Math Problems Vault

Problem 2.18.1

Find the domain of convergence and the domain of absolute convergence of the series of functions.

?

(1)

$\sum_{n = 1}^{\infty } \frac{1}{x^{n}}$ ;

(2)

$\sum_{n = 1}^{\infty } \frac{1}{n} \sin \frac{\pi x}{n}$

(3)

$\sum_{n = 1}^{\infty } e^{-n x}$

(4)

$\sum_{n = 1}^{\infty } \frac{\cos \pi n x}{n \ln^{2}(n+1)}$ ;

(5)

$\sum_{n = 1}^{\infty } \frac{1}{n(x+2)^{n}}$ ;

(6)

$\sum_{n = 1}^{\infty } \frac{\ln^{n} x}{n^{2}}$ .

Problem 2.18.2

Find the domain of convergence and the domain of absolute convergence of the series of functions.

?

(1)

$\sum_{n = 1}^{\infty }\left(5-x^{2}\right)^{n}$ ;

(2)

$\sum_{n = 1}^{\infty } \frac{1}{n^{x}}$ ;

(3)

$\sum_{n = 1}^{\infty } n^{-\ln x^{2}}$ ;

(4)

$\sum_{n = 1}^{\infty } n^{2} e^{n x^{2}}$ ;

(5)

$\sum_{n = 1}^{\infty } \frac{n}{\ln^{n}(x+2)}$ ;

(6)

$\sum_{n = 1}^{\infty } \frac{\operatorname {tg}^{n} x}{n^{2}+4}$ .

Problem 2.18.3

Find the domain of convergence and the domain of absolute convergence of the series of functions.

?

(1)

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

(2)

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

(3)

$\sum_{n = 1}^{\infty } \sqrt{\frac{n+2}{n}}\left(\frac{x^{2}-5 x+6}{x^{2}+5 x+6}\right)^{n}$ ;

(4)

$\sum_{n = 1}^{\infty } \frac{1}{\sqrt{n}}\left(\frac{2 x}{x^{2}+1}\right)^{n}$ ;

(5)

$\sum_{n = 1}^{\infty } n^{2}\left(\frac{2 x-3}{4}\right)^{n}$ ;

(6)

$\sum_{n = 1}^{\infty } \frac{2^{n} \sin^{n} x}{n(n+1)}$ .

Problem 2.18.4

Find the domain of convergence and the domain of absolute convergence of the series of functions.

?

(1)

$\sum_{n = 1}^{\infty } \frac{\cos n x}{\sqrt[3]{n}}$

(2)

$\sum_{n = 1}^{\infty } \frac{(-1)^{n}}{x^{2}+\sqrt{n}}$ ;

(3)

$\sum_{n = 1}^{\infty } e^{-n x} \sin n x$ ;

(4)

$\sum_{n = 1}^{\infty } \frac{(-1)^{n}}{\sqrt{n}} e^{-n \sin x}$ ;

(5)

$\sum_{n = 1}^{\infty }\left(\frac{n}{3} \ln \left(1+\frac{x}{n}\right)\right)^{n}$ ;

(6)

$\sum_{n = 1}^{\infty }\left(\frac{n}{2}\right)^{n}\left(e^{x / n}-1\right)^{n}$ .

Problem 2.18.5

Find the domain of convergence and the domain of absolute convergence of the series of functions.

?

(1)

$\sum_{n = 1}^{\infty } \sin \pi n x$ ;

(2)

$\sum_{n = 1}^{\infty }\left(1+\frac{x}{n}\right)^{n} n^{-x}$ ;

(3)

$\sum_{n = 1}^{\infty } \frac{1}{x^{2 n}+1}$ ;

(4)

$\sum_{n = 1}^{\infty }\left(\frac{x^{2}}{n}+x\right)^{n}$

(5)

$\sum_{n = 1}^{\infty }\left(\cos \frac{\pi x}{n}\right)^{n^{3}}$ ;

(6)

$\sum_{n = 1}^{\infty } \frac{x^{n}}{1-x^{n}}$ .

Problem 2.18.6

Find the domain of convergence and the domain of absolute convergence of the series of functions.

?

(1)

$\sum_{n = 1}^{\infty } \sin \left(\pi \sqrt{n^{2}+x^{2}}\right)$ ;

(2)

$\sum_{n = 1}^{\infty } \frac{\sin n \sin n^{2}}{n+\sqrt{x}}$

(3)

$\sum_{n = 1}^{\infty } \underbrace{\sin \sin \ldots \sin }_{n \text{ times }} x$ ;

(4)

$\sum_{n = 1}^{\infty } \frac{x(x-1) \ldots (x-(n-1))}{n !}$ ;

(5)

$\sum_{n = 1}^{\infty } \frac{n !}{\left(x^{2}+1\right)\left(x^{2}+1\right) \ldots \left(x^{2}+n\right)}$ ;

(6)

$\sum_{n = 1}^{\infty } \frac{x^{n}}{(1+x)\left(1+x^{2}\right) \ldots \left(1+x^{n}\right)}$ .

Problem 2.18.7

Starting from the definition of uniform convergence, prove the uniform convergence of the series of functions on the given interval:

?

(1)

$\sum_{n = 1}^{\infty } x^{n}, \quad -q \leqslant x \leqslant q, \quad 0 < q < 1$

(2)

$\sum_{n = 1}^{\infty }\left(\frac{x^{n-1}}{n}-\frac{x^{n}}{n+1}\right), \quad -1 \leqslant x \leqslant 1$ ;

(3)

$\sum_{n = 1}^{\infty } \frac{1}{(x+n)(x+n+1)}, \quad 0 \leqslant x < +\infty$ ;

(4)

$\sum_{n = 1}^{\infty }\left(\frac{\sin n x}{\sqrt{n}}-\frac{\sin (n+1) x}{\sqrt{n+1}}\right),-\infty < x < +\infty$ ;

(5)

$\sum_{n = 1}^{\infty } \frac{1}{x^{2}+n \sqrt{n}}, \quad -\infty < x < +\infty$ ;

(6)

$\sum_{n = 1}^{\infty } \frac{x}{3^{n} \sqrt{1+n x^{2}}}, \quad 0 \leqslant x \leqslant 2$ .

Problem 2.18.8

Using the Weierstrass test, prove the uniform convergence of the series of functions on the given interval.

