Series

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?

Do the series

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

For which $p>0$ do the series

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

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

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

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

Prove that for $a>1$ the series

converges.

Do the double series

Verify that the sum of the series

(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 $-1$.

Let $a_{n}>0, a_{n} \uparrow$. Prove that the boundedness of the sequence $\left\{ a_{n}\right\}$ is equivalent to the convergence of the series $\sum \arccos^{2}\left(\frac{a_{n}}{a_{n+1}}\right)$.

Let $a_{n} \downarrow 0$. Prove that

Let $a_{n} \downarrow 0, b_{n} \downarrow 0$ and $\sum a_{n}=\sum b_{n}=+\infty$. Does the series $\sum \min \left\{ a_{n}, b_{n}\right\}$ always diverge?

In the solutions of many of the subsequent problems, Abel's test for series is used: if the sequences $\left\{ a_{n}\right\}$ and $\left\{ b_{n}\right\}$ are such that the sequence $\left\{ a_{n} B_{n}\right\}$ converges (here $B_{n}=b_{1}+\ldots +b_{n}$), then the series $\sum a_{n} b_{n}$ and $\sum \left(a_{n}-a_{n+1}\right) B_{n}$ converge or diverge simultaneously. In the case when the series $\sum b_{n}$ converges, another version of this statement may prove useful: if the sequence $\left\{ a_{n} \beta_{n}\right\}$ converges (here $\beta_{n}=b_{n}+b_{n+1}+\ldots$), then the series $\sum a_{n} b_{n}$ and $\sum \left(a_{n}-a_{n+1}\right) \beta_{n}$ converge or diverge simultaneously. Proofs of these statements are easily obtained by applying the Abel transformation (see § 2 of Ch. I) to the sums $\sum_{1 \leqslant n \leqslant N} a_{n} b_{n}$. Abel's test is especially convenient to use when the sequence $\left\{ a_{n}\right\}$ is monotone.

Let $a_{n} \downarrow 0, p>-1$. Prove that the series $\sum \left(a_{n}-a_{n+1}\right) n^{p+1}$ and $\sum n^{p} a_{n}$ converge or diverge simultaneously, and in the case of convergence $a_{n}=o\left(n^{-p-1}\right)$. In particular, the series $\sum a_{n}$ and $\sum \left(a_{n}-a_{n+1}\right) n$ converge or diverge simultaneously, and in the case of convergence $a_{n}=o\left(\frac{1}{n}\right)$.

Let $a_{n} \downarrow 0$. Prove that the series $\sum \frac{a_{n}}{n}$ and $\sum \left(a_{n}-a_{n+1}\right) \ln n$ converge or diverge simultaneously, and in the case of convergence $a_{n}=o\left(\frac{1}{\ln n}\right)$.

Let $a_{n} \geqslant 0$ for all $n \in \mathbb {N}$ and $\sum a_{n} \ln n<\infty$. For $p>0$ set $b_{n}=\sum_{k \geqslant n^{p}} a_{k}$. Prove that $\sum \frac{b_{n}}{n}<\infty$.

Let $p>1$ and $a_{n} \downarrow 0$. Prove that the convergence of the series $\sum a_{n}^{p-1} n^{-1 / p}$ implies the convergence of the series $\sum a_{n}^{p}$. Is monotonicity essential?

Let $a_{n} \downarrow 0$. Prove that

Let $a_{n} \downarrow 0$ and the series $\sum x_{n}$ converge. Prove that the series $\sum a_{n} x_{n}$ converges and $\sum_{k \geqslant n} a_{k} x_{k}=o\left(a_{n}\right)$.

Let $x_{n}>0, x_{n} \rightarrow 0$. Prove that there exists a sequence $\left\{ a_{n}\right\}$ decreasing to zero such that $\sum a_{n}=\infty$ and $\sum a_{n} x_{n}<\infty$.

Let $p>0, q, Q \in \mathbb {N}, Q>1$. Let $\left\{ a_{n}\right\}$ be a nonnegative sequence and $A_{n}=a_{1}+\ldots +a_{n}$. Prove the equivalence of the following statements:

1. $\sum \frac{a_{n}}{n^{p}}<+\infty$;

2. $\sum \frac{A_{Q^{n}}}{Q^{p n}}<+\infty$.

