Math Problems Vault

Problem 2.17.1

Find the limit function $f(x)$ of the sequence $\left\{ f_{n}(x)\right\}$ on the set $E$ .

?

(1)

$f_{n}(x) = x^{n}-3 x^{n+2}+2 x^{n+3}, E = [0 ; 1]$ ;

(2)

$f_{n}(x) = x^{4} \cos \frac{1}{n x}, E = (0 ;+\infty )$ ;

(3)

$f_{n}(x) = \frac{n x^{2}}{x+3 n+2}, E = [0 ;+\infty )$ ;

(4)

$f_{n}(x) = \sqrt{x^{2}+1 / \sqrt{n}}, E = R$

(5)

$f_{n}(x) = (x-1) \operatorname {arctg} x^{n}, E = (0 ;+\infty )$ ;

(6)

$f_{n}(x) = \sqrt[n]{1+x^{n}}, E = [0 ; 2]$ .

Problem 2.17.2

Find the limit function $f(x)$ of the sequence $\left\{ f_{n}(x)\right\}$ on the set $E$ .

?

(1)

$f_{n}(x) = n^{3} x^{2} e^{-n x}, E = [0 ;+\infty )$ ;

(2)

$f_{n}(x) = n\left(\sqrt{x^{2}+1 / n}-x\right), E = (0 ;+\infty )$ ;

(3)

$f_{n}(x) = n\left(x^{1 / n}-1\right), E = [1 ; 3]$ ;

(4)

$f_{n}(x) = n \operatorname {arcctg} n x^{2}, E = (0 ;+\infty )$ ;

(5)

$f_{n}(x) = n\left(x^{1 / n}-x^{1 / 2 n}\right), E = (0 ;+\infty )$ ;

(6)

$f_{n}(x) = \sqrt[n]{1+x^{n}+\left(x^{2} / 2\right)^{n}}, E = [0 ;+\infty )$ .

Problem 2.17.3

Prove that the sequence $\left\{ f_{n}(x)\right\}$ converges uniformly on the set $E$ .

?

(1)

$f_{n}(x) = e^{-n x^{2}}, E = [1 ;+\infty )$ ;

(2)

$f_{n}(x) = x^{2 n}, E = [0 ; \delta ], 0 < \delta < 1$ ;

(3)

$f_{n}(x) = \sqrt{n} \sin \frac{x}{n \sqrt{n}}, E = R$ ;

(4)

$f_{n}(x) = \frac{\sin n \sqrt{x}}{\ln (n+1)}, E = [0 ;+\infty )$ ;

(5)

$f_{n}(x) = \frac{n}{x^{2}+n^{2}} \operatorname {arctg} \sqrt{n x}, E = [0 ;+\infty )$ ;

(6)

$f_{n}(x) = \ln \left(1+\frac{\cos n x}{\sqrt{n+x}}\right), E = [0 ;+\infty )$ .

Problem 2.17.4

Prove that the sequence $\left\{ f_{n}(x)\right\}$ converges uniformly on the set $E$ .

?

(1)

$f_{n}(x) = \frac{n x^{2}}{n+x}, E = [1 ;+\infty )$ ;

(2)

$f_{n}(x) = \sqrt{x^{2}+1 / n}, E = R$

(3)

$f_{n}(x) = e^{-(x-n)^{2}}, E = [-1 ; 1]$ ;

(4)

$f_{n}(x) = \frac{n x}{1+n^{3} x^{2}}, E = [1 ;+\infty )$ ;

(5)

$f_{n}(x) = n^{3 / 4} x e^{-\sqrt{n x}}, E = [0 ;+\infty )$ ;

(6)

$f_{n}(x) = \frac{n^{2} x^{2}}{1+n^{2} x^{4}} \sin \frac{x^{2}}{\sqrt{n}}, E = [1 ;+\infty )$ .

Problem 2.17.5

Investigate the sequence $\left\{ f_{n}(x)\right\}$ for convergence and uniform convergence on the set $E$ .

?

