Properties of Numerical Series Related to Monotonicity

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.