {a^{n}}{n!}=0$.

Prove the following identity:$\lim \limits_{n \to \infty } nq^{n}=0$, if $\left|q\right|<1$.

Prove the following identity:$\lim \limits_{n \to \infty } \sqrt[n]{a}=1 \quad (a>0)$.

Prove the following identity:$\lim \limits_{n \to \infty } \dfrac {\log_{a}n}{n}=0 \quad (a>1)$.

Prove the following identity:$\lim \limits_{n \to \infty } \sqrt[n]{n}=1$.

Prove the following identity:$\lim \limits_{n \to \infty } \dfrac {1}{\sqrt[n]{n!}}=0$.

When $n$ is sufficiently large, which of the following expressions is larger?

Prove that

Prove that the sequence

is monotonically increasing and bounded above, while the sequence

is monotonically decreasing and bounded below. From this, deduce that these sequences have a common limit:

Prove that

For what values of the exponent $n$ does the expression $\left(1+\dfrac {1}{n}\right)^{n}$ differ from the number $e$ by less than $0.001$?

Let $p_{n}(n=1,2,\ldots )$ be an arbitrary sequence tending to $+\infty$, and let $q_{n}(n=1,2,\ldots )$ be an arbitrary sequence tending to $-\infty$. Prove that

Given that

prove that

From this, deduce the formula

where $0<\theta_{n}<1$, and compute the number $e$ to an accuracy of $10^{-5}$.

Prove that the number $e$ is irrational.

Prove the inequality

Prove the inequalities:

Prove that

where $\ln a$ is the logarithm of the number $a$ taken to base $e=2.718\ldots$.

Using the theorem on the existence of the limit of a monotone and bounded sequence, prove the convergence of the sequence

where $p_{i}(i=0,1,2,\ldots )$ are nonnegative integers, and from $p_{1}$ onward not exceeding 9.

Using the theorem on the existence of the limit of a monotone and bounded sequence, prove the convergence of the sequence$x_{n}=\dfrac {10}{1} \cdot \dfrac {11}{3} \cdots \dfrac {n+9}{2n-1}$.

Using the theorem on the existence of the limit of a monotone and bounded sequence, prove the convergence of the sequence$x_{n}=\left(1-\dfrac {1}{2}\right)\left(1-\dfrac {1}{4}\right)\cdots \left(1-\dfrac {1}{2^{n}}\right)$.

Using the theorem on the existence of the limit of a monotone and bounded sequence, prove the convergence of the sequence$x_{n}=\left(1+\dfrac {1}{2}\right)\left(1+\dfrac {1}{4}\right)\cdots \left(1+\dfrac {1}{2^{n}}\right)$.

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

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 if the sequence $x_{n}(n=1,2,\ldots )$ converges, then any of its subsequences $x_{p_{n}}$ also converges, and has the same limit:

Prove that if some subsequence of a monotone sequence converges, then the monotone sequence itself is convergent.

Prove that if

then

Let $x_{n} \to a$, then what is the limit

Prove that a convergent sequence is bounded.

Prove that a convergent sequence either attains its supremum, or attains its infimum, or attains both. Give examples of each of these three types of sequences.

Prove that a sequence $x_{n}(n=1,2,\ldots )$ tending to $+\infty$ necessarily attains its infimum.

Find the largest term of the sequence $x_{n}$ $(n=1,2,\ldots )$, where$x_{n}=\dfrac {n^{2}}{2^{n}}$.

Find the largest term of the sequence $x_{n}$ $(n=1,2,\ldots )$, where$x_{n}=\dfrac {\sqrt{n}}{100+n}$.

Find the largest term of the sequence $x_{n}$ $(n=1,2,\ldots )$, where$x_{n}=\dfrac {1000^{n}}{n!}$.

Find the smallest term of the sequence $x_{n}$ $(n=1,2,\ldots )$, if$x_{n}=n^{2}-9n-100$.

Find the smallest term of the sequence $x_{n}$ $(n=1,2,\ldots )$, if$x_{n}=n+\dfrac {100}{n}$.

Find $\inf \left\{ x_{n}\right\}$, $\sup \left\{ x_{n}\right\}$, $\varliminf \limits_{n \to \infty } x_{n}$ and $\varlimsup \limits_{n \to \infty } x_{n}$ for the sequence $x_{n}$ $(n=1,2,\ldots )$, given:$x_{n}=1-\dfrac {1}{n}$.

