Computing Limits

$\lim_{n \rightarrow \infty } \sum_{0 \leqslant k \leqslant 2 n} 2^{-k} \cos \sqrt{\frac{k}{n}}$;

$\lim_{n \rightarrow \infty } \sum_{0 \leqslant k \leqslant 2 n} 2^{-n k /(n+k)}$.

Using integration by parts, prove that

Using the inequalities $W_{2 k+1}<W_{2 k}<W_{2 k-1}$, obtain Wallis's formula

Find the limit $\lim S_{n}$.

What is the nature of monotonicity of the sequence $\left\{ S_{n}\right\}$?

Is it true that $S_{n} \leqslant 2$ for every $n \in \mathbb {N}$?

$\lim \sum_{0 \leqslant k \leqslant 2 n} \frac{k}{k^{2}+n^{2}}$;

$\lim \frac{1}{n^{2}} \sum_{0 \leqslant k \leqslant n} \sqrt{k(n-k)}$;

$\lim \sum_{0 \leqslant k \leqslant 2 n} \frac{k}{k+n^{2}}$;

$\lim \sum_{n \leqslant k \leqslant 2 n} \sin \frac{\pi }{k}$;

$\lim \sum_{1 \leqslant k<n} \frac{1}{\sqrt{k(n-k)}}$;

$\lim \sum_{1<k<n} \frac{1}{(n-k) \ln k}$.

$x_{n}=\frac{n^{3 n / 2}}{n!} \prod_{1 \leqslant k \leqslant n} \sin \frac{k}{n^{3 / 2}}$;

$x_{n}=\frac{2^{n}}{\sqrt{n}} \prod_{1 \leqslant k \leqslant n} \sin \frac{\pi k}{2 n}$.

$\lim \sin (2 \pi e n!)$;

$\lim n \sin (2 \pi e n!)$;

$\lim n^{2} \sin (2 \pi e n!)$;

$\lim n^{p} \sin \left(\pi (\sqrt{2}+1)^{n}\right)$.

$x_{n} \rightarrow l_{a}=\frac{1}{2}(1+\sqrt{1+4 a})$;

$l_{a}-x_{n} \sim \frac{C_{a}}{\left(2 l_{a}\right)^{n}}$ for some $C_{a}>0$. What is $C_{2}$ equal to?

Let $p>1$ and let $\left\{ c_{n}\right\}$ be a nonnegative sequence. Set $\alpha_{n}=\sqrt[p]{c_{1}+\sqrt[p]{c_{2}+\ldots +\sqrt[p]{c_{n}}}}$. Prove that convergence of the sequence $\left\{ \alpha_{n}\right\}$ is equivalent to boundedness above of the sequence $\left\{ \frac{1}{p^{n}} \ln c_{n}\right\}$.

For which nonnegative sequences $\left\{ c_{n}\right\}$ does the sequence $\left\{ \beta_{n}\right\}$ converge, where $\beta_{n}=c_{1}+\sqrt{c_{2}+\sqrt[3]{c_{3}+\ldots +\sqrt[n]{c_{n}}}}$?

Let $c>0$, and let $\left\{ p_{n}\right\}$ be a sequence of numbers greater than one. Set $\gamma_{n}=\sqrt[p_{1}]{c+\sqrt[p_{2}]{c+\ldots +\sqrt[p_{n}]{c}}}$. Prove that convergence of the sequence $\left\{ \gamma_{n}\right\}$ is equivalent to boundedness above of the sequence $\left\{ \frac{\ln n}{p_{1} \cdot p_{2} \cdots p_{n}}\right\}$.

$3=\sqrt{1+2 \sqrt{1+3 \sqrt{1+4 \sqrt{1+\ldots }}}}$;

$3=\sqrt{5+\sqrt{6+2 \sqrt{7+3 \sqrt{8+4 \sqrt{9+\ldots }}}}}$;

$\cos x=\sqrt{\sin^{2} \frac{x}{2}+\cos \frac{3 x}{2} \sqrt{\sin^{2} \frac{x}{4}+\cos \frac{3 x}{4} \sqrt{\sin^{2} \frac{x}{8}+\ldots }}}$ for $x \in \left[-\frac{\pi }{2}, \frac{\pi }{2}\right]$;

$9=\sqrt{3 u_{2} u_{4}+u_{4} \sqrt{3 u_{4} u_{6}+u_{6} \sqrt{3 u_{6} u_{8}+\ldots }}}$, where $\left\{ u_{n}\right\}$ is the sequence of Fibonacci numbers, i.e. $u_{1}=u_{2}=1$ and $u_{n+1}=u_{n}+u_{n-1}$ for $n \geqslant 2$;

$2 x^{2}=\sqrt{2 x T_{0} T_{1}+T_{1} \sqrt{2 x T_{1} T_{2}+T_{2} \sqrt{2 x T_{2} T_{3}+\ldots }}}$ for $x \geqslant 1$; here $\left\{ T_{n}\right\}$ is the sequence of values of the Chebyshev polynomials at the point $x$ (see problem I.2.20);

$2=\sqrt[3]{2+1 \cdot 2 \sqrt[3]{3+2 \cdot 3 \sqrt[3]{4+3 \cdot 4 \sqrt[3]{5+\ldots }}}}$.

Prove that $\overline{\lim }\left(\frac{x_{1}+x_{n+1}}{x_{n}}\right)^{n} \geqslant e$ for any positive sequence $\left\{ x_{n}\right\}$.

For any number $x \geqslant e$ exhibit a positive sequence $\left\{ x_{n}\right\}$ such that $\left(\frac{x_{1}+x_{n+1}}{x_{n}}\right)^{n} \rightarrow x$.

Let a sequence of real numbers $\left\{ x_{n}\right\}$ be such that $x_{n+m} \leqslant x_{n}+x_{m}$ for all $n, m \in \mathbb {N}$. Prove that $\frac{x_{n}}{n} \rightarrow \inf \frac{x_{m}}{m}$.

Let a sequence of complex numbers $\left\{ x_{n}\right\}$ be such that $\left|x_{n+m}-x_{n}-x_{m}\right| \leqslant A$ for all $n, m \in \mathbb {N}$. Prove that a finite limit $C=\lim \frac{x_{n}}{n}$ exists, and $\left|\frac{x_{n}}{n}-C\right| \leqslant \frac{A}{n}$.

Describe all sequences of complex numbers $\left\{ x_{n}\right\}$ satisfying the inequality $\left|x_{n+m}-x_{n}-x_{m}\right| \leqslant \frac{1}{n+m}$ for all $n, m \in \mathbb {N}$.

Let a sequence of real numbers $\left\{ x_{n}\right\}$ be such that $x_{n+1}+x_{n} \rightarrow 0$. Prove that the set of its partial limits is either infinite or contains at most two points.

Prove that if the sequence $\left\{ x_{n}+ +x_{n+1}+\ldots +x_{n+100}\right\}$ converges, then the set of partial limits of the sequence $\left\{ x_{n}\right\}$ can be represented as the union $\Delta_{1} \cup \ldots \cup \Delta_{101}$, where $\Delta_{j}$ are closed intervals (possibly degenerate or empty) in $\overline{\mathbb {R}}$.