Properties of Uniformly Convergent Functional Sequences and Series

Prove the continuity of the sum of the series of functions $\sum_{n = 1}^{\infty } u_{n}(x)$ on the set $E$:

Prove that the function $f(x) = \sum_{n = 1}^{\infty } n e^{-n x}$ is continuous for $x > 0$, and compute $\int_{\ln 2}^{\ln 5} f(x) d x$.

Prove that the function $f(x) = \sum_{n = 1}^{\infty } \frac{1}{n^{2}(n+1)^{2}+x^{2}}$ is continuous on $R$, and compute $\int_{0}^{+\infty } f(x) d x$.

Prove that the function $f(x) = \sum_{n = 1}^{\infty } \frac{\cos^{2} n x}{n(n+1)}$ is continuous on $R$, and compute $\int_{0}^{2 \pi } f(x) d x$.

Prove that the series $\sum_{n = 1}^{\infty }\left((n+1) x e^{-(n+1) x}-n x e^{-n x}\right)$ converges nonuniformly on the segment $[0 ; 1]$, but its sum is a function continuous on this segment.

Prove that the series $\sum_{n = 1}^{\infty }\left(x^{2(n+1)}-x^{2 n}\right)$ converges nonuniformly on the segment $[-1 ; 1]$, but it can be integrated term by term on this segment.

It is known that the sum $S(x)$ of the series $\sum_{n = 1}^{\infty } a_{n} \sin n x$, where $\left\{ a_{n}\right\}$ is a given numerical sequence, is an odd periodic function with period $2 \pi$, such that

Is it true that this series converges nonuniformly on $R$?

Find the set $E$ of all values of $x$ for which the function $f(x)$ is defined, and investigate it for continuity on $E$, if:

Find the set $E$ of all values of $x$ for which the function $f(x)$ is defined, and investigate it for differentiability on the set $E$, if:

Prove that the series $\sum_{n = 1}^{\infty } \frac{\sin n x}{n^{2} \ln^{2}(n+1)}$ can be differentiated term by term on $R$.

Prove that the function $f(x) = \sum_{n = 1}^{\infty } \frac{1}{(n-x)^{2}}$ is continuous on $R$, except at the points $x = n, n \in N$.

Prove that the function $f(x) = \sum_{n = 1}^{\infty } \frac{\cos n x}{n^{3}}$ has a continuous derivative on $R$.

Prove that the series $f(x) = \sum_{n = 1}^{\infty } e^{-(x-n)^{2}}$ can be differentiated term by term on the segment $[-1 ; 1]$ any number of times.

Prove that the Riemann zeta function $\zeta (x) = \sum_{n = 1}^{\infty } \frac{1}{n^{x}}$ is continuous on the set $(1 ;+\infty )$ and has derivatives of every order on this set.

Prove that the function $f(x) = \sum_{n = 1}^{\infty } e^{-n^{2} x}$ is infinitely differentiable for $x > 0$.

Prove that the sum of the series $\sum_{n = 1}^{\infty } \frac{e^{-n x}}{1+n^{2}}$ is continuous for $x \geqslant 0$ and is differentiable for $x > 0$.

Show that the sequence $\left\{ f_{n}(x)\right\}$, where $f_{n}(x) = n x e^{-n x^{2}}$, converges on the segment $[0 ; 1]$, but

Show that the sequence $\left\{ f_{n}(x)\right\}$, where $f_{n}(x) = \frac{1}{n} \operatorname {arctg} x^{n}$, converges uniformly on $R$, but

Show that the sequence $\left\{ f_{n}(x)\right\}$, where

converges uniformly on $R$ to the function $f$, and that the equality

holds for all $x$, except $x = 0$.

Show that the sequence $\left\{ f_{n}(x)\right\}$, where

converges uniformly on $R$, but $\left(\lim_{n \rightarrow \infty } f_{n}(x)\right)^{\prime } \neq \lim_{n \rightarrow \infty } f_{n}^{\prime }(x)$.

Show that the sequence $\left\{ f_{n}(x)\right\}$, where

converges uniformly on $R$ to a differentiable function $f(x)$, but

Show that the sequence $\left\{ f_{n}(x)\right\}$, where

converges on the segment $[0 ; 1]$ to a continuous function $f(x)$, but

Show that the sequence $\left\{ f_{n}(x)\right\}$, where $f_{n}(x) = n x(1-x)^{n}$, converges nonuniformly on the segment $[0 ; 1]$, however

Determine whether the equality holds

Can the series $\sum_{n = 1}^{\infty } \operatorname {arctg} \frac{x}{n^{2}}$ be differentiated term by term on $R$?