(1)

$f_{n}(x) = \frac{\cos \sqrt{n x}}{\sqrt{n+2 x}}, E = [0 ;+\infty )$ ;

(2)

$f_{n}(x) = \sin \left(n e^{-n x}\right), E = [1 ;+\infty )$ ;

(3)

$f_{n}(x) = \frac{\ln n x}{n x^{2}}, E = [1 ;+\infty )$ ;

(4)

$f_{n}(x) = \frac{4 n \sqrt{n x}}{3+4 n^{2} x}, E = [\delta ;+\infty ), \delta > 0$ ;

(5)

$f_{n}(x) = x e^{-n x} \ln^{2} n, E = [0 ;+\infty )$ ;

(6)

$f_{n}(x) = n^{3 / 2}\left(1-\frac{\sqrt[4]{x}}{n}\right), E = [0 ;+\infty )$ .

Problem 2.17.6

Investigate the sequence $\left\{ f_{n}(x)\right\}$ for convergence and uniform convergence on the set $E$ .

?

(1)

$f_{n}(x) = \frac{x+x n^{3}+x^{3} n^{6}}{1+x^{2} n^{6}}, E = [1 ;+\infty )$ ;

(2)

$f_{n}(x) = n \int_{0}^{x} \sin \frac{\pi t^{n}}{2} d t, E = [0 ; \alpha ], 0 < \alpha < 1$ ;

(3)

$f_{n}(x) = \operatorname {tg}\left(\frac{n-1}{n} x\right), E = \left(0 ; \frac{\pi }{4}\right)$ ;

(4)

$f_{n}(x) = x^{n}-x^{n+2}, E = [0 ; 1]$ ;

(5)

$f_{n}(x) = \sqrt[4]{x^{4}+1 / n^{\alpha }}, \alpha > 0, E = R$ ;

(6)

$f_{n}(x) = \sin \frac{1+n x}{2 n}, E = R$ .

Problem 2.17.7

Investigate the sequence $\left\{ f_{n}(x)\right\}$ for convergence and uniform convergence on the set $E$ .

?

(1)

$f_{n}(x) = \sin \frac{x}{n^{\alpha }}, \alpha > 0, E = R$ ;

(2)

$f_{n}(x) = x^{n}+x^{2 n}-2 x^{3 n}, E = [0 ; 1]$ ;

(3)

$f_{n}(x) = \sin^{n} x, E = (0 ; \pi / 2)$ ;

(4)

$f_{n}(x) = \frac{n}{x} \ln \left(1+\frac{x}{n}\right), E = (0 ; 10)$ ;

(5)

$f_{n}(x) = n x(1-x)^{n}, E = [0 ; 1]$

(6)

$f_{n}(x) = n\left(x^{1 / n}-1\right), E = [1 ; a], 1 < a < +\infty$ .

Problem 2.17.8

Investigate the sequence $\left\{ f_{n}(x)\right\}$ for convergence and uniform convergence on the sets $E_{1}$ and $E_{2}$ .

?

(1)

$f_{n}(x) = \frac{x}{x+n}, E_{1} = [0 ; a], 0 < a < +\infty , E_{2} = [0 ;+\infty )$ ;

(2)

$f_{n}(x) = \frac{n x^{2}}{1+2 n+x}, E_{1} = [0 ; 1], E_{2} = [1 ;+\infty )$ ;

(3)

$f_{n}(x) = \frac{n x^{2}}{1+n^{2} x^{4}}, E_{1} = [0 ; 1], E_{2} = [1 ;+\infty )$ ;

(4)

$f_{n}(x) = \operatorname {arctg} \frac{n}{x}, E_{1} = (0 ; a], 0 < a < +\infty , E_{2} = (0 ;+\infty )$ ;

(5)

$f_{n}(x) = n\left(\frac{x}{\sqrt{n}}-\operatorname {arctg} \frac{x}{\sqrt{n}}\right), E_{1} = [0 ; 1], E_{2} = (1 ;+\infty )$ ;

(6)

$f_{n}(x) = \frac{(n+x)^{2}}{x^{2}+n^{2}-n x}, E_{1} = [0 ; 2), E_{2} = (2 ;+\infty )$ ;

(7)

$f_{n}(x) = \frac{x n}{n+x^{n}}, E_{1} = (0 ; 1), E_{2} = (1 ;+\infty )$ ;

(8)

$f_{n}(x) = \frac{\operatorname {arctg} n^{2} x}{x}, E_{1} = (0 ; 1), E_{2} = (1 ;+\infty )$ ;

(9)

$f_{n}(x) = \operatorname {arctg} \frac{1+n x}{x^{2}+n}, E_{1} = (0 ; 1), E_{2} = (1 ;+\infty )$ ;

(10)

$f_{n}(x) = x \operatorname {arcctg} \frac{x}{n^{2}}, E_{1} = (0 ; 1), E_{2} = (1 ;+\infty )$ ;

(11)

$f_{n}(x) = n \operatorname {arctg} \frac{\ln x}{n}, E_{1} = (e ; 5), E_{2} = (5 ;+\infty )$ .

