Math Problems Vault

Problem 4.1

Let there be defined on a set $E$ functions $f_{n}(z), n \in \mathbb {N}$ , and $f(z)$ such that for all $n \in \mathbb {N}$ and all $z \in E$ the inequality $\left|f_{n}(z)-f(z)\right| \leqslant a_{n}$ holds, where $a_{n} \rightarrow 0$ as $n \rightarrow \infty$ . Prove that $f_{n}(z) \rightrightarrows f(z), z \in E$ .

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Problem 4.2

Prove that on every closed set $E$ lying in the disk $\left|z\right| < 1$ , the sequence $\left\{ \frac{1}{1+z^{n}}\right\}$ converges uniformly to the function $f(z) = 1$ , and on every closed set $E$ lying in the domain $\left|z\right| > 1$ , this sequence converges uniformly to the function $f(z) = 0$ .

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Problem 4.3

Prove that the sequence $\left\{ n z e^{-n^{2} z^{2}}\right\}$ converges uniformly in the angle $\left|\arg z\right| \leqslant \alpha$ , for any $\alpha$ where $0 \leqslant \alpha < \pi / 4$ , to the function $f(z) = 0$ .

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Problem 4.4

Let the sequence $\left\{ f_{n}(z)\right\}$ converge uniformly on a set $E$ to the function $f(z)$ , and let the functions $f_{n}(z)$ be continuous on the set $E$ . Prove that the function $f(z)$ is also continuous on this set.

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Problem 4.5

Let the series $\sum_{n = 1}^{\infty }\left|u_{n}(z)\right|$ converge uniformly on a set $E$ , and let the functions $v_{n}(z)$ be defined on the set $E$ and satisfy the inequalities $\left|v_{n}(z)\right| \leqslant \left|u_{n}(z)\right|(z \in E)$ . Prove that the series $\sum_{n = 1}^{\infty } v_{n}(z)$ converges uniformly on the set $E$ .

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Problem 4.6

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

Prove the uniform convergence of the series on the set $E$ :

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(1)

$\sum_{n = 1}^{\infty } \frac{1}{n^{2}} z^{-2 n}, E = \left\{ z:\left|z\right| \geqslant 1\right\}$ ;

(2)

$\sum_{n = 1}^{\infty } n^{3} e^{-n z}, E = \left\{ z: \mathrm{Re}\left(z\right) \geqslant \delta > 0\right\}$ ;

(3)

$\sum_{n = 1}^{\infty } \frac{2^{n}}{z^{n}+z^{-n}}, E = \left\{ z:\left|z\right| \leqslant \rho < \frac{1}{2}\right\}$ ;

(4)

$\sum_{n = 1}^{\infty } 2^{-n} \cos n z, E = \left\{ z:\left|\mathrm{Im}\left(z\right)\right| \leqslant \delta < \ln 2\right\}$ .

Problem 4.7

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

Let $\lim_{n \rightarrow \infty } \sqrt[n]{\left|a_{n}\right|} = 1$ . Prove the uniform convergence of the series on the set $E$ :

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(1)

$\sum_{n = 1}^{\infty } a_{n} z^{n}, E = \left\{ z:\left|z\right| \leqslant \rho < 1\right\}$ ;

(2)

$\sum_{n = 1}^{\infty } a_{n} e^{-n z}, E = \left\{ z: \mathrm{Re}\left(z\right) \geqslant \delta > 0\right\}$ ;

(3)

$\sum_{n = 1}^{\infty } a_{n} e^{-n^{2} z}, E = \left\{ z: \mathrm{Re}\left(z\right) \geqslant \delta > 0\right\}$ ;

(4)

$\sum_{n = 1}^{\infty } \frac{a_{n}}{2^{n}} \cos n z, E = \left\{ z:\left|\mathrm{Im}\left(z\right)\right| \leqslant \delta < \ln 2\right\}$ .