Find $\inf \left\{ x_{n}\right\}$, $\sup \left\{ x_{n}\right\}$, $\varliminf \limits_{n \to \infty } x_{n}$ and $\varlimsup \limits_{n \to \infty } x_{n}$ for the sequence $x_{n}$ $(n=1,2,\ldots )$, given:$x_{n}=\dfrac {(-1)^{n}}{n}+\dfrac {1+(-1)^{n}}{2}$.

Find $\inf \left\{ x_{n}\right\}$, $\sup \left\{ x_{n}\right\}$, $\varliminf \limits_{n \to \infty } x_{n}$ and $\varlimsup \limits_{n \to \infty } x_{n}$ for the sequence $x_{n}$ $(n=1,2,\ldots )$, given:$x_{n}=1+\dfrac {n}{n+1}\cos \dfrac {n\pi }{2}$.

Find $\inf \left\{ x_{n}\right\}$, $\sup \left\{ x_{n}\right\}$, $\varliminf \limits_{n \to \infty } x_{n}$ and $\varlimsup \limits_{n \to \infty } x_{n}$ for the sequence $x_{n}$ $(n=1,2,\ldots )$, given:$x_{n}=1+2(-1)^{n+1}+3(-1)^{\frac{n(n-1)}{2}}$.

Find $\inf \left\{ x_{n}\right\}$, $\sup \left\{ x_{n}\right\}$, $\varliminf \limits_{n \to \infty } x_{n}$ and $\varlimsup \limits_{n \to \infty } x_{n}$ for the sequence $x_{n}$ $(n=1,2,\ldots )$, given:$x_{n}=\dfrac {n-1}{n+1}\cos \dfrac {2n\pi }{3}$.

Find $\inf \left\{ x_{n}\right\}$, $\sup \left\{ x_{n}\right\}$, $\varliminf \limits_{n \to \infty } x_{n}$ and $\varlimsup \limits_{n \to \infty } x_{n}$ for the sequence $x_{n}$ $(n=1,2,\ldots )$, given:$x_{n}=(-1)^{n}n$.

Find $\inf \left\{ x_{n}\right\}$, $\sup \left\{ x_{n}\right\}$, $\varliminf \limits_{n \to \infty } x_{n}$ and $\varlimsup \limits_{n \to \infty } x_{n}$ for the sequence $x_{n}$ $(n=1,2,\ldots )$, given:$x_{n}=-n\left[2+(-1)^{n}\right]$.

Find $\inf \left\{ x_{n}\right\}$, $\sup \left\{ x_{n}\right\}$, $\varliminf \limits_{n \to \infty } x_{n}$ and $\varlimsup \limits_{n \to \infty } x_{n}$ for the sequence $x_{n}$ $(n=1,2,\ldots )$, given:$x_{n}=n(-1)^{n}$.

Find $\inf \left\{ x_{n}\right\}$, $\sup \left\{ x_{n}\right\}$, $\varliminf \limits_{n \to \infty } x_{n}$ and $\varlimsup \limits_{n \to \infty } x_{n}$ for the sequence $x_{n}$ $(n=1,2,\ldots )$, given:$x_{n}=1+n\sin \dfrac {n\pi }{2}$.

Find $\inf \left\{ x_{n}\right\}$, $\sup \left\{ x_{n}\right\}$, $\varliminf \limits_{n \to \infty } x_{n}$ and $\varlimsup \limits_{n \to \infty } x_{n}$ for the sequence $x_{n}$ $(n=1,2,\ldots )$, given:$x_{n}=\dfrac {1}{n-10.2}$.

Find $\varliminf \limits_{n \to \infty } x_{n}$ and $\varlimsup \limits_{n \to \infty } x_{n}$, given:$x_{n}=\dfrac {n^{2}}{1+n^{2}}\cos \dfrac {2n\pi }{3}$.

Find $\varliminf \limits_{n \to \infty } x_{n}$ and $\varlimsup \limits_{n \to \infty } x_{n}$, given:$x_{n}=\left(1+\dfrac {1}{n}\right)^{n} \cdot (-1)^{n}+\sin \dfrac {n\pi }{4}$.

Find $\varliminf \limits_{n \to \infty } x_{n}$ and $\varlimsup \limits_{n \to \infty } x_{n}$, given:$x_{n}=\dfrac {n}{n+1}\sin^{2}\dfrac {n\pi }{4}$.