?

(1)

$\sum_{n = 1}^{\infty } \frac{x^{n}}{n^{2}}, \quad -1 \leqslant x \leqslant 1$ ;

(2)

$\sum_{n = 1}^{\infty } \frac{1}{(n+x)^{2}}, \quad 0 \leqslant x < +\infty$ ;

(3)

$\sum_{n = 1}^{\infty } \frac{x^{2}}{1+n^{3 / 2} x^{2}}, \quad -\infty < x < +\infty$ ;

(4)

$\sum_{n = 1}^{\infty } \frac{\sin^{2} 2 n x}{\sqrt[3]{n^{4}+x^{2}}}, \quad -\infty < x < +\infty$ ;

(5)

$\sum_{n = 1}^{\infty } \frac{\operatorname {arctg} n x}{x^{4}+n \sqrt[3]{n}}, \quad -\infty < x < +\infty$ ;

(6)

$\sum_{n = 1}^{\infty } 2^{-n} \cos \pi n x, \quad -\infty < x < +\infty$ .

Problem 2.18.9

Using the Weierstrass test, prove the uniform convergence of the series of functions on the given interval.

?

(1)

$\sum_{n = 1}^{\infty } e^{-\sqrt{n x}}, \quad 1 \leqslant x < +\infty$ ;

(2)

$\sum_{n = 1}^{\infty } \frac{(3 x)^{n}}{n \sqrt{n+x}}, \quad 0 \leqslant x \leqslant \frac{1}{3}$ ;

(3)

$\sum_{n = 1}^{\infty } \frac{(x-1)^{n}}{(3 n+1) 3^{n}}, \quad -1 \leqslant x \leqslant 3$

(4)

$\sum_{n = 1}^{\infty } \frac{(x+2)^{n} \cos^{2} n x}{\sqrt{n^{3}+x^{4}}}, \quad -3 \leqslant x \leqslant -1$ ;

(5)

$\sum_{n = 1}^{\infty } \frac{(n+2)^{3}(2 x)^{2 n}}{x^{2}+3 n+4}, \quad -\frac{1}{4} \leqslant x \leqslant \frac{1}{4}$

(6)

$\sum_{n = 1}^{\infty } \frac{n \operatorname {arctg} 2 n^{2} x}{\sqrt[3]{n^{7}+n+x}}, 0 \leqslant x < +\infty$ .

Problem 2.18.10

Using the Weierstrass test, prove the uniform convergence of the series of functions on the given interval.

?

(1)

$\sum_{n = 1}^{\infty } \frac{\cos n x \sin (1 /(n x))}{4+\ln^{2} n x}, 2 \leqslant x < +\infty$ ;

(2)

$\sum_{n = 1}^{\infty } n^{3} e^{-n^{2} x}, \delta < x < +\infty , \quad \delta > 0$

(3)

$\sum_{n = 1}^{\infty } \frac{R^{n} x^{n}}{1+x^{2 n}}, R > 0, \quad -\alpha < x < \alpha , \alpha < \min \left(1 ; \frac{1}{R}\right)$ ;

(4)

$\sum_{n = 1}^{\infty } n^{-x} \ln^{2} n, \quad \delta < x < +\infty , \quad \delta > 1$ ;

(5)

$\sum_{n = 1}^{\infty } \frac{x}{4+n^{3} x^{2}}, \quad 0 \leqslant x < +\infty$ ;

(6)

$\sum_{n = 1}^{\infty } \frac{n x}{1+n^{5} x^{2}}, \quad -\infty < x < +\infty$

Problem 2.18.11

Using the Weierstrass test, prove the uniform convergence of the series of functions on the given interval.

?

(1)

$\sum_{n = 1}^{\infty } \operatorname {arctg} \frac{x}{x^{2}+n^{3}}, \quad -\infty < x < +\infty$ ;

(2)

$\sum_{n = 1}^{\infty } \frac{n \ln (1+n x)}{x^{n}}, \quad 1+\alpha \leqslant x < +\infty , \quad \alpha > 0$

(3)

$\sum_{n = 1}^{\infty } \frac{x}{1+n^{2} x^{4}} \operatorname {arctg} \frac{x}{n}, \quad -\infty < x < +\infty$ ;

(4)

$\sum_{n = 1}^{\infty }\left(\operatorname {arctg} \frac{\sqrt{x}}{x+n}\right)^{3}, \quad 0 \leqslant x < +\infty$ ;

(5)

$\sum_{n = 1}^{\infty } \frac{n^{2}}{n+1} \frac{x^{2} \sin x}{1+n^{5} x^{4}}, \quad -\infty < x < +\infty$ ;

(6)

$\sum_{n = 1}^{\infty } \sin^{2} \frac{\sqrt{x}}{1+n^{2} x}, \quad 0 \leqslant x < +\infty$ .

Problem 2.18.12

Using the Weierstrass test, prove the uniform convergence of the series of functions on the given interval.

?

(1)

$\sum_{n = 1}^{\infty } \sin \frac{1}{n x} \ln \left(1+\frac{x}{\sqrt{n}}\right), \quad 0 < x < +\infty$ ;

(2)

$\sum_{n = 1}^{\infty } n x e^{-x^{2} n^{5}}, \quad -\infty < x < +\infty$ ;

(3)

$\sum_{n = 1}^{\infty } \frac{\cos n x \sin (x / n)}{x^{2}+\ln^{3}(n+1)}, \quad -\infty < x < +\infty$ ;

(4)

$\sum_{n = 1}^{\infty } e^{-n^{6} x^{2}} \sin n x, \quad -\infty < x < +\infty$ ;

(5)

$\sum_{n = 1}^{\infty } \frac{x}{1+n^{4} x^{3}}, \quad 0 \leqslant x < +\infty$ ;

(6)

$\sum_{n = 1}^{\infty } \frac{\sin n x}{1+n^{8} x^{3}}, \quad 0 \leqslant x < +\infty$ .

Problem 2.18.13

Investigate the series of functions for convergence and uniform convergence on the given interval.

?

