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Questions tagged [uniform-convergence]

For sequences of functions, uniform convergence is a mode of convergence stronger than pointwise convergence, preserving certain properties such as continuity. This tag should be used with the tag [convergence].

2
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38 views

Uniform convergence of $f_n(x)=nx/(1+n^{2}x^{2})$?

We have to show that $f_n(x)=\frac{nx}{1+n^{2}x^{2}}$ is uniform convergent on $[a,\infty),a \gt 0$ but not on $[0,\infty)$ I am trying to prove uniform covegence on $[a, \infty]$ by using the result ...
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0answers
40 views

$f_n\to f,$ uniformly on $S$ if and only if $f$ is continuous on $S$ and $\left| f_{k+n}(x)-f(x) \right|<\epsilon$

Let $\{f_n\}$ be a sequence of continuous functions, defined on a compact set $S$ and assume that $\{f_n\}$ converges pointwise on $S$ to a limit function $f.$ Prove that $f_n\to f,$ uniformly on $S$, ...
2
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1answer
85 views

What exactly is the contradiction in proving that $h_n(x)$ does not converge uniformly on any bounded interval?

I am currently going through this pdf https://www.csie.ntu.edu.tw/~b89089/book/Apostol/ch9.pdf and in Exercise 9.2b, page 3, we have the following question. Prove that $h_n(x)$ does not converges ...
1
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1answer
43 views

Alternative proof of Abel's Theorem for Uniform convergence of series of functions

The following is the Abel's Theorem. I proved it my own way. Let $g_n(x)$ be a sequence of real-valued functions such that $g_{n+1}(x)\leq g_{n}(x),\forall \,x\in T$ and $n\in\Bbb{N}.$ If $\{g_{n}\...
1
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1answer
28 views

Does there exist a sequence of holomorphic functions uniformly convergent to $\bar{z}^3$ on boundary of annulus

Is there a sequence $\{f_n\}$ holomorphic on D(0,2) such that $f_n\to \bar{z}^3$ uniformly on $\{|z|=1\}\cup\{|z|=\frac{1}{2}\}$. I notice that on $\{|z|=1\}$, $\bar{z}^3=\frac{1}{z^3}$, $\{|z|=\frac{...
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3answers
57 views

Uniform Convergence of $\sum_{n=1}^{\infty}e^{-nx^{2}}\sin(nx)$ in $[0,\infty)$

Consider the series $f(x)=\sum_{n=1}^{\infty}e^{-nx^{2}}\sin(nx)$: a) Prove that this series converges uniformly on $[a,\infty)$, for each $a>0$ b) Does the series converge uniformly on $[0,\...
2
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1answer
50 views

Does $\sum^{\infty}_{n=1}xe^{-nx}$ converge uniformly on $[0,\infty)$?

I was trying to solve this problem with a friend but we got stuck. TRIAL We claimed that the series does not converge uniformly which implies that it is not uniformly Cauchy, i.e., $\exists\,\...
2
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2answers
35 views

relaxation on Dini's Theorem

Dini's Theorem states: Let $[a,b]$ be a compact intervall. Let $f,f_{n}: [a,b] \xrightarrow{} \mathbb{R}$, $ n \in \mathbb{N}$, functions with $f$ and $f_{n}$ are continuos for all $n$ $\lim_\limits{...
1
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1answer
27 views

Prove that $f_n(x)=\sin{\sqrt{x+4n^2\pi ^2}}\,,\;x\geq 0$ is equicontinous on $[0,+\infty)$

Prove that $f_n(x)=\sin{\sqrt{x+4n^2\pi ^2}}\,,\;x\geq 0$ is equicontinous on $[0,+\infty)$. converges pointwise to $0$ on $[0,+\infty)$. MY TRIAL \begin{align}\sqrt{x+4n^2\pi ^2}&=\sqrt{x+4n^...
2
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1answer
33 views

Using Definition to show that $x^n$ on [0,1] is not uniform convergent

I know that by continuty transfer property we can show above very easily . But I wanted to it by defination. My attempt: Uniform Convergence $f_n(x)\to f(x)$ on E means $\forall \epsilon>0 ,\exists ...
3
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1answer
41 views