Problem 4.8

Find the radius of convergence of the power series:

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(1)

$\sum_{n = 1}^{\infty }\left(\frac{n+3}{n+6}\right)^{n^{2}} \cdot z^{n}$ ;

(2)

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

(3)

$\sum_{n = 1}^{\infty } 6^{n} z^{5 n}$ ;

(4)

$\sum_{n = 1}^{\infty } n!e^{-n^{\alpha }} z^{n}, \quad \alpha > 1$ ;

(5)

$\sum_{n = 1}^{\infty } n\left(\cos \frac{1}{n}\right)^{2 n^{3}} \cdot z^{n}$ ;

(6)

$\sum_{n = 1}^{\infty } \frac{(2+i \sqrt{5})^{n}}{(3-i \sqrt{7})^{2 n}} z^{3 n}$ .

Problem 4.9

(1)(2)(3)

Let $R$ be the radius of convergence of the series $\sum_{n = 0}^{\infty } c_{n} z^{n}$ . Find the radius of convergence of the series:

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(1)

$\sum_{n = 0}^{\infty } c_{n}^{m} z^{n}, m \in \mathbb {N}$ ;

(2)

$\sum_{n = 0}^{\infty } c_{n}^{m} z^{m n}, m \in \mathbb {N}$ ;

(3)

$\sum_{n = 0}^{\infty } \frac{c_{n}}{1+\left|c_{n}\right|} z^{n}$ .

Problem 4.10

Prove that the power series $\sum_{n = 1}^{\infty } c_{n} z^{n}, \quad \sum_{n = 1}^{\infty } \frac{c_{n}}{n+1} z^{n+1}$ and $\sum_{n = 1}^{\infty } n c_{n} z^{n-1}$ have the same radius of convergence.

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Problem 4.11

Denote the radii of convergence of the power series

\sum _{n = 0}^{\infty } a_{n} z^{n}, \quad \sum _{n = 0}^{\infty } b_{n} z^{n}, \quad \sum _{n = 0}^{\infty }\left(a_{n}+b_{n}\right) z^{n}, \quad \sum _{n = 0}^{\infty } a_{n} b_{n} z^{n}

by $R_{1}, R_{2}, R_{3}, R_{4}$ , respectively. Prove that

R_{3} \geqslant \min \left(R_{1}, R_{2}\right), \quad R_{4} \geqslant R_{1} \cdot R_{2}

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Problem 4.12

Prove that if the inequality

\left|\frac{c_{n}}{c_{n+1}}\right| \geqslant R\left(1+\frac{\alpha }{n}\right), \quad n > n_{0}

holds, where $\alpha > 0$ , then the power series $\sum_{n = 0}^{\infty } c_{n} z^{n}$ converges at all points of the circle of its disk of convergence.

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Problem 4.13

Let all the numbers $c_{n}$ be positive, $c_{k-1} > c_{k}(k \in \mathbb {N})$ , and $c_{n} \rightarrow 0$ as $n \rightarrow \infty$ . Prove that the power series $\sum_{n = 0}^{\infty } c_{n} z^{n}$ converges at all points of the circle $\left|z\right| = 1$ , except possibly the point $z = 1$ .

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Problem 4.14

Determine at which points of the circle of the disk of convergence the following series converge:

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(1)

$z+\frac{2}{1 \cdot 3} z^{2}+\frac{2 \cdot 4}{1 \cdot 3 \cdot 5} z^{3}+\ldots$ ;

(2)

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

(3)

$\sum_{n = 2}^{\infty } \frac{z^{n}}{n \ln^{2} n}$ ;

(4)

$\sum_{n = 1}^{\infty } \frac{(2 n)!}{(n!)^{2}} z^{n}$ ;

(5)

$\sum_{n = 1}^{\infty } \frac{(3 n)!}{(2 n)!n!}(-1)^{n} z^{2 n}$ ;

(6)

$\sum_{n = 2}^{\infty } \frac{(-1)^{n}}{n \ln n} z^{3 n}$ ;

(7)

$\sum_{n = 1}^{\infty } \frac{e^{\pi i n^{2} / 2}}{n} z^{n}$ ;

(8)

$\sum_{n = 1}^{\infty } \frac{e^{\pi i n^{2} / 2}}{\sqrt{n}} z^{n}$ .

Problem 4.15

Prove Abel's second theorem: if the power series $\sum_{n = 0}^{\infty } c_{n} z^{n}$ converges at a point $z_{0}$ , then it converges uniformly on the entire straight-line segment joining the point $z = 0$ to the point $z_{0}$ .

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