(1)

$\sum_{n = 1}^{\infty } \frac{x^{2} \sin (n \sqrt{x})}{1+n^{3} x^{4}}, 0 \leqslant x < +\infty$ ;

(2)

$\sum_{n = 1}^{\infty } \frac{n^{2} x}{\left(n^{2}+1\right)\left(1+n^{4} x^{2}\right) \operatorname {arctg}(1+x)}, 0 < x < +\infty$ ;

(3)

$\sum_{n = 1}^{\infty } \frac{\sqrt{x} \cos n x}{n\left(2 n x^{2}+1\right)}, \quad 0 < x < +\infty$ ;

(4)

$\sum_{n = 1}^{\infty }\left(\operatorname {arctg} \frac{x}{x^{2}+n^{2}}\right)^{2}, \quad 0 \leqslant x < +\infty$ ;

(5)

$\sum_{n = 1}^{\infty } \frac{x}{n}\left(1+\frac{1}{n}-x\right)^{n}, \quad 0 \leqslant x \leqslant 1$ ;

(6)

$\sum_{n = 1}^{\infty } \frac{x \sin (x+n)}{n^{2} x^{2}+n+1}, 0 \leqslant x < +\infty$ .

Problem 2.18.14

Investigate the series of functions for convergence and uniform convergence on the given interval.

?

(1)

$\sum_{n = 1}^{\infty } \frac{x e^{-x^{2} n}}{\sqrt{n \ln^{3}(n+1)}}, \quad -\infty < x < +\infty$ ;

(2)

$\sum_{n = 1}^{\infty } \frac{\sin \left(x^{2} / n\right)}{x^{2} \sqrt{n}+1}, \quad -\infty < x < +\infty$ ;

(3)

$\sum_{n = 1}^{\infty } \frac{\sin (x / n) \sin 2 n x}{x^{2}+4 n}, \quad -\infty < x < +\infty$ ;

(4)

$\sum_{n = 1}^{\infty } \frac{\cos \left(\left(n^{2}+1\right) x / n\right)-\cos \left(\left(n^{2}-1\right) x / n\right)}{x^{2}+n},-\infty < x < +\infty$ ;

(5)

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

(6)

$\sum_{n = 1}^{\infty }\left(\frac{x^{2}}{1+n x^{3}}\right)^{3}, 0 \leqslant x < +\infty$ .

Problem 2.18.15

Investigate the series of functions for convergence and uniform convergence on the given interval.

?

(1)

$\sum_{n = 1}^{\infty } \frac{\sqrt[3]{x}}{n\left(1+2 n x^{2}\right)}, \quad -\infty < x < +\infty$ ;

(2)

$\sum_{n = 1}^{\infty }\left(\frac{x^{2}}{1+n^{2} x^{5}}\right)^{2}, 0 \leqslant x < +\infty$ ;

(3)

$\sum_{n = 1}^{\infty } \frac{x^{2}}{(1+n x)^{4}}, 0 \leqslant x < +\infty$ ;

(4)

$\sum_{n = 1}^{\infty } \frac{\operatorname {arctg}(x / \ln (n+2))}{\left(n \ln (n+1)+x^{2}\right)^{2}}, \quad 0 \leqslant x < +\infty$ ;

(5)

$\sum_{n = 1}^{\infty } \frac{\sin (n / x) \sin (x / n)}{1+n x^{2}}, 0 < x < +\infty$ ;

(6)

$\sum_{n = 1}^{\infty } \frac{\cos n x \sin (x / n)}{1+\sqrt{n} x^{4}}, \quad 0 \leqslant x < +\infty$ .

Problem 2.18.16

Investigate the series of functions for convergence and uniform convergence on the given interval.

?

(1)

$\sum_{n = 1}^{\infty }\left(\frac{x \sin x}{1+n x^{3}}\right)^{2}, \quad 0 \leqslant x < +\infty$ ;

(2)

$\sum_{n = 1}^{\infty } \frac{\sin^{2} x}{x+n^{3} x^{2}}, \quad 0 < x < +\infty$ ;

(3)

$\sum_{n = 1}^{\infty } \sin \frac{1}{n x} \sin^{2} \frac{n x}{n^{3}+x}, \quad 0 < x < +\infty$ ;

(4)

$\sum_{n = 1}^{\infty } \sin \frac{1}{\sqrt{n}} \operatorname {arctg} \frac{2 x}{x^{2}+n^{2}}, \quad -\infty < x < +\infty$ ;

(5)

$\sum_{n = 1}^{\infty }\left(\frac{x \sin (x / \sqrt{n})}{x^{3}+n}\right)^{2}, \quad 0 \leqslant x < +\infty$ ;

(6)

$\sum_{n = 1}^{\infty } \frac{\cos (x / n) \operatorname {arctg}(x / \sqrt{n})}{n x+x^{2}}, 0 < x < +\infty$ .

Problem 2.18.17

Investigate the series of functions for convergence and uniform convergence on the given interval.

?

(1)

$\sum_{n = 1}^{\infty } n x^{2} e^{-n x} \sin \frac{1}{n}, \quad 0 \leqslant x < +\infty$ ;

(2)

$\sum_{n = 1}^{\infty } \frac{e^{-x / n} \cos n x}{x^{2}+n^{2} x}, \quad \alpha \leqslant x < +\infty , \quad \alpha > 0$ ;

(3)

$\sum_{n = 1}^{\infty } \frac{\cos (x \sqrt{n}) \ln \left(1+x^{2} / n\right)}{\sqrt{1+n x^{4}}},-\infty < x < +\infty$ ;

(4)

$\sum_{n = 1}^{\infty } \frac{x^{3} \sin^{2} n x}{2+n^{3} x^{6}}, \quad 0 \leqslant x < +\infty$ ;

(5)

$\sum_{n = 1}^{\infty } \ln \left(1+\frac{n x^{2}}{2+n^{3} x^{2}}\right), \quad -\infty < x < +\infty$ ;

(6)

$\sum_{n = 1}^{\infty } \frac{\sqrt{x}}{n} \frac{\sin n x}{2 n x^{2}+1}, \quad 0 \leqslant x < +\infty$ .

Problem 2.18.18

Investigate the series of functions for convergence and uniform convergence on the given interval.

?