Problem 2.17.9

Investigate the sequence $\left\{ f_{n}(x)\right\}$ for convergence and uniform convergence on the sets $E_{1}$ and $E_{2}$ .

?

(1)

$f_{n}(x) = \sqrt[n]{x^{2}+n x+1}, E_{1} = (0 ; 1), E_{2} = (1 ;+\infty )$ ;

(2)

$f_{n}(x) = e^{-x^{2}-n x}, E_{1} = (0 ; 1), E_{2} = (1 ;+\infty )$ ;

(3)

$f_{n}(x) = e^{-(x-n)^{2}}, E_{1} = [-2 ; 2], E_{2} = R$ ;

(4)

$f_{n}(x) = \frac{x}{n} \ln \frac{x}{n}, E_{1} = (0 ; 2), E_{2} = (0 ;+\infty )$ ;

(5)

$f_{n}(x) = \frac{n x^{2}}{n^{3}+x^{3}}, E_{1} = [0 ; 1], E_{2} = [0 ;+\infty )$ ;

(6)

$f_{n}(x) = \frac{1}{x^{3}} \cos \frac{x}{n}, E_{1} = (0 ; 1), E_{2} = (1 ;+\infty )$ ;

(7)

$f_{n}(x) = \sqrt{n}(\sqrt{1+n x}-\sqrt{n x}), E_{1} = (0 ; 1), E_{2} = (1 ;+\infty )$ ;

(8)

$f_{n}(x) = x^{2} \sqrt{1+1 /(n x)}, E_{1} = (0 ; 1), E_{2} = (1 ;+\infty )$ ;

(9)

$f_{n}(x) = \frac{x^{2} \sqrt{n}-x \sqrt{n+1}-1}{x \sqrt{n}-\sqrt{n+1}}, E_{1} = \left(0 ; \frac{1}{2}\right), E_{2} = \left(\frac{1}{2} ; 1\right)$ ;

(10)

$f_{n}(x) = \frac{\sqrt{1+n^{2} x}}{x n}, E_{1} = (0 ; 1), E_{2} = (1 ;+\infty )$ ;

(11)

$f_{n}(x) = n(1-\sqrt{1+x / n}), E_{1} = (0 ; 1), E_{2} = (1 ;+\infty )$ .

Problem 2.17.10

Investigate the sequence $\left\{ f_{n}(x)\right\}$ for convergence and uniform convergence on the sets $E_{1}$ and $E_{2}$ .

?

(1)

$f_{n}(x) = n \operatorname {arctg} \frac{1}{n x}, E_{1} = (0 ; 2), E_{2} = (2 ;+\infty )$ ;

(2)

$f_{n}(x) = \operatorname {arctg}\left(x / n-n^{2}\right), E_{1} = (0 ; 1), E_{2} = (1 ;+\infty )$ ;

(3)

$f_{n}(x) = \frac{1}{x^{2}} \sqrt{1+\frac{x}{n}}, E_{1} = (0 ; 1), E_{2} = (1 ;+\infty )$ ;

(4)

$f_{n}(x) = \operatorname {ch} e^{-n x}, E_{1} = (0 ; 1), E_{2} = (1 ;+\infty )$ ;

(5)

$f_{n}(x) = \frac{\ln n^{2} x}{n^{2} x}, E_{1} = (0 ; 1), E_{2} = (1 ;+\infty )$ ;

(6)

$f_{n}(x) = n x^{2} e^{-(n+1) x^{2}}, E_{1} = [0 ; 1], E_{2} = [\delta ; 1], 0 < \delta < 1$ ;

(7)

$f_{n}(x) = \operatorname {sh}\left(n e^{-n x}\right), E_{1} = (0 ; 1), E_{2} = (1 ;+\infty )$ ;