Can the series $\sum_{n = 1}^{\infty }\left(x^{1 /(2 n-1)}-x^{1 /(2 n+1)}\right)$ be integrated term by term on the segment $[0 ; 1]$?

Find all values of $\alpha$ for which, for the sequence $\left\{ f_{n}(x)\right\}$, where $f_{n}(x) = n^{\alpha } x e^{-n x}$, the equality holds

Can a sequence $\left\{ f_{n}(x)\right\}$ of discontinuous functions converge uniformly to a continuous function?

Let the functions $f_{n}(x), n \in N$, be defined and uniformly bounded on the set $R$, and, moreover, let $f_{n}(x) \rightrightarrows f(x), x \in R$. Does it follow that

Let the function $f$ be defined on $R$, infinitely differentiable, and

Prove that $\lim_{n \rightarrow \infty } f^{(n)}(x) = C e^{x}$, where $C =$ const.

Let the function $f$ be infinitely differentiable on $R$, and let the sequence $\left\{ f^{(n)}(x)\right\}$ converge uniformly to $\varphi (x)$ on every finite interval $(a ; b)$. Prove that $\varphi (x) = C e^{x}$, where $C =$ const.

Show that the sequence $\left\{ f_{n}(x)\right\}$, where

converges on $R$ to a continuous function, but nonuniformly.

Prove that if the functions $f_{n}(x), n \in N$, are continuous on the set $E$ and $f_{n}(x) \rightrightarrows f(x), x \in E$, then

for every sequence $\left\{ x_{n}\right\}$ such that $x_{n} \in E, x_{n} \rightarrow x$ as $n \rightarrow \infty$, where $x \in E$.

Let $\left\{ f_{n}(x)\right\}$ be a sequence of functions, monotonic on the segment $[a ; b]$, converging to a function $f(x)$ continuous on $[a ; b]$. Prove that

Let the functions $f_{n}(x), n \in N$, be integrable on the segment $[a ; b]$, and let the sequence $\left\{ f_{n}(x)\right\}$ be uniformly bounded on $[a ; b]$, i.e.,

Show that from the sequence $\left\{ F_{n}(x)\right\}$, where

a subsequence $\left\{ F_{n_{k}}(x)\right\}$ can be extracted that converges uniformly on the segment $[a ; b]$.

Show that the series $\sum_{n = 1}^{\infty } \frac{x\left(n(n+1) x^{2}-1\right)}{\left(1+n^{2} x^{2}\right)\left(1+(n+1)^{2}x^{2}\right)}$ converges nonuniformly on the segment $[-1 ; 1]$, but its sum is continuous on this segment.

Show that the series $\sum_{n = 1}^{\infty } \frac{\sin 2^{n} \pi x}{2^{n}}$ converges uniformly on $R$, but it cannot be differentiated term by term on any interval.

Prove that:

Let $r_{n}, n \in N$, be the rational numbers of the segment $[0 ; 1]$. Show that the function

is continuous on the segment $[0 ; 1]$, differentiable at irrational points, and not differentiable at rational points of this segment.

Find the limit:

Let the functions $u_{n}(x), n \in N$, be continuous on the segment $[a ; b]$ and $u_{n}(x) > 0, x \in [a ; b], n \in N$. Show that if the series $\sum_{n = 1}^{\infty } u_{n}(x)$ converges on $[a ; b]$ to a function $f(x)$ continuous on this segment, then

Let $\varphi (x)$ be a periodic function with period 1 such that $\varphi (x) = \left|x\right|$ for $x \in [-1 / 2 ; 1 / 2]$. Show that the sum of the series

is continuous on $R$, but is not differentiable at any point $x \in R$.

A sequence $\left\{ f_{n}(x)\right\}$ is called equicontinuous on the segment $[a ; b]$ if

$\forall \varepsilon > 0 \quad \exists \delta > 0 \quad \forall x^{\prime }, x^{\prime \prime } \in [a ; b]\left|x^{\prime }-x^{\prime \prime }\right| < \delta \quad \forall n \in N:\left|f_{n}\left(x^{\prime }\right)-f_{n}\left(x^{\prime \prime }\right)\right| < \varepsilon$.

Prove that if the sequence $\left\{ f_{n}(x)\right\}$ of continuous functions converges uniformly on the segment $[a ; b]$, then it is uniformly bounded (see problem 35) and equicontinuous on the segment $[a ; b]$.

Prove that if the sequence $\left\{ f_{n}(x)\right\}$ is uniformly bounded and equicontinuous (see problem 43) on the segment $[a ; b]$, then from it one can extract a subsequence that converges uniformly on this segment (Arzelà's theorem).