(1)

$\sum_{n = 1}^{\infty } e^{-n x}, \quad 0 < x < +\infty$ ;

(2)

$\sum_{n = 1}^{\infty } \frac{1}{(1+n x)^{2}}, \quad 0 < x < +\infty$ ;

(3)

$\sum_{n = 1}^{\infty }\left(\operatorname {arctg} \frac{x}{n}\right)^{2}, \quad -\infty < x < +\infty$ ;

(4)

$\sum_{n = 1}^{\infty } \sin^{2} \frac{1}{1+n x}, \quad 0 < x < +\infty$ ;

(5)

$\sum_{n = 1}^{\infty } e^{-n \operatorname {tg} x}, \quad 0 < x < \frac{\pi }{2}$ ;

(6)

$\sum_{n = 1}^{\infty }\left(1+\frac{n \sin x}{x}\right)^{-2}, \quad 0 < x < \pi$ .

Problem 2.18.19

Investigate the series of functions for convergence and uniform convergence on the given interval.

?

(1)

$\sum_{n = 1}^{\infty } \frac{\sin n x}{(1+n x) \sqrt{n x}}, \quad 0 < x < \pi$ ;

(2)

$\sum_{n = 1}^{\infty } e^{-n^{2} x^{2}} \sin n x, \quad 0 \leqslant x \leqslant 1$ ;

(3)

$\sum_{n = 1}^{\infty } \frac{\sqrt{n x}}{1+x^{3} n^{2}}, \quad 0 \leqslant x < +\infty$ ;

(4)

$\sum_{n = 1}^{\infty } \operatorname {arctg} \frac{x^{3}}{n \ln^{2}(n+1)}, \quad 1 \leqslant x < +\infty$ ;

(5)

$\sum_{n = 1}^{\infty }(1-x) x^{n}, \quad 0 \leqslant x \leqslant 1$ ;

(6)

$\sum_{n = 1}^{\infty } \frac{x^{n}}{n !}, \quad 0 < x < +\infty$ .

Problem 2.18.20

Investigate the series of functions for convergence and uniform convergence on the given interval.

?

(1)

$\sum_{n = 1}^{\infty } 2^{n} \sin \frac{x}{3^{n}}, \quad 0 \leqslant x < +\infty$ ;

(2)

$\sum_{n = 1}^{\infty } \frac{x}{(1+(n-1) x)(1+n x)}, \quad 0 < x < +\infty$ ;

(3)

$\sum_{n = 1}^{\infty } \frac{\operatorname {ctg}(\pi x / n)}{2^{n}}, 0 < x < 1$ ;

(4)

$\sum_{n = 1}^{\infty } x^{n}\left(1-x^{n}\right), \quad \frac{1}{2} < x < 1$ ;

(5)

$\sum_{n = 1}^{\infty } \frac{x-1}{x^{n}}, \quad 1 \leqslant x+\infty$ ;

(6)

$\sum_{n = 1}^{\infty } \frac{n}{x^{2}} e^{-n^{2} / x}, \quad 0 < x < +\infty$ .

Problem 2.18.21

Investigate the series of functions for convergence and uniform convergence on the given interval.

?

(1)

$\sum_{n = 1}^{\infty } \frac{(-1)^{n}}{x+\sqrt{n}}, \quad 0 \leqslant x < +\infty$ ;

(2)

$\sum_{n = 1}^{\infty } \frac{1}{(x+2 n-1)(x+2 n+1)}, 0 \leqslant x < +\infty$ ;

(3)

$\sum_{n = 1}^{\infty } \frac{(-1)^{n+1}}{2 n+\sin x}, \quad -\infty < x < +\infty$ ;

(4)

$\sum_{n = 1}^{\infty }(-1)^{n} n^{-x}, \quad \delta \leqslant x < +\infty , \quad \delta > 0$ ;

(5)

$\sum_{n = 1}^{\infty }(-1)^{n} \frac{x^{n}}{n}, \quad 0 \leqslant x \leqslant 1$ ;

(6)

$\sum_{n = 1}^{\infty } \frac{(-1)^{n+1} e^{-n x}}{n}, 0 \leqslant x < +\infty$ .

Problem 2.18.22

Investigate the series of functions for convergence and uniform convergence on the given interval.

?

(1)

$\sum_{n = 1}^{\infty } \frac{\sin n x \sin x}{\sqrt[3]{n+x}} x, 0 \leqslant x < +\infty$ ;

(2)

$\sum_{n = 1}^{\infty } \frac{(-1)^{n}}{n} \frac{x^{n}}{x^{n}+1}, \quad 1 \leqslant x < +\infty$ ;

(3)

$\sum_{n = 1}^{\infty } \frac{(-1)^{n}}{\sqrt{n}} \cdot \operatorname {arctg} x^{n}, \quad 1 \leqslant x < +\infty$ ;

(4)

$\sum_{n = 1}^{\infty } \frac{n|x|^{3 / 2}}{\left(1+n^{3} x^{2}\right)^{3}}, \quad -\infty < x < +\infty$

(5)

$\sum_{n = 1}^{\infty } \frac{x}{1+n^{3} x^{3}}, \quad 0 \leqslant x < +\infty$ ;

(6)

$\sum_{n = 1}^{\infty } x^{n} \sin \frac{1}{2^{n}}, \quad -2 < x < 2$ .

Problem 2.18.23

Investigate the series of functions for convergence and uniform convergence on the given interval.

?

(1)

$\sum_{n = 1}^{\infty } \frac{\sin (\sqrt{x} / n)}{\sqrt{x^{2}+n^{2}}}, 0 \leqslant x < +\infty$ ;

(2)

$\sum_{n = 1}^{\infty } \frac{n x^{2}}{4+n^{4} x^{3}}, \quad 0 \leqslant x < +\infty$ ;

(3)

$\sum_{n = 1}^{\infty } \frac{n \sqrt{x}}{1+n^{3} x^{3}}, \quad 0 \leqslant x < +\infty$ ;

(4)

$\sum_{n = 1}^{\infty } \frac{x^{n}}{4^{n}} \ln \left(1+\frac{x^{2}}{n^{2}}\right), \quad -4 \leqslant x \leqslant 4$ ;

(5)

$\sum_{n = 1}^{\infty } \frac{x^{n}}{n(n+1)} \operatorname {arctg} \frac{x}{2^{n}}, \quad -2 \leqslant x \leqslant 2$ ;

(6)

$\sum_{n = 1}^{\infty } \frac{\operatorname {sh}\left(2 x /\left(x^{2}+n^{2}\right)\right) \sin \left(\sqrt{n+1} / n^{2}\right)}{(1+1 / n)^{n+1}},-\infty < x < +\infty$ .