(8)

$f_{n}(x) = n^{2}(\operatorname {ch}(x / n)-1), E_{1} = (0 ; 1), E_{2} = (1 ;+\infty )$ ;

(9)

$f_{n}(x) = n \operatorname {sh} \frac{x}{n+x}, E_{1} = (0 ; 1), E_{2} = (1 ;+\infty )$ ;

(10)

$f_{n}(x) = \operatorname {ch}\left(\frac{\ln n}{n}+\frac{x}{n}\right), E_{1} = (0 ; 1), E_{2} = (1 ;+\infty )$ ;

(11)

$f_{n}(x) = n^{2}\left(n \operatorname {sh} \frac{1}{n x}-\frac{1}{x}\right), E_{1} = (0 ; 1), E_{2} = (1 ;+\infty )$ ;

(12)

$f_{n}(x) = \frac{\operatorname {th}(1+n x)}{x}, E_{1} = (0 ; 1), E_{2} = (1 ;+\infty )$ .

Problem 2.17.11

Investigate the sequence $\left\{ f_{n}(x)\right\}$ for convergence and uniform convergence on the sets $E_{1}$ and $E_{2}$ .

?

(1)

$f_{n}(x) = \frac{\sqrt{n}+\operatorname {arctg} n x}{\sqrt{n} x}, E_{1} = (0 ; 1), E_{2} = (1 ;+\infty )$ ;

(2)

$f_{n}(x) = \cos (1 /(1+\left|\ln n x\right|)), E_{1} = (0 ; 2), E_{2} = (2 ;+\infty )$ ;

(3)

$f_{n}(x) = \sqrt{n} \sin (x / \sqrt{n}), E_{1} = [0 ; \pi ], E_{2} = [\pi ;+\infty )$ ;

(4)

$f_{n}(x) = \cos (1 /(n x)), E_{1} = (0 ; \pi ), E_{2} = (\pi ;+\infty )$ ;

(5)

$f_{n}(x) = n^{2} x^{2} e^{-n x}, E_{1} = [0 ;+\infty ), E_{2} = [\delta ;+\infty ), \delta > 0$ ;

(6)

$f_{n}(x) = n \operatorname {arctg} x^{n}, E_{1} = [0 ; 1), E_{2} = (0 ; \delta ), 0 < \delta < 1$ ;

(7)

$f_{n}(x) = e^{x} \cos (1 /(n x)), E_{1} = (1 ; 2), E_{2} = (2 ;+\infty )$ ;

(8)

$f_{n}(x) = n \sin (1 /(n x+1)), E_{1} = (0 ; 1), E_{2} = (1 ;+\infty )$ ;

(9)

$f_{n}(x) = n \sin^{2}(x / \sqrt{n}), E_{1} = (0 ; 1), E_{2} = (1 ;+\infty )$ ;

(10)

$f_{n}(x) = \frac{1}{n \sqrt{x}} \ln \frac{e^{n x}}{n x}, E_{1} = (0 ; 4), E_{2} = (4 ;+\infty )$ .

Problem 2.17.12

Investigate the sequence $\left\{ f_{n}(x)\right\}$ for convergence and uniform convergence on the sets $E_{1}$ and $E_{2}$ .

?

(1)

$f_{n}(x) = \operatorname {arctg} \frac{n x}{1+n^{2} x^{2}}, E_{1} = (0 ;+\infty ), E_{2} = (\delta ;+\infty ), \delta > 0$ ;

(2)

$f_{n}(x) = e^{n(1 / x-1)}, E_{1} = (1 ;+\infty ), E_{2} = (\delta ;+\infty ), \delta > 1$ ;

(3)

$f_{n}(x) = \frac{1+\ln n x}{n x}, E_{1} = (0 ; 1), E_{2} = (1 ;+\infty )$ ;

(4)

$f_{n}(x) = n \operatorname {arcctg} n x^{2}, E_{1} = (0 ; 1), E_{2} = (1 ;+\infty )$ ;

(5)

$f_{n}(x) = \ln \left(x^{2}+1 / n\right), E_{1} = (0 ;+\infty ), E_{2} = (a ;+\infty ), a > 0$ ;

(6)