3. $\sum \frac{A_{n} q}{n^{1+p q}}<+\infty$;

Prove that the series $\sum \frac{x_{n}}{n^{1+\varepsilon }}$ converges if $\varepsilon >0$ and the sequence $\left\{ x_{n}\right\}$ is such that $\frac{1}{n}\left(x_{1}+\ldots +x_{n}\right) \rightarrow 0$. Is the convergence always absolute? Does the series $\sum \frac{x_{n}}{n}$ always converge?

Let $a_{n}>0$ and $S_{n}=a_{1}+\ldots +a_{n}, S_{n} \rightarrow +\infty$. Prove that

Let $a_{n}>0, \sum a_{n}<+\infty$ and $\alpha_{n}=\sum_{k \geqslant n} a_{k}$. Prove that

Let $p \geqslant 1, a_{n} \geqslant 0$ and $r_{n}=\sum_{k \geqslant n} a_{k}^{p}<+\infty$. Prove that

Let $\left\{ a_{n}\right\}$ be a positive monotone sequence, $a_{n} \neq 1$ for all indices $n$, and $a_{n} \rightarrow a<+\infty$. Prove that the series

diverges if $a=0$ or $a=1$, and converges in the remaining cases. What happens when $a=+\infty$?

Let $f$ be a positive, strictly increasing function on the interval $[1,+\infty )$, with $f(x) \rightarrow +\infty$ as $x \rightarrow +\infty$. Prove that the series

converge simultaneously ($f^{-1}$ is the function inverse to $f$).

Let the function $f$ decrease to zero on the interval $[0,+\infty )$; let the sequence $\left\{ a_{n}\right\}$ be nonnegative and bounded, with $\sum a_{n}=+\infty$. Prove that the series

converge or diverge simultaneously.

Let $a_{n} \downarrow 0$ and $\sum a_{n}=A$. For an arbitrary set $\nu \subset \mathbb {N}$, call the quantity $A(\nu )=\sum_{n \in \nu } a_{n}$ the sum of the partial series corresponding to $\nu$ (by definition $A(\varnothing )=0$). Prove the equivalence of the following statements:

1. the sums of the partial series fill the interval $\left[0, A\right]$;

2. $a_{n} \leqslant a_{n+1}+a_{n+2}+\ldots$ for all indices $n$.

Does there exist a series $\sum a_{n}$ such that $a_{n} \downarrow 0$ and the sums of its partial series (see 2.19) fill the Cantor set?

Let $a_{n} \downarrow 0$ and $\sum a_{n}=+\infty$. Prove that there exists a sequence of indices $\left\{ n_{k}\right\}$ such that $n_{k}<n_{k+1}, a_{n_{k}}<\frac{1}{k}$ and the series $\sum a_{n_{k}}$ diverges. Can monotonicity of $\left\{ a_{n}\right\}$ be dropped?

Let $\left\{ t_{n}\right\}$ be a sequence of positive numbers.

Does there exist a sequence $\left\{ a_{n}\right\} , a_{n} \neq 0$, such that the series $\sum a_{n}$ and $\sum \frac{1}{n^{2} a_{n}}$ converge? Can such a positive sequence $\left\{ a_{n}\right\}$ be chosen?

Let $f \in C^{1}([1,+\infty ))$ and $\int_{1}^{\infty }\left|f^{\prime }(x)\right| d x<+\infty$. Prove that the convergence of the series $\sum f(n)$ is equivalent to the convergence of the integral $\int_{1}^{\infty } f(x) d x$.

Let $a_{n}>0$ and $\sum a_{n}<+\infty$. Prove that

Let $0<a_{n}<1$. Prove that the convergence of the series $\sum \frac{a_{n}}{\ln a_{n}}$ implies the convergence of the series $\sum \frac{a_{n}}{\ln (1+n)}$. Is the converse statement true if $a_{n} \downarrow 0$? Can monotonicity be dropped?

Let $a_{n} \geqslant 0$ and $\sum \frac{a_{n}}{n}<+\infty$. Prove that there exists a sequence of indices $\left\{ n_{k}\right\}$ of density 1 (see problem II.2.14) such that $a_{n_{k}} \rightarrow 0$.

Prove the equivalence of the two statements:

1. $\sum \left|a_{n}\right|<+\infty$;

2. for every sequence of indices $\left\{ n_{k}\right\}$ of density 0 (see problem II.2.14) the series $\sum a_{n_{k}}$ converges.