Problem 2.18.24

Prove that the series $\sum_{n = 1}^{\infty }(-1)^{n} \frac{x^{2}+n}{n^{2}}$ converges uniformly on any finite segment, but does not converge absolutely for any value of $x$ .

?

Problem 2.18.25

Prove that the series $\sum_{n = 1}^{\infty } \frac{(-1)^{n} x^{2}}{\left(1+x^{2}\right)^{n}}$ converges uniformly on the entire real axis, while the series $\sum_{n = 1}^{\infty } \frac{x^{2}}{\left(1+x^{2}\right)^{n}}$ , composed of the absolute values of the terms of the original series, converges nonuniformly.

?

Problem 2.18.26

Prove that if $\left\{ a_{n}\right\}$ is a numerical sequence such that $a_{k+1} \leqslant a_{k}(k \in N)$ and $\lim_{n \rightarrow \infty } a_{n} = 0$ , then the series

\sum _{n = 1}^{\infty } a_{n} \sin n x \quad \text{ and } \quad \sum _{n = 1}^{\infty } a_{n} \cos n x

converge uniformly on any segment not containing the points $x_{k} = 2 k \pi , k \in Z$ .

?

Problem 2.18.27

Prove that the series $\sum_{n = 1}^{\infty } \frac{x^{4}}{\left(1+x^{4}\right)^{n}}$ converges uniformly on any segment not containing the point $x = 0$ .

?

Problem 2.18.28

Investigate the series for convergence and uniform convergence on the given interval:

?

(1)

$\sum_{n = 1}^{\infty } \frac{x^{3}}{\left(1+x^{3}\right)^{n}}, \quad 0 < x < +\infty$ ;

(2)

$\sum_{n = 1}^{\infty } \frac{(-1)^{n(n-1) / 2}}{\sqrt[3]{n+x^{2}}}, \quad -a \leqslant x < a, \quad a > 0$ ;

(3)

$\sum_{n = 1}^{\infty } \frac{\cos (\pi n / 3)}{\sqrt{n^{2}+x^{4}}}, \quad -\infty < x < +\infty$ ;

(4)

$\sum_{n = 1}^{\infty } \frac{(-1)^{[\sqrt{n}]}}{\sqrt{n^{2}+n x^{2}}}, \quad -\infty < x < +\infty$ ;

(5)

$\sum_{n = 1}^{\infty } \frac{\sin n x}{2 n+n^{2} x^{2}}, \quad 0 < x < 1$ ;

(6)

$\sum_{n = 1}^{\infty }\left(1+\frac{x}{n}\right)^{n} \frac{(-1)^{n}}{\sqrt[5]{n+x^{2}}}, \quad 0 \leqslant x \leqslant 2$ .

Problem 2.18.29

Investigate the series for convergence and uniform convergence on the sets $E_{1}$ and $E_{2}$ .

?

(1)

$\sum_{n = 1}^{\infty } \sin \left(\frac{\pi }{2} x^{n}\right), \quad E_{1} = (0 ; \alpha ), \quad 0 < \alpha < 1, \quad E_{2} = (0 ; 1)$ ;

(2)

$\sum_{n = 1}^{\infty } e^{-n \operatorname {arctg} x}, \quad E_{1} = (\delta ;+\infty ), \quad \delta > 0, \quad \bar{E}_{2} = (0 ;+\infty )$ ;

(3)

$\sum_{n = 1}^{\infty } \operatorname {arctg} \frac{x}{n^{2}}, \quad E_{1} = [0 ; a], \quad a > 0, \quad E_{2} = [0 ;+\infty )$ ;

(4)

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

(5)

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

(6)

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

(7)

$\sum_{n = 1}^{\infty } \frac{n x^{2}}{1+n^{5 / 2} x^{6}}, \quad E_{1} = (0 ; 1), \quad E_{2} = (1 ;+\infty )$ ;

(8)

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

(9)

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

(10)

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

Problem 2.18.30

Investigate the series for convergence and uniform convergence on the sets $E_{1}$ and $E_{2}$ .

?

(1)

$\sum_{n = 1}^{\infty } \operatorname {sh}\left(\frac{\sqrt{x}}{n \ln^{3}(n+2)}\right), \quad E_{1} = (0 ; 1), \quad E_{2} = (1 ;+\infty )$ ;

(2)

$\sum_{n = 1}^{\infty } e^{-n \arcsin x}, \quad E_{1} = \left(0 ; \frac{1}{2}\right), \quad E_{2} = \left(\frac{1}{2} ; 1\right)$ ;

(3)

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

(4)

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

(5)

$\sum_{n = 1}^{\infty } e^{-n \sin^{2} x}, \quad E_{1} = \left(0 ; \frac{\pi }{4}\right], E_{2} = \left(\frac{\pi }{4} ; \frac{\pi }{2}\right]$ ;

(6)

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

(7)

$\sum_{n = 1}^{\infty } x e^{n} \sin \frac{x}{5^{n}}, \quad E_{1} = (0 ; 1), \quad E_{2} = (1 ;+\infty )$ ;

(8)

$\sum_{n = 1}^{\infty } \frac{\operatorname {sh}(1 /(x n)) \cos (x n)}{1+x n}, \quad E_{1} = (0 ; 1), \quad E_{2} = (1 ;+\infty )$ ;

(9)

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

(10)

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

Problem 2.18.31

Investigate the series for convergence and uniform convergence on the sets $E_{1}$ and $E_{2}$ .

?