$f_{n}(x) = \sin \left(e^{-n x}+1 / \sqrt{n}\right), E_{1} = [\alpha ;+\infty ), \alpha > 0, E_{2} = (0 ;+\infty )$ ;

(7)

$f_{n}(x) = \sqrt{1+n e^{-n x}}, E_{1} = (0 ; 1), E_{2} = (1 ;+\infty )$ ;

(8)

$f_{n}(x) = e^{\left(x^{2}+n x+1\right) / n}, E_{1} = (0 ; 1), E_{2} = (1 ;+\infty )$ ;

(9)

$f_{n}(x) = 2^{\left(x^{2}-n x+1\right) / n}, E_{1} = (0 ; 1), E_{2} = (1 ;+\infty )$ ;

(10)

$f_{n}(x) = n\left(1-e^{x / n}\right), E_{1} = (0 ; 1), E_{2} = (1 ;+\infty )$ ;

(11)

$f_{n}(x) = e^{x^{2}-n^{2}}\left(x^{2}+n^{2}\right), E_{1} = (0 ; 1), E_{2} = (1 ;+\infty )$ ;

(12)

$f_{n}(x) = n^{2}\left(e^{-1 /(n x)}-1\right)+n / x, E_{1} = (0 ; 1), E_{2} = (1 ;+\infty )$ .

Problem 2.17.13

Investigate the sequence $\left\{ f_{n}(x)\right\}$ for convergence and uniform convergence on the sets $E_{1}$ and $E_{2}$ .

?

(1)

$f_{n}(x) = \sin \left(\frac{\pi }{2} e^{x / n}\right), E_{1} = (0 ; \alpha ), \alpha > 0, E_{2} = (0 ;+\infty )$ ;

(2)

$f_{n}(x) = \cos \left(\frac{\pi }{2} x^{n}\right), E_{1} = (0 ; \alpha ), 0 < \alpha < 1, E_{2} = (0 ; 1)$ ;

(3)

$f_{n}(x) = \operatorname {arctg} \frac{1-x^{n}}{1+x^{n}}, E_{1} = \left(0 ; \frac{1}{2}\right), E_{2} = \left(\frac{1}{2} ; 1\right)$ ;

(4)

$f_{n}(x) = \frac{x^{6}+x^{3} n^{2}+x^{2} n^{4}}{x^{4}+n^{4}}, \quad E_{1} = (0 ; a), 0 < a < +\infty , \quad E_{2} = (0 ;+\infty )$ ;

(5)

$f_{n}(x) = \frac{1+x^{2} n^{6}+\sqrt{x^{3} n^{6}}}{x+x^{3} n^{6}}, E_{1} = (a ;+\infty ), a > 0, E_{2} = (0 ;+\infty )$ ;

(6)

$f_{n}(x) = \operatorname {arctg} 2 n x-\operatorname {arctg} n x, E_{1} = (0 ; 1), E_{2} = (1 ;+\infty )$ ;

(7)

$f_{n}(x) = n^{2}\left(1-\cos \frac{1}{x n}\right), E_{1} = (0 ; 1), E_{2} = (1 ;+\infty )$ ;

(8)

$f_{n}(x) = \operatorname {arctg} \frac{n(x-1)}{x}, E_{1} = (0 ; 2), E_{2} = (2 ;+\infty )$ ;

(9)

$f_{n}(x) = \ln \left(2+\frac{n x}{1+n^{2} x^{2}}\right), E_{1} = \left(0 ; \frac{1}{2}\right), E_{2} = \left(\frac{1}{2} ;+\infty \right)$ .

Problem 2.17.14

Investigate the sequence $\left\{ f_{n}(x)\right\}$ for convergence and uniform convergence on the sets $E_{1}$ and $E_{2}$ .

?

