Functional Series

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

converge

Let $\lambda_{n} \rightarrow 0$. Does the series $\sum \lambda_{n} e^{-\left|x-n\right|}$ converge uniformly on $\mathbb {R}$? Estimate $\sup_{x}\left|\sum \lambda_{n} e^{-\left|x-n\right|}\right|$ under the condition that $\left|\lambda_{n}\right| \leqslant 1$ for every $n \in \mathbb {N}$.

Does the series

converge uniformly on $[0,+\infty )$? Estimate $\sup_{x \geqslant 0} S(x)$.

Does the series $\sum \min \left(x n!, \frac{1}{x n!}\right)$ converge uniformly on $(0,+\infty )$? Give a uniform estimate of its sum for $x>0$.

For which $p>0$ does the series $\sum \frac{1}{n^{p}} \min \left(n^{p} x^{n}, \frac{1}{n^{p} x^{n}}\right)$ converge uniformly on $(0,+\infty )$?

Let $a_{n}>0$ for $n \in \mathbb {N}$. Prove that the series $\sum \frac{x^{n}}{\left(1+x^{n a_{n}}\right)^{n}}$

Let $a_{n} \in \mathbb {R}, b_{n}>0, a_{n}=O\left(b_{n}\right), \sum b_{n}=+\infty$ and let the series $B(t)=\sum_{n \geqslant 0} b_{n} t^{n}$ converge for $\left|t\right|<1$. Prove that the series $A(t)=\sum_{n \geqslant 0} a_{n} t^{n}$ converges for $\left|t\right|<1$ and

(compare with problem II.2.2). In particular, if $a_{n} \sim c b_{n}, c \neq 0$, then $A(t) \sim c B(t)$ as $t \rightarrow 1-0$; if $a_{n}=o\left(b_{n}\right)$, then $A(t)=o(B(t))$ as $t \rightarrow 1-0$.

Prove that if the series $\sum_{n \geqslant 0} a_{n}$ converges, then

(Abel's theorem).

If the partial sums $S_{n}$ of the series $\sum_{n \geqslant 0} a_{n}$ are such that the finite limit $\lim \frac{1}{n}\left(S_{1}+\ldots +S_{n}\right)=l$ exists, then the series $A(t)=\sum_{n \geqslant 0} a_{n} t^{n}$ converges for $\left|t\right|<1$ and $A(t) \underset {t \rightarrow 1-0}{\longrightarrow } l$ (Frobenius's theorem).

Thus Abel's method of summing series (computing the limit $\lim_{t \rightarrow 1-0} \sum_{n \geqslant 0} a_{n} t^{n}$) is stronger than the method of arithmetic means (Cesàro's method).

Let $b_{n} \geqslant 0, \sum b_{n}=+\infty$ and let the series $B(t)=\sum_{n \geqslant 0} b_{n} t^{n}$ converge for $\left|t\right|<1$. If the sequence $\left\{ a_{n}\right\}_{n \geqslant 0}$ is such that the finite limit

exists, then the series $A(t)=\sum_{n \geqslant 0} a_{n} t^{n}$ converges for $\left|t\right|<1$ and $\frac{A(t)}{B(t)} \xrightarrow [t \rightarrow 1-0]{ } l$.

Let the sequence $\left\{ a_{n}\right\}$ be such that $a_{n}>-1$ for $n \in \mathbb {N}$ and $\Pi \left(1+a_{n}\right)=A \in (0,+\infty )$. Prove that for $\left|t\right|<1$ the product $A(t)=\prod \left(1+a_{n} t^{n}\right)$ converges and, if $\sum a_{n}^{2}<+\infty$, then $A(t) \rightarrow A$ as $t \rightarrow 1-0$.

Can the condition $\sum a_{n}^{2}<+\infty$ be dropped?

Let $a_{n} \in \mathbb {R}$ for $n \in \mathbb {N}$ and let the product $P(t)=\prod \left(1+t a_{n}\right)$ converge for at least two nonzero real values of $t$. Prove that the series $\sum a_{n}$ and $\sum a_{n}^{2}$ converge.

Let $\left\{ v_{k}\right\}$ be a strictly increasing sequence of positive numbers, $\left\{ c_{k}\right\}_{k \geqslant 0} \subset \mathbb {R}$. Consider the generalized power series $c_{0}+c_{1} t^{\nu_{1}}+c_{2} t^{\nu_{2}}+\ldots$ for $t \geqslant 0$. Prove that

Let $f(x)=\sum a_{n}\left|\sin n x\right|$, where $a_{n} \geqslant 0$ and $\sum a_{n}<+\infty$. Prove that $f \in \operatorname {Lip}_{1}$ if and only if $\sum n a_{n}<+\infty$.

Let $\alpha , x \in \mathbb {R}$. Prove that

Let the series $\sum a_{n}$ converge and $\sum a_{n} \cos \frac{x}{n} \rightarrow L \in \mathbb {R}$ as $x \rightarrow +\infty$. Prove that $a_{n}=0$ for all $n \in \mathbb {N}$.

Do the series $\sum \frac{1}{n} \sin \frac{x}{n}$ and $\sum \frac{1}{n} \sin \frac{x}{n^{2000}}$ converge uniformly on $[0,+\infty )$?

Let $c_{n} \geqslant 0$ for all $n \in \mathbb {N}$. Prove that uniform convergence of the series $\sum c_{n} \sin \frac{2 \pi x}{4^{n}}$ on $[0,+\infty )$ implies convergence of the series $\sum c_{n}$.

Let $f(x)=\sum \frac{1}{n} \sin \frac{x}{4^{n}}$. Prove that

Does there exist a uniformly convergent series on $\mathbb {R}$ of the form $\sum c_{n} \sin \frac{x}{m_{n}}$, where $c_{n}>0$ and $\sum c_{n}=+\infty$, and $\left\{ m_{n}\right\}$ is an increasing sequence of natural numbers?

Let $-\infty \leqslant a<b \leqslant +\infty$ and let $F, f_{n}, g_{n} \in C((a, b))$, $f_{n} \geqslant 0, g_{n} \geqslant 0(n \in \mathbb {N})$. Prove that if $\sum g_{n} \leqslant F \leqslant \sum f_{n}$, then

In particular, if $F=\sum f_{n}$, then $\int_{a}^{b} F(x) d x=\sum \int_{a}^{b} f_{n}(x) d x$.