(1)

$\sum_{n = 1}^{\infty }\left(1-\cos \sqrt[3]{\frac{x}{n^{2}}}\right), \quad E_{1} = (0 ; 1), \quad \bar{E}_{2} = (1 ;+\infty )$ ;

(2)

$\sum_{n = 1}^{\infty } 2^{-n x} \operatorname {arctg}\left(n^{2} x\right), \quad E_{1} = (0 ; \delta ), \quad E_{2} = (\delta ;+\infty ), \quad \delta > 0$ ;

(3)

$\sum_{n = 1}^{\infty } \frac{\operatorname {arctg}(1 /(n x)) \cos n x}{4+\ln^{2} 2 n x}, \quad E_{1} = (0 ; 1), \quad E_{2} = (1 ;+\infty )$ ;

(4)

$\sum_{n = 1}^{\infty } \frac{x}{n^{2}+\cos (n /(x+1))}, \quad E_{1} = (0 ; \delta ), \quad E_{2} = (\delta ;+\infty ), \quad \delta > 0$ ;

(5)

$\sum_{n = 1}^{\infty } 2^{n} \operatorname {tg} \frac{1}{3^{n} x+1}, \quad E_{1} = (0 ; \delta ), \quad E_{2} = (\delta ;+\infty ), \quad \delta > 0 ;$

(6)

$\sum_{n = 1}^{\infty } \frac{1}{n+\sqrt{(\ln x) / n}} \sin \frac{1}{\sqrt{n}}, \quad E_{1} = (0 ; e), \quad E_{2} = (e ;+\infty )$ ;

(7)

$\sum_{n = 1}^{\infty } \sqrt[n]{\operatorname {ch}(x \sqrt{n})}\left(\frac{x}{n}\right)^{2}, \quad E_{1} = (0 ; 1), \quad E_{2} = (1 ;+\infty )$ ;

(8)

$\sum_{n = 1}^{\infty } \ln \left(1+\frac{e^{x}}{n}\right) \sin \frac{x^{3}}{\sqrt{n}}, \quad E_{1} = (0 ; 1), \quad E_{2} = (1 ;+\infty )$ ;

(9)

$\sum_{n = 1}^{\infty } \operatorname {sh} \frac{e^{x}}{n} \sin \frac{x}{\sqrt{n}}, \quad E_{1} = (0 ; 1), \quad E_{2} = (1 ;+\infty )$ ;

(10)

$\sum_{n = 1}^{\infty } \sqrt{x n} \operatorname {arctg} \frac{x}{(x-n)^{2}+n x}, \quad E_{1} = (0 ; 1), \quad E_{2} = (1 ;+\infty )$ .

Problem 2.18.32

Investigate the series for convergence and uniform convergence on the sets $E_{1}$ and $E_{2}$ .

?

(1)

$\sum_{n = 1}^{\infty } \frac{\sqrt{n}}{2+x \sqrt{n^{3}}} \sin \sqrt[3]{\frac{x}{n}}, \quad E_{1} = \left(\frac{1}{2} ;+\infty \right), E_{2} = \left(0 ; \frac{1}{2}\right)$ ;

(2)

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

(3)

$\sum_{n = 1}^{\infty } \frac{\sin n x}{e^{n^{2} x}}, \quad E_{1} = (\delta ;+\infty ), \quad \delta > 0, \quad E_{2} = (0 ;+\infty )$ ;

(4)

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

(5)

$\sum_{n = 1}^{\infty } \frac{\sqrt{x}}{n^{2}} \sin \frac{x}{n^{2}}, \quad E_{1} = (0 ; \delta ), \quad \delta > 0, \quad E_{2} = (0 ;+\infty )$ ;

(6)

$\sum_{n = 1}^{\infty } \sqrt{\frac{n}{x}} \sin \frac{x}{4 n^{2}}, \quad E_{1} = (0 ; \delta ), \quad \delta > 0, \quad E_{2} = (\delta ;+\infty )$ ;

(7)

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

(8)

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

(9)

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

(10)

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

(11)

$\sum_{n = 1}^{\infty } \frac{x^{3} \sqrt[6]{n}}{x^{3}+n} \sin \frac{x}{\sqrt[3]{n}}, \quad E_{1} = (0 ; 1), \quad E_{2} = (1 ;+\infty )$ ;

(12)

$\sum_{n = 1}^{\infty } \sqrt{n x} \sin \frac{1}{(1-n x)^{2}+n x}, \quad E_{1} = (0 ; 1), \quad E_{2} = (1 ;+\infty )$ .

Problem 2.18.33

Investigate the series for convergence and uniform convergence on the sets $E_{1}$ and $E_{2}$ .

?

(1)

$\sum_{n = 1}^{\infty } \frac{\sin n x}{n}, \quad E_{1} = [0 ; 2 \pi ], \quad E_{2} = [\delta ; 2 \pi -\delta ], \quad 0 < \delta < 2 \pi -\delta$ ;

(2)

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

(3)

$\sum_{n = 1}^{\infty } \frac{x}{\sqrt{n}} e^{-x^{2} n}, \quad E_{1} = (\delta ;+\infty ), \quad \delta > 0, \quad E_{2} = (0 ;+\infty )$ ;

(4)

$\sum_{n = 1}^{\infty } \frac{n x}{(1+x)(1+2 x) \ldots (1+n x)}, \quad E_{1} = [0 ; \delta ], \quad E_{2} = [\delta ;+\infty ), \quad \delta > 0$ ;

(5)

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

(6)

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

(7)

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

(8)

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

(9)

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

(10)

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

(11)

$\sum_{n = 1}^{\infty } \frac{x^{1 / 4} n^{2}}{x+n^{2}} \operatorname {arctg} \sqrt{\frac{x}{n^{3}}}, \quad E_{1} = (0 ; 1), \quad E_{2} = (1 ;+\infty )$ ;

(12)

$\sum_{n = 1}^{\infty } x e^{-n^{2} x^{2}} \operatorname {arctg} n x, \quad E_{1} = (0 ; 1), \quad E_{2} = (1 ;+\infty )$ .

Problem 2.18.34

Investigate the series for convergence and uniform convergence on the sets $E_{1}$ and $E_{2}$ .

?

(1)

$\sum_{n = 1}^{\infty } \frac{x \sin (x / n)}{\sqrt{n+x^{2}}}, \quad E_{1} = (0 ;+\infty ), \quad E_{2} = (0 ; \delta ), \quad \delta > 0$ ;

(2)

$\sum_{n = 1}^{\infty } \frac{n^{2}}{x} e^{-n^{2} / x}, \quad E_{1} = (0 ;+\infty ), \quad E_{2} = (0 ; \delta ), \quad \delta > 0$ ;

(3)

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

(4)

$\sum_{n = 1}^{\infty } e^{-x^{2} n^{6}} \sin \left(n^{3} x\right), \quad E_{1} = (0 ;+\infty ), \quad E_{2} = (\delta ;+\infty ), \quad \delta > 0$ ;

(5)

$\sum_{n = 1}^{\infty } \frac{1+n^{2} x^{2}}{x+n^{3}} \ln \left(1+\frac{x^{2}}{n}\right), \quad E_{1} = (0 ; 1), \quad E_{2} = (1 ;+\infty )$ ;

(6)

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

(7)

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

(8)

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

(9)

$\sum_{n = 1}^{\infty } \frac{1}{n^{3} \ln (n /(n+x))}, \quad E_{1} = (0 ; 1), \quad E_{2} = (1 ;+\infty )$ ;

(10)

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

(11)

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

(12)

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

Problem 2.18.35

Investigate the series for convergence and uniform convergence on the sets $E_{1}$ and $E_{2}$ .