(1)

$f_{n}(x) = \frac{n^{2} x^{2}+n x+1}{n^{2} x^{2}+2}, E_{1} = (0 ; 1), E_{2} = (1 ;+\infty )$ ;

(2)

$f_{n}(x) = \ln \frac{n^{2} x+n \sqrt{x}+1}{n^{2} x-n \sqrt{x}+1}, E_{1} = (0 ; 1), E_{2} = (1 ;+\infty )$ ;

(3)

$f_{n}(x) = \operatorname {arctg} \frac{n x-1}{n x+1}, E_{1} = (0 ; 1), E_{2} = (1 ;+\infty )$ ;

(4)

$f_{n}(x) = \ln \left(1+\sin \frac{x \sqrt{n}}{x^{2}+n}\right), E_{1} = (0 ; 1), E_{2} = (1 ;+\infty )$ ;

(5)

$f_{n}(x) = n^{2}\left(e^{x^{n}}-\cos x^{n}\right), E_{1} = [0 ; 1), E_{2} = [0 ; \delta ], 0 < \delta < 1$ ;

(6)

$f_{n}(x) = n \ln \left(1+\frac{1}{n x}\right), E_{1} = (0 ; 2), E_{2} = (2 ;+\infty )$ ;

(7)

$f_{n}(x) = \arcsin \frac{1+n x}{1+2 n x}, E_{1} = (0 ; 1), E_{2} = (1 ;+\infty )$ ;

(8)

$f_{n}(x) = n \operatorname {arctg} \frac{\operatorname {tg} x}{n}, E_{1} = \left(0 ; \frac{\pi }{4}\right), E_{2} = \left(\frac{\pi }{4} ; \frac{\pi }{2}\right)$ ;

(9)

$f_{n}(x) = \frac{x \ln \left(1+e^{n / x}\right)}{n}, E_{1} = (0 ; 1), E_{2} = (1 ;+\infty )$ ;

(10)

$f_{n}(x) = \frac{\ln \left(1+n^{x}\right)}{x+\ln n}, E_{1} = (0 ; 1), E_{2} = (1 ;+\infty )$ ;

(11)

$f_{n}(x) = n \operatorname {sh} \frac{1}{x \ln n+n}, E_{1} = (0 ; 2), E_{2} = (2 ;+\infty )$ .

Problem 2.17.15

Investigate the sequence $\left\{ f_{n}(x)\right\}$ for convergence and uniform convergence on the sets $E_{1}$ and $E_{2}$ .

?

(1)

$f_{n}(x) = \sin \left(\sqrt{1+(n x)^{2}}-n x\right), E_{1} = (0 ; 1), E_{2} = (1 ;+\infty )$ ;

(2)

$f_{n}(x) = \sqrt{(n x)^{3}+n x^{2}+1}-\sqrt{(n x)^{3}+1}, \quad E_{1} = (0 ; 1), \quad E_{2} = (1 ;+\infty )$ ;

(3)

$f_{n}(x) = \sin \frac{x}{e^{-n}+e^{n} x^{2}}, E_{1} = (0 ; 1), E_{2} = (1 ;+\infty )$ ; $a > 0$ ;

(4)

$f_{n}(x) = 2 \ln \left(e^{x}+n\right)-\ln \left(e^{2 x}+n^{2}\right), E_{1} = [0 ;+\infty ), E_{2} = [0 ; a]$ ,

(5)

$f_{n}(x) = n^{8}\left(\operatorname {ch} x^{n}-1+\frac{1}{x^{n}}\right), E_{1} = [0 ; 1), E_{2} = [0 ; a], 0 < a < 1$ ;

(6)

$f_{n}(x) = \ln \left(\frac{6 e^{2 x}+3 n^{2}+n e^{x}}{n^{2}+2 e^{2 x}}\right), E_{1} = [0 ; a], a > 0, E_{2} = [0 ;+\infty )$ ;

(7)

$f_{n}(x) = n \ln \left(1+\frac{1}{n x^{2}}\right), E_{1} = (0 ; 1), E_{2} = (1 ;+\infty )$ ;

(8)

$f_{n}(x) = \sqrt{n} \ln (1+\sqrt{x / n}), E_{1} = (0 ; 1], E_{2} = (1 ;+\infty )$ ;

(9)

$f_{n}(x) = 1-\frac{n}{n+x \ln n}, E_{1} = (0 ; 1), E_{2} = (1 ;+\infty )$ ;

(10)

$f_{n}(x) = n \ln \left(1+\operatorname {arctg} \frac{x^{2}}{n}\right), E_{1} = (0 ; 1), E_{2} = (1 ;+\infty )$ ;

(11)

$f_{n}(x) = \ln \left(x+\frac{\ln^{2} x}{n x+1}\right), E_{1} = (0 ; 1), E_{2} = (1 ;+\infty )$ .