Let the series $\sum a_{n}$ converge absolutely and let $\sum a_{k n}=0$ for every $k \in \mathbb {N}$. Prove that $a_{n}=0$ for every $n$. Is this statement true for series that converge but not absolutely?

Prove that if the series $\sum a_{n} x_{n}$ converges for every sequence $\left\{ x_{n}\right\}$ tending to zero, then $\sum \left|a_{n}\right|<+\infty$.

Prove that the following two properties of the sequence $\left\{ c_{n}\right\}$ are equivalent:

1. $\sum \left|c_{n}-c_{n+1}\right|<+\infty$;

2. if the series $\sum a_{n}$ converges, then the series $\sum c_{n} a_{n}$ also converges.

Let the function $\varphi$ be monotone and bounded on the interval $[0,+\infty )$. Prove that for every convergent series $\sum a_{n}$

Construct a positive sequence $\left\{ a_{n}\right\}$ with the properties:

1. $a_{n} \rightarrow 0$;

2. $\sum a_{n}=+\infty$;

3. if $a_{n_{k}} \downarrow 0$, where $\left\{ n_{k}\right\}$ is a strictly increasing sequence of indices, then $\sum a_{n_{k}} \leqslant 2$.

Construct a positive sequence $\left\{ a_{n}\right\}$ with the properties:

1. $a_{n} \rightarrow 0$;

2. $\sum a_{n}=+\infty$;

3. if a strictly increasing sequence of indices $\left\{ n_{k}\right\}$ is such that $\sup \left\{ a_{n_{k}} / a_{n_{j}} \mid k, j \in \mathbb {N}, k \geqslant j\right\} <+\infty$, then the series $\sum a_{n_{k}}$ converges.

Let $\left\{ a_{n}\right\}$ be a sequence of positive numbers. Set $A_{n}=a_{1}+\ldots +a_{n}, \alpha_{n}=\frac{1}{a_{n}}+\frac{1}{a_{n+1}}+\ldots$ Prove that the following six statements are equivalent:

1. $A_{n}=O\left(a_{n}\right)$;

2. $\alpha_{n}=O\left(\frac{1}{a_{n}}\right)$;

3. $\frac{A_{n}}{A_{n+1}} \leqslant q_{1}<1$ for all $n \in \mathbb {N}$;

4. $\alpha_{1}<+\infty$ and $\frac{\alpha_{n+1}}{\alpha_{n}} \leqslant q_{2}<1$ for all $n \in \mathbb {N}$;

5. $a_{1}^{p}+\ldots +a_{n}^{p}=O\left(a_{n}^{p}\right)$, if $p>0$;

6. $\frac{1}{a_{n}^{p}}+\frac{1}{a_{n+1}^{p}}+\ldots =O\left(\frac{1}{a_{n}^{p}}\right)$, if $p>0$.

From statements a)-f) follows the statement g) for every $Q>1$ there exists an index $L=L_{Q}$ such that $a_{n+L} \geqslant Q a_{n}$ for all $n \in \mathbb {N}$, i.e., the sequence $\left\{ a_{n}\right\}$ splits into $L$ subsequences $\left\{ a_{n}^{(l)}\right\} =\left\{ a_{n L+l}\right\}_{n \geqslant 1}(l=0,1, \ldots , L-1)$, each of which grows no slower than a geometric progression: $a_{n+1}^{(l)} \geqslant Q a_{n}^{(l)}$. Prove that if the sequence $\left\{ a_{n}\right\}$ is monotone, then g) $\Rightarrow$ a). Is this statement true for non-monotone sequences?

Let $\left\{ a_{n}\right\}$ be a sequence of positive numbers. Set $A_{n}=a_{1}+\ldots +a_{n}, \alpha_{n}=\frac{1}{a_{n}}+\frac{1}{a_{n+1}}+\ldots$ Prove that the following seven statements are equivalent:

1. $A_{n-1}=o\left(a_{n}\right)$;

2. $\alpha_{n+1}=o\left(\frac{1}{a_{n}}\right)$;

3. $A_{n-1}=o\left(A_{n}\right)$;

4. $\alpha_{1}<+\infty$ and $\alpha_{n+1}=o\left(\alpha_{n}\right)$;

5. $a_{1}^{p}+\ldots +a_{n-1}^{p}=o\left(a_{n}^{p}\right)$, if $p>0$;

6. $\frac{1}{a_{n+1}^{p}}+\frac{1}{a_{n+1}^{p}}+\ldots =o\left(\frac{1}{a_{n}^{p}}\right)$, if $p>0$;

7. $a_{n-1}=o\left(a_{n}\right)$.