?

(1)

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

(2)

$\sum_{n = 1}^{\infty } e^{-n^{2} x^{2}} \operatorname {arctg}(n x), \quad E_{1} = (0 ;+\infty ), \quad E_{2} = (\delta ;+\infty ), \quad \delta > 0$ ;

(3)

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

(4)

$\sum_{n = 1}^{\infty } \frac{x \sin \sqrt{x / n}}{n+x}, \quad E_{1} = (0 ; \delta ), \quad E_{2} = (\delta ;+\infty ), \quad \delta > 0$ ;

(5)

$\sum_{n = 1}^{\infty } n x^{n} \operatorname {tg} \frac{x}{4^{n}}, \quad E_{1} = [-2 ; 2], \quad E_{2} = (-3 ; 3)$ ;

(6)

$\sum_{n = 1}^{\infty } \frac{x^{n}}{n^{2}+3^{n}} \sin \frac{x}{2^{n}}, \quad E_{1} = [-5 ; 5], \quad E_{2} = (-6 ; 6)$ ;

(7)

$\sum_{n = 1}^{\infty } x^{2} 2^{n} \operatorname {tg} \frac{x}{4^{n}}, \quad E_{1} = (0 ; 1), \quad E_{2} = (1 ;+\infty )$ ;

(8)

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

(9)

$\sum_{n = 1}^{\infty } \operatorname {sh} \frac{x}{\sqrt{n}} \sin \frac{\sqrt{x}}{n+1}, \quad E_{1} = (0 ; 1), \quad E_{2} = (1 ;+\infty )$ ;

(10)

$\sum_{n = 1}^{\infty } \frac{x}{n+\sqrt{n x}} \operatorname {arctg} \frac{n x^{2}}{n^{3}+x^{3}}, \quad E_{1} = (0 ; 1), \quad E_{2} = (1 ;+\infty )$ .

Problem 2.18.36

Investigate the series for convergence and uniform convergence on the sets $E_{1}$ and $E_{2}$ .

?

(1)

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

(2)

$\sum_{n = 1}^{\infty } \frac{x^{2} n^{2}}{x^{4}+n^{4}} \sin \frac{n}{x}, \quad E_{1} = (0 ; 1), \quad E_{2} = (1 ;+\infty )$ ;

(3)

$\sum_{n = 1}^{\infty } \frac{n}{1+n^{2} x^{2}} \operatorname {tg} \sqrt{\frac{x}{n}}, \quad E_{1} = \left[0 ; \frac{1}{2}\right], E_{2} = \left[\frac{1}{2} ; 1\right]$ ;

(4)

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

(5)

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

(6)

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

(7)

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

(8)

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

(9)

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

(10)

$\sum_{n = 1}^{\infty } \frac{\sqrt{n}}{2+x^{3} n^{3 / 2}} \sin \sqrt[5]{\frac{x}{n}}, \quad E_{1} = (0 ; 1), \quad E_{2} = (1 ;+\infty )$ ;

(11)

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

(12)

$\sum_{n = 1}^{\infty }\left(e^{x / \sqrt{n}}-1\right) \operatorname {arctg} \frac{x^{2}}{n+1}, \quad E_{1} = (0 ; 1), \quad E_{2} = (1 ;+\infty )$ .

Problem 2.18.37

Investigate the series for convergence and uniform convergence on the sets $E_{1}$ and $E_{2}$ .

?

(1)

$\sum_{n = 1}^{\infty } \frac{\ln (n x)}{1+n^{3} \ln^{2} x}, \quad E_{1} = \left(0 ; \frac{1}{2}\right), \quad E_{2} = \left(\frac{1}{2} ; 1\right)$ ;

(2)

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

(3)

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

(4)

$\sum_{n = 1}^{\infty } \frac{1}{n^{3} \ln (n /(n+x))}, \quad E_{1} = (0 ; 1), \quad E_{2} = (1 ;+\infty )$ ;

(5)

$\sum_{n = 1}^{\infty }(-1)^{n} \ln \frac{3+x \sqrt{n^{5}}}{2+x \sqrt{n^{5}}}, \quad E_{1} = [1 ;+\infty ), \quad E_{2} = (0 ;+\infty )$ ;

(6)

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

(7)

$\sum_{n = 1}^{\infty } \frac{\cos (x / n)}{2^{n-x}+2^{n / x}}, \quad E_{1} = [10 ;+\infty ), \quad E_{2} = (-\infty ;-10]$ ;

(8)

$\sum_{n = 1}^{\infty } \frac{n \sin \sqrt{n}}{1+n^{3} x^{3}}, \quad E_{1} = (\alpha ;+\infty ), \quad \alpha > 0, \quad E_{2} = (0 ;+\infty )$ ;

(9)

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

(10)

$\sum_{n = 1}^{\infty }\left(1-\cos \sqrt{\frac{x^{5}}{n}}\right) \operatorname {arctg} \frac{e^{x}}{\sqrt{n}}, \quad E_{1} = (0 ; 1), \quad E_{2} = (1 ;+\infty )$ ;

(11)

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

(12)

$\sum_{n = 1}^{\infty } \frac{1}{\sqrt{3+n x^{2}}} \ln \left(1+\frac{2 n^{2} x^{4}}{1+n^{3} x^{6}}\right), \quad E_{1} = (0 ; 1), \quad E_{2} = (1 ;+\infty )$ .

Problem 2.18.38

Investigate the series for convergence and uniform convergence on the sets $E_{1}$ and $E_{2}$ .

?