Problem 2.17.16

Investigate the sequence $\left\{ f_{n}(x)\right\}$ for convergence and uniform convergence on the sets $E_{1}$ and $E_{2}$ .

?

(1)

$f_{n}(x) = \arcsin \frac{x^{n}}{1+x^{n}}, E_{1} = [0 ; a], 0 < a < 1, E_{2} = [0 ; 1)$ ;

(2)

$f_{n}(x) = \ln \frac{\sqrt{e^{x}+n^{2}}}{1+n e^{-x / 2}}, E_{1} = [0 ; a], a > 0, E_{2} = [0 ;+\infty )$ ;

(3)

$f_{n}(x) = \sqrt{n}\left((1+1 / n)^{x}-(1-1 / n)^{x}\right), E_{1} = (0 ; 1), E_{2} = (1 ;+\infty )$ ;

(4)

$f_{n}(x) = (1-x / n)^{n}, E_{1} = [-a ; a], a > 0, E_{2} = R$ ;

(5)

$f_{n}(x) = n\left(x^{1 / n}-x^{1 /(2 n)}\right), E_{1} = (1 / 2 ; 1), E_{2} = (1 ;+\infty )$ ;

(6)

$f_{n}(x) = \sqrt{1+x^{n}}, E_{1} = [0 ; 2], E_{2} = [2 ;+\infty )$ ;

(7)

$f_{n}(x) = \ln (\sin x+1 / n), E_{1} = (0 ; \pi / 6), E_{2} = (\pi / 6 ; \pi / 2)$ ;

(8)

$f_{n}(x) = \frac{\ln \left(n+\sqrt{x^{2}+n^{2}}\right)}{\ln n}, E_{1} = (0 ; 1), E_{2} = (1 ;+\infty )$ ;

(9)

$f_{n}(x) = (n+\ln x) \operatorname {arctg} \frac{1}{n x}, E_{1} = (0 ; 1), E_{2} = (1 ;+\infty )$ ;

(10)

$f_{n}(x) = n^{2} \ln (1+x / n)-x \sqrt{n^{2}-1}, E_{1} = (0 ; 1), E_{2} = (1 ;+\infty )$ ;

(11)

$f_{n}(x) = \frac{x \ln (1+n)-x^{2} \ln n-1}{\ln (1+n)-x \ln n}, E_{1} = \left(0 ; \frac{1}{2}\right), E_{2} = \left(\frac{1}{2} ; 1\right)$ .

Problem 2.17.17

Investigate the sequence $\left\{ f_{n}(x)\right\}$ for convergence and uniform convergence on the sets $E_{1} = (0 ; 1)$ and $E_{2} = (1 ;+\infty )$ :

?

(1)

$f_{n}(x) = n \ln \left(1+\frac{2^{x}}{n}\right)$ ;

(2)

$f_{n}(x) = \frac{1+\ln n x}{n x}$ ;

(3)

$f_{n}(x) = \frac{1}{x+1} \operatorname {ch} \frac{x+e^{-n x}}{n}$ ;

(4)

$f_{n}(x) = n^{2}\left(\left(x+\frac{1}{n^{2}}\right)^{4}-x^{4}\right)$ ;

(5)

$f_{n}(x) = \sin \left(\pi x 2^{-x / n}\right)$ ;

(6)

$f_{n}(x) = n^{2} \ln \left(1+x / n^{2}\right)$ ;

(7)

$f_{n}(x) = \cos \left(n e^{-n x}\right)$ ;

(8)

$f_{n}(x) = \frac{1}{x^{2}} \sqrt{1+\frac{x}{n}}$

(9)

$f_{n}(x) = \frac{1}{x^{3}} \cos \frac{x}{n}$

(10)

$f_{n}(x) = n^{2}\left(x-n \sin \frac{x}{n}\right)$ .

Problem 2.17.18

Investigate the sequence $\left\{ \frac{x^{n}}{1+x^{n}}\right\}$ for convergence and uniform convergence on the sets $E_{1}, E_{2}, E_{3}, E_{4}, E_{5}$ , where $E_{1} = [0 ; \delta ], \quad 0 < \delta < 1, E_{2} = [0 ; 1), E_{3} = [1-\alpha ; 1+\alpha ], \quad 0 < \alpha < 1$ , $E_{4} = [1+\alpha ;+\infty ), E_{5} = (1 ;+\infty )$ .