From statements a)-g) follows the statement h) $\sqrt[n]{a_{n}} \rightarrow +\infty$.

Show by examples that h) $\centernot \Longrightarrow$ a) even for monotone sequences; however h) $\Rightarrow$ a) if $\sqrt[n]{a_{n}} \uparrow +\infty$.

Let $p>1, a_{n} \geqslant 0$ and $A_{n}=\frac{1}{n}\left(a_{1}+\ldots +a_{n}\right)(n \in \mathbb {N})$. Prove that

Let $a_{n}>0$ for all $n \in \mathbb {N}$ and let the series $\sum a_{n}$ converge. Prove that the series $\sum \sqrt[n]{a_{1} a_{2} \ldots a_{n}}$ converges (note that the series $\sum \frac{1}{n}\left(a_{1}+\ldots +a_{n}\right)$ obviously diverges).

Let the function $f$ be defined on $\mathbb {R}$. Prove that the following properties of the function $f$ are equivalent:

1. $f(x)=O(x)$ as $x \rightarrow 0$;

2. for any absolutely convergent series $\sum a_{n}$ the series $\sum f\left(a_{n}\right)$ converges absolutely;

3. for any absolutely convergent series $\sum a_{n}$ the series $\sum f\left(a_{n}\right)$ converges.

Let the function $f$ be defined on $\mathbb {R}$. Prove that if for any convergent series $\sum a_{n}$ the series $\sum f\left(a_{n}\right)$ converges, then $f(x)=C x$ in some neighborhood of zero.

Let the countable set $Q$ be contained in the interval $(0,1)$, $\inf Q=0, \sup Q=1$. Prove that for any number $x \in (0,1)$ one can enumerate the set $Q=\left\{ q_{1}, q_{2}, \ldots \right\}$ in such a way that $x=\sum \frac{q_{n}}{2^{n}}$.

$\sum 3^{n} \sin^{3} \frac{x}{3^{n}}$.

$\sum (-1)^{n} \frac{\cos^{3}\left(3^{n} x\right)}{3^{n}}$.

$\sum \operatorname {arctg} \frac{1}{n^{2}+n+1}$.

$\sum \operatorname {arctg} \frac{1}{2 n^{2}}$.

$\sum_{n \geqslant 1} \frac{1}{C_{n+r}^{1+r}}(r \in \mathbb {N})$.

$\sum_{n \geqslant 0} \frac{(n!)^{2} z^{n}}{(2 n+1)!} \quad (|z|<4)$.

$\sum n(\operatorname {ch} 2 n-e) e^{-n^{2}}$.

$\sum_{n \geqslant 2}\left(\frac{1}{n} \sum_{m \geqslant 2} m^{-n}\right)$.

$\sum \frac{(-1)^{n}}{n} \ln n$.

$\sum \frac{(-1)^{n}}{n}\left[\log_{2} n\right]$.

$\sum \left(\frac{1}{2 n-1}-\ln \left(1+\frac{1}{2 n-1}\right)\right)$.

$\sum \left(\frac{1}{2 n}-\ln \left(1+\frac{1}{2 n}\right)\right)$.

$\sum \left(1-\left(n+\frac{1}{2}\right) \ln \left(1+\frac{1}{n}\right)\right)$.

$\sum \frac{1}{2^{n}} \operatorname {tg} \frac{1}{2^{n}}$.

$\sum \frac{1}{n} \int_{2 \pi n}^{\infty } \frac{\sin x}{x} d x$.

$\sum \frac{1}{n} \int_{\pi n}^{\infty } \frac{\sin x}{x} d x$.

Prove that

In particular,

Prove that

Here $p_{1}=2, p_{2}=3, \ldots$ are the prime numbers, numbered in increasing order.

Prove that for $\left|t\right|<1$

Find the sum of the series $\sum_{n \geqslant 1} \frac{a_{n}(m)}{n}$, where $m \in \mathbb {N}, m>1$ and $a_{n}(m)=1$ if $n$ is not divisible by $m$, $a_{n}(m)=1-m$ if $n$ is divisible by $m$.

For which $p, q>0$ and $x, y \in \mathbb {R}$ does the series

converge