(1)

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

(2)

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

(3)

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

(4)

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

(5)

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

(6)

$\sum_{n = 1}^{\infty } \frac{n x}{\sqrt{n}+x} \sin^{2} \frac{\ln x}{n}, \quad E_{1} = (1 ; 2), \quad E_{2} = (2 ;+\infty )$ ;

(7)

$\sum_{n = 1}^{\infty } \frac{x^{4}}{n+x+1} \operatorname {arctg} \frac{x^{2}}{\sqrt{n}}, \quad E_{1} = (0 ; 1), \quad E_{2} = (1 ;+\infty )$ ;

(8)

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

(9)

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

(10)

$\sum_{n = 1}^{\infty } \sqrt[3]{n}\left(\cos \frac{1}{x n}-1\right) \cos x n, \quad E_{1} = (0 ; 1), \quad E_{2} = (1 ;+\infty )$ ;

(11)

$\sum_{n = 1}^{\infty }\left(1-\cos \sqrt{\frac{x}{n}}\right) \operatorname {arctg} \frac{x}{\sqrt{n}}, \quad E_{1} = (0 ; 1), \quad E_{2} = (1 ;+\infty )$ .

Problem 2.18.39

Find all values of $\alpha$ for which the series $\sum_{n = 1}^{\infty } x^{\alpha } e^{-n x^{2}}$ converges uniformly on the interval $(0 ;+\infty )$ .

?

Problem 2.18.40

(1)(2)

Investigate the series for uniform convergence on the set $[0 ;+\infty )$ :

?

(1)

$\sum_{n = 0}^{\infty } \frac{x^{n}}{n !} e^{-x}$

(2)

$\sum_{n = 0}^{\infty }(-1)^{n} \frac{x^{2 n}}{(2 n) !}$ .

Problem 2.18.41

(1)(2)(3)(4)

Let $\varlimsup_{n \rightarrow \infty } \sqrt[n]{\left|a_{n}\right|} = 1$ . Prove that the series $\sum_{n = 1}^{\infty } a_{n} u_{n}(x)$ converges uniformly on the set $E$ , if:

?

(1)

$u_{n}(x) = x^{n}, E = [-\alpha ; \alpha ], 0 < \alpha < 1$ ;

(2)

$u_{n}(x) = e^{-n x}, E = [\delta ;+\infty ), \delta > 0$ ;

(3)

$u_{n}(x) = \frac{\cos n x}{2^{n}}, E = (-\infty ;+\infty )$ ;

(4)

$u_{n}(x) = e^{-n^{2} x}, E = [\delta ;+\infty ), \delta > 0$ .

Problem 2.18.42

Prove that if the series $\sum_{n = 1}^{\infty } u_{n}(x)$ converges uniformly on the set $E$ , and the function $\varphi (x)$ is bounded on this set, then the series $\sum_{n = 1}^{\infty } \varphi (x) u_{n}(x)$ also converges uniformly on the set $E$ .

?

Problem 2.18.43

Prove that if the series $\sum_{n = 1}^{\infty }\left|v_{n}(x)\right|$ converges uniformly on the set $E$ , and the functions $u_{n}(x)$ are defined on $E$ and satisfy the condition

\left|u_{n}(x)\right| \leqslant \left|v_{n}(x)\right|, \quad x \in E

then the series $\sum_{n = 1}^{\infty } u_{n}(x)$ converges on the set $E$ absolutely and uniformly.

?

Problem 2.18.44

Prove that if the series $\sum_{n = 1}^{\infty }\left|u_{n}(x)\right|$ converges uniformly on the set $E$ , then the series $\sum_{n = 1}^{\infty } u_{n}(x)$ also converges uniformly on $E$ .

?

Problem 2.18.45

Does the absolute and uniform convergence of the series $\sum_{n = 1}^{\infty } u_{n}(x)$ on the set $E$ imply the uniform convergence of the series $\sum_{n = 1}^{\infty }\left|u_{n}(x)\right|$ on $E$ ? Consider the example

\sum _{n = 1}^{\infty }(-1)^{n} x^{n}(1-x), \quad E = [0 ; 1]

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Problem 2.18.46

Prove that if the series $\sum_{n = 1}^{\infty } u_{n}(x)$ converges absolutely at the points $a$ and $b$ , and the functions $u_{n}(x), n \in N$ , are monotonic on the segment $[a ; b]$ , then this series converges absolutely and uniformly on $[a ; b]$ .

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Problem 2.18.47

Prove that if the series $\sum_{n = 1}^{\infty } \frac{1}{\left|a_{n}\right|}$ converges, then the series $\sum_{n = 1}^{\infty } \frac{1}{x-a_{n}}$ converges absolutely and uniformly on any closed bounded set not containing the points $x = a_{n}, n \in N$ .

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Problem 2.18.48

Prove that if the series $\sum_{n = 1}^{\infty } a_{n}$ converges, then the Dirichlet series $\sum_{n = 1}^{\infty } \frac{a_{n}}{n^{x}}$ converges uniformly on the set $[0 ;+\infty )$ .

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Problem 2.18.49

Prove that if the series $\sum_{n = 1}^{\infty } a_{n}$ converges, then the series $\sum_{n = 1}^{\infty } a_{n} e^{-n x}$ converges uniformly on the set $[0 ;+\infty )$ .

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Problem 2.18.50

Let the functions $u_{n}(x), n \in N$ , be continuous on the segment $[0 ; 1]$ , $u_{n}(x) \rightrightarrows 0, x \in [0 ; 1]$ , and let the series $\sum_{n = 1}^{\infty } u_{n}(x)$ converge for every $x \in [0 ; 1]$ . Does it follow that this series converges uniformly on the segment $[0 ; 1]$ ?

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Problem 2.18.51

Prove that the series $\sum_{n = 1}^{\infty } u_{n}(x)$ , where

u_{n}(x) = \begin{cases} 0, & \text{ if } 0 \leqslant x \leqslant 2^{-(n+1)}, \\ (1 / n) \sin ^{2}\left(2^{n+1} \pi x\right), & \text{ if } 2^{-(n+1)} < x < 2^{-n}, \\ 0, & \text{ if } 2^{-n} \leqslant x \leqslant 1, \end{cases}

converges absolutely and uniformly on the segment $[0 ; 1]$ , but cannot be majorized by a convergent numerical series with nonnegative terms, i.e., there cannot exist a convergent series $\sum_{n = 1}^{\infty } a_{n}, a_{n} \geqslant 0 (n \in N)$ , such that for all $n \in N$ the inequalities hold

u_{n}(x) \leqslant a_{n}, \quad x \in [0 ; 1].

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