Prob. 10 (a), Sec. 26, in Munkres' TOPOLOGY, 2nd ed: A Partial Converse To The Uniform Limit Theorem

Here is Prob. 10 (a), Sec. 26, in the book Topology by James R. Munkres, 2nd edition: Prove the following partial converse to the uniform limit theorem: Theorem. Let $f_n \colon X \to \...
1
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1answer
33 views

Uniform Convergence of $\sum_{n=1}^{\infty} \frac{\sin(nx)}{\sqrt{n}}, [0, 2\pi]$ by using Cauchy's test or Weierstrass-M test

$\sum_{n=1}^{\infty} \frac{\sin(nx)}{\sqrt{n}}, [0, 2\pi]$ I have been thinking for hours and I do not know how to approach whether it uniformly converges or not. Of course, I did search, but there ...
3
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1answer
25 views

If $\{f_n\}$ converges uniformly on $(a,b)$, $\{f_{n}(a)\}$ and $\{f_{n}(b)\}$ converge pointwise, then it converges uniformly on $[a,b]$

Prove that if $\{f_n\}$ converges uniformly on $(a,b)$, $\{f_{n}(a)\}$ and $\{f_{n}(b)\}$ converge pointwise. I want to show that $\{f_n\}$ converges uniformly on $[a,b]$ MY TRIAL: Let $\epsilon>...
0
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1answer
33 views

Does the series $s_n(x)=\sum^{n}_{k=0}\frac{e^{-kx}}{1+kx}$ converge uniformly over $ [1,2]?$

Does the series \begin{align}s_n(x)=\sum^{n}_{k=0}\frac{e^{-kx}}{1+kx},\;x\in [1,2]\end{align} converge uniformly? MY TRIAL: I thought, by intuition, that $s_n(x)$ does not converge uniformly, which ...
3
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0answers
77 views

Find region of convergence of double power series

How can i calculate the region of convergence of this double power series ? $$ S(x,y)=\displaystyle{ \sum_{n=0}^{\infty} \sum_{k=0}^{\infty} \frac{(n-\frac12)!\>(k-\frac12)!\>(\frac{n}{2}+k-\...
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1answer
26 views

Condition on Pointwise convergent sequnce on complex valued function to be uniform convergent

I Know that Pointwise convergence of continous sequnce of real valued function can be said to uniform converges BY Dini's Theorem By assuming Some condition like Monotonicity and compactness . As ...
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0answers
13 views

Convergence of two binomials with equal n as their parameters converge

If we have two binomials $B(n,p_1)$ and $B(n,p_2)$, what statements can we make about the similarity between them as $\mid p_1 - p_2\mid \rightarrow 0$? If $p_1$ were constant and $p_2$ monotonically ...
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0answers
28 views

Infinite sum smoothness Proof

There is a function: $V(n) = \sum_{i=1}^\infty f(b_i(n))b_i^{'}(n)- f(a_i(n))a_i^{'}(n)$, where $f$,$b$ and $a$ are $C^\infty$ smooth functions. I know that we can prove that the function $V(n)$ ...
0
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1answer
23 views

If $f(x)=\sum^{\infty}_{n=0}a_n x^n$ converges for $x=50$, can $f$ be uniformly continuous over $[0,10]$?

I want to show that if \begin{align}f(x)=\sum^{\infty}_{n=0}a_n x^n\end{align} converges for $x=50,$ then $f(x)$ is uniformly continuous on $[0,10].$ MY TRIAL Since the series \begin{align}f(x)=\sum^...
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0answers
21 views

Uniform convergence of logarithm on a compact set.