Find $\varliminf \limits_{n \to \infty } x_{n}$ and $\varlimsup \limits_{n \to \infty } x_{n}$, given:$x_{n}=\sqrt[n]{1+2^{n} \cdot (-1)^{n}}$.

Find $\varliminf \limits_{n \to \infty } x_{n}$ and $\varlimsup \limits_{n \to \infty } x_{n}$, given:$x_{n}=\cos^{n}\dfrac {2n\pi }{3}$.

Find the limit points of the sequence$\dfrac {1}{2}, \dfrac {1}{2}, \dfrac {1}{4}, \dfrac {3}{4}, \dfrac {1}{8}, \dfrac {7}{8}, \ldots , \dfrac {1}{2^{n}}, \dfrac {2^{n}-1}{2^{n}}, \ldots$.

Find the limit points of the sequence$1, \dfrac {1}{2}, 1+\dfrac {1}{2}, \dfrac {1}{3}, 1+\dfrac {1}{3}, \dfrac {1}{2}+\dfrac {1}{3}, \dfrac {1}{4}, 1+\dfrac {1}{4}, \dfrac {1}{2}+\dfrac {1}{4}, \dfrac {1}{3}+\dfrac {1}{4}, \dfrac {1}{5}, \ldots , \dfrac {1}{n}, 1+\dfrac {1}{n}, \dfrac {1}{2}+\dfrac {1}{n}, \dfrac {1}{n-1}+\dfrac {1}{n}, \dfrac {1}{n+1}, \ldots$.

Find the limit points of the sequence$\dfrac {1}{2}, \dfrac {1}{3}, \dfrac {2}{3}, \dfrac {1}{4}, \dfrac {2}{4}, \dfrac {3}{4}, \dfrac {1}{5}, \dfrac {2}{5}, \dfrac {3}{5}, \dfrac {4}{5}, \ldots$.

Find the limit points of the sequence$x_{n}=3\left(1-\dfrac {1}{n}\right)+2(-1)^{n}$.

Find the limit points of the sequence$x_{n}=\dfrac {1}{2}\left[(a+b)+(-1)^{n}(a-b)\right]$.

Give an example of a sequence having the given numbers

as its limit points.

Give an example of a sequence for which every term of the given sequence

is a limit point of it. What other limit points must the given sequence then necessarily have?

Give examples of sequences:

Prove that the sequences $x_{n}$ and $y_{n}=x_{n}\cdot \sqrt[n]{n} (n=1,2,\ldots )$ have the same limit points.

Prove that from a bounded sequence $x_{n} (n=1,2,\ldots )$ one can always select a convergent subsequence $x_{p_{n}} (n=1,2,\ldots )$.

Prove that if the sequence $x_{n} (n=1,2,\ldots )$ is unbounded, then there exists a subsequence $x_{p_{n}} (n=1,2,\ldots )$ such that

Suppose the sequence $x_{n} (n=1,2,\ldots )$ converges, while the sequence $y_{n} (n=1,2,\ldots )$ diverges. Can one then determine the convergence or divergence of the sequences

Suppose the sequences $x_{n}$ and $y_{n}$ both diverge $(n=1,2,\ldots )$. Can one conclude that the sequences

Suppose $\lim \limits_{n\to \infty }x_{n}=0$ and $y_{n} (n=1,2,\ldots )$ is an arbitrary sequence. Can one conclude that $\lim \limits_{n\to \infty }x_{n}y_{n}=0$? Give an appropriate example.

Suppose:

Does it follow that either $\lim \limits_{n\to \infty }x_{n}=0$ or $\lim \limits_{n\to \infty }y_{n}=0$?

Prove

Suppose $x_{n}\geqslant 0$ and $y_{n}\geqslant 0 (n=1,2,\ldots )$. Prove:

Prove: if $\lim \limits_{n\to \infty }x_{n}$ exists, then for any sequence $y_{n} (n=1,2,\ldots )$, we have

Prove: if for some nonnegative sequence $x_{n} (x_{n}\geqslant 0,n=1,2,\ldots )$, every sequence $y_{n} (n=1,2,\ldots )$ makes at least one of the following two identities hold:

Prove: if $x_{n}>0 (n=1,2,\ldots )$ and