Math Problems Vault

Problem IV.1.1

Does the series $\sum \frac{\varepsilon_{n}}{n}$ converge, where $\varepsilon_{n}=0$ if the decimal representation of $n$ contains the digit 9, and $\varepsilon_{n}=1$ otherwise?

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Remark.

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Recall that $\sum a_{n}$ is the notation for the series $\sum_{n \geqslant 1} a_{n}$ .

Problem IV.1.2

Do the series

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(a)

$\sum \frac{\left|\cos 2^{n}\right|}{n}$ ;

(b)

$\quad \sum \frac{\left|\sin (n+\ln n)\right|}{n}$ ;

(c)

$\quad \sum \frac{\left|\sin n^{2}\right|}{n}$ ;

(d)

$\quad \sum \frac{\left|\operatorname {tg} n\right|}{n\left|\operatorname {tg} n\right|}$ ;

(e)

$\sum \frac{\operatorname {tg}^{2} n}{n^{1+\left|\operatorname {tg} n\right|}}$ ;

(f)

$\quad \sum \frac{\left|\cos \sqrt{n}\right|}{n}$ converge?

Problem IV.1.3

Prove that the series $\sum \frac{\cos (b \ln n)}{n^{a}}$ converges only for $a>1$ .

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Problem IV.1.4

(a)(b)(c)

For which $p>0$ do the series

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(a)

$\sum_{n \geqslant 2} \frac{(\ln n)^{-2 p}}{n^{p}+p^{-\sqrt[p]{\ln n}}}$ ;

(b)

$\quad \sum \frac{(-1)^{\left[\sqrt{n}\right]}}{n^{p}}$ ;

(c)

$\quad \sum \arccos^{p} \frac{\nu (n)}{\nu (n+1)}$ converge?

Here $\nu (n)$ is the number of digits in the decimal representation of the number $n$ .

Problem IV.1.5

(a)(b)(c)(d)

Let $\tau (n)$ be the number of divisors of $n$ , $p_{n}$ the $n$ -th prime number, $\varphi (n)$ the number of integers less than $n$ that are coprime to $n$ . Do the series

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(a)

$\sum \frac{\tau (n)}{n^{2}}$ ;

(b)

$\sum \frac{1}{p_{n}}$ ;

(c)

$\quad \sum \frac{\varphi (n)}{n^{2}}$ ;

(d)

$\sum \frac{1}{n p_{n}-(n-1) p_{n-1}}$ converge?

Problem IV.1.6

Let $\varphi \in C(\left[a, b\right])$ and

a_{n}=\int _{a}^{b} \frac{\varphi (t) d t}{t+\ln n} \quad \text{ for } \quad n \geqslant N>e^{-a}

For which functions $\varphi$ does the series $\sum_{n \geqslant N} a_{n}$ converge?

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Problem IV.1.7

(a)(b)

Let $\varphi \in C(\left[-\pi , \pi \right])$ . For which functions $\varphi$ do the series

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(a)

$\sum_{n \geqslant 3} \int_{-\pi }^{\pi } \frac{\varphi (t) d t}{\cos t+\ln n}$ ;

(b)

$\sum_{n \geqslant 3} \int_{-\pi }^{\pi } \frac{\varphi (t) d t}{\sin t+\ln n}$ converge?

Problem IV.1.8

Prove that for $a>1$ the series

\sum \frac{1}{n^{a}} \frac{1}{\left|\sin (\pi \sqrt{2} n)\right|}

converges.

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Problem IV.1.9

Do the double series

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(a)

$\sum_{n, m \geqslant 1} \frac{1}{n^{2}+m^{2}}$ ;

(b)

$\quad \sum_{n, m \geqslant 1} \frac{\min (n, m)}{n^{3}+m^{3}}$ ;

(c)

$\quad \sum_{n, m \geqslant 1} \frac{\max (n, m)}{n^{4}+m^{4}}$ ;

(d)

$\sum_{n, m \geqslant 1} \frac{\varepsilon (n, m)}{n^{2}+m^{2}}$ ;

(e)

$\quad \sum_{n, m \geqslant 1} \frac{\gcd (n, m)}{n^{3}+m^{3}}$ ;

(f)

$\quad \sum_{n, m \geqslant 1} \frac{\operatorname {lcm}(n, m)}{n^{4}+m^{4}}$ ;

(g)

$\sum_{n, m \geqslant 1} \frac{n m}{n^{5}+m^{5}}$ ;

(h)

$\sum_{n, m \geqslant 1} \frac{1}{(\operatorname {lcm}(n, m))^{p}}, p>1$

converge?

Here $\operatorname {lcm}(n, m)$ is the least common multiple of $n$ and $m$ ; $\gcd (n, m)$ is the greatest common divisor of $n$ and $m$ ; $\varepsilon (n, m)=1$ if $n$ and $m$ are coprime, and $\varepsilon (n, m)=0$ otherwise.

Problem IV.1.10

Verify that the sum of the series

\begin{aligned} & 1+\frac{1}{2}+\frac{1}{2}-1+\frac{1}{4}+\frac{1}{4}-\frac{1}{2}+\frac{1}{4}+\frac{1}{4}-\frac{1}{2}+ \\ & +\frac{1}{8}+\frac{1}{8}-\frac{1}{4}+\frac{1}{8}+\frac{1}{8}-\frac{1}{4}+\frac{1}{8}+\frac{1}{8}-\frac{1}{4}+\frac{1}{8}+\frac{1}{8}-\frac{1}{4}+\ldots \end{aligned}

(the group of terms $\frac{1}{2^{n+1}}+\frac{1}{2^{n+1}}-\frac{1}{2^{n}}$ repeats $2^{n}$ times) equals 1. Exhibit a rearrangement after which the sum of the series becomes equal to $-1$ .

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