?

Problem 2.17.19

Find all values of $\alpha$ for which the sequence $\left\{ f_{n}(x)\right\}$ :

converges on the set $E$ ;
converges uniformly on the set $E$ .

Indicate the limit function of this sequence:

?

(1)

$f_{n}(x) = \frac{\ln n x}{n^{\alpha } x}, \quad E = (0 ; 1)$ ;

(2)

$f_{n}(x) = \frac{n^{\alpha } x}{x^{2}+n^{2}}, \quad E = (0 ;+\infty )$ ;

(3)

$f_{n}(x) = \frac{n^{\alpha } x}{x^{2}+n}, \quad E = R$

(4)

$f_{n}(x) = n^{\alpha } x e^{-n x^{2}}, \quad E = [0 ;+\infty )$ ;

(5)

$f_{n}(x) = \frac{x^{2} e^{-(n+2) x^{2}}}{n^{\alpha }}, E = [0 ;+\infty )$ ;

(6)

$f_{n}(x) = n^{\alpha } \operatorname {arctg}\left(1 / x^{n}\right), \quad E = (0 ; 1)$ ;

(7)

$f_{n}(x) = n^{\alpha } \ln \left(1+\frac{1}{n x}\right), \quad E = (0 ;+\infty )$ ;

(8)

$f_{n}(x) = x \operatorname {arctg} n^{\alpha } x, \quad E = [0 ;+\infty )$ ;

(9)

$f_{n}(x) = n^{\alpha }(\sqrt{x+1 / n}-\sqrt{x}), \quad E = (0 ;+\infty )$ .

Problem 2.17.20

Investigate the sequence $\left\{ f_{n}(x)\right\}$ for convergence and uniform convergence on the set $E = [0 ; 1]$ , where

f_{n}(x) = \begin{cases} n^{2} x, & \text{ if } 0 \leqslant x \leqslant 1 / n \\ n^{2}(2 / n-x), & \text{ if } 1 / n < x < 2 / n \\ 0, & \text{ if } x \geqslant 2 / n \end{cases}

?

Problem 2.17.21

Prove that if the sequences $\left\{ f_{n}(x)\right\}$ and $\left\{ g_{n}(x)\right\}$ converge uniformly on the set $E$ to $f(x)$ and $g(x)$ , respectively, then for any $\alpha$ and $\beta (\alpha \in R, \beta \in R)$ the sequence $\left\{ \alpha f_{n}(x)+\beta g_{n}(x)\right\}$ converges uniformly to $\alpha f(x)+\beta g(x)$ .

?

Problem 2.17.22

Prove that if the sequence $\left\{ f_{n}(x)\right\}$ converges uniformly on the set $E$ to the function $f(x)$ , and the function $g(x)$ is bounded on this set, then the sequence $\left\{ g(x) f_{n}(x)\right\}$ converges uniformly to $g(x) f(x)$ .

?

Problem 2.17.23

Prove that if $f(x)$ is an arbitrary function defined on the segment $[a ; b]$ , then the sequence $\left\{ f_{n}(x)\right\}$ , where $f_{n}(x) = \frac{[n f(x)]}{n}([a]$ is the integer part of $a)$ , converges uniformly to $f(x)$ on the segment $[a ; b]$ .

?

Problem 2.17.24

Prove that if the function $f(x)$ has a continuous derivative on the interval $(a ; b)$ , then the sequence $\left\{ f_{n}(x)\right\}$ , where $f_{n}(x) = n(f(x+1 / n)-f(x))$ , converges uniformly to $f^{\prime }(x)$ on the segment $\left[a_{1} ; b_{1}\right], a < a_{1} < b_{1} < b$ .

?

Problem 2.17.25

Prove that if a sequence of polynomials of degree at most $n$ converges uniformly on the interval $(a ; b)$ , then the limit function of this sequence is a polynomial of degree at most $n$ .

?

Problem 2.17.26

Prove that if the function $f(x)$ is continuous on $R$ , then the sequence $\left\{ f_{n}(x)\right\}$ , where $f_{n}(x) = \sum_{k = 0}^{n-1} \frac{1}{n} f\left(x+\frac{k}{n}\right)$ , converges uniformly on any finite segment $[a ; b]$ .

?