Given that $\phi$ and $\phi_n,n \geq 1$ are continuous functions from $\mathbb{R}$ into $\mathbb{C}$ such that $ \phi(0) = \phi_n(0) = 1$, $\phi(x) \neq 0$ and $\phi_n(x) \neq 0$ for any $x$. Suppose ...
0
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0answers
51 views

Show that $\sum^{\infty}_{n=1}(-1)^{n+1}\dfrac{1}{n+x^4}$ is uniformly convergent on $\Bbb{R}$

Show that the following series is uniformly convergent on $\Bbb{R}$ \begin{align}\sum^{\infty}_{n=1}(-1)^{n+1}\dfrac{1}{n+x^4}\end{align} MY TRIAL I tried using the alternating series test before ...
2
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1answer
70 views

Show this function series converge uniformly to a derivative function on $\mathbb{R}$.

We define $f_n(x) = \frac{n x^2 \sin(nx)}{n^4 + x^4}$. We have to prove that $$ \sum_{n=1}^\infty f_n $$ converges uniformly to a derivative function on $\mathbb{R}$. To prove that, it suffices to ...
3
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1answer
30 views

Prove that $x\mapsto \sum^{\infty}_{n=0}f_n(x)=\sum^{\infty}_{n=0}\frac{x^2}{(1+x^2)^n}$ does not converge uniformly on $[-1,1]$

For each $n,$ we define \begin{align} f_n:\Bbb{R}\to \Bbb{R} \end{align} \begin{align} x\mapsto f_n(x)=\frac{x^2}{(1+x^2)^n}\end{align} We consider the function \begin{align} f:\Bbb{R}\to \Bbb{R} \...
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0answers
27 views

Completeness - Topology of countably many seminorms on $C^{k}$

Question: For the space $C^{k}(U)$ of $k$ times differentiable functions on $U$, where $U \subset \mathbb{R}^n$, and with topology induced by the family $||f||_{\alpha,K}=\sup\limits_{x \in K}\big|D^...
0
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1answer
52 views

Uniform convergence of integrable functions $f_n$ implies limit is integrable. [duplicate]

Let $(X,\Sigma,\mu)$ be a measure space with $\mu(X)<\infty$. Let $\{f_n\}$ converge uniformly to $f$ with $f_n$ integrable for all $n$. Show that $f$ is integrable and $$ \int f\ d\mu = \lim_{n\...
4
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0answers
35 views

Uniform convergence of convolution of a distribution with a test function

For an exercise I have to show the following: Let $u_j \to u$ in $\mathcal{D'(\mathbb{R}^n)}$ and let $\phi_j \to \phi$ in $C^{\infty}_0(\mathbb{R}^n)$. Show that $$ \lim_{j\to \infty} u_j * \phi_j ...
0
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1answer
24 views

Determine whether a sequence of functions converges uniformly

The function $f_n(x)=3n^3(x-1/n)^2$ for $x\in[0,2]$ is given, and I need to show whether the sequence of functions $(f_n)_{n\in\Bbb{N}}$ converges uniformly to $f(x)=\lim\limits_{n \to \infty}f_n(x)$, ...
0
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1answer
26 views

Differentiability and Uniform convergence on unbounded intervals

Let $I$ be a bounded interval of $\mathbb{R}$, and let $\{f_{n}:I\to\mathbb{R}\}$ be a sequence of differentiable functions. Suppose that a sequence $\{f_{n}(x_{0})\}$ converges for some $x_{0}\in I$ ...
2
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1answer
49 views

Prove that $t\mapsto \frac{d}{dt}\left[\rho(t)\right]^{-1}=-\rho(t)^{-1}\rho'(t)\rho(t)^{-1}$

Let $\rho:(a,b)\to \operatorname{ISO}\left(\Bbb{R}^n\right)$ be differentiable. I want to prove that \begin{align}\frac{d}{dt}\left[\rho(t)\right]^{-1}=-\rho(t)^{-1}\rho'(t)\rho(t)^{-1}\end{align} $\...
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3answers
26 views

Uniform Convergence Example

Let $f_n(x)=x^n$ on $[0, 1]$. The pointwise limit of this sequence is $f(x)=\left\{\begin{matrix} 1,& \text{if $x=1$} \\ 0,& \text{if $0\leq x<1$.} \end{matrix}\right.$ Now, it is said ...
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0answers
55 views

Test series of function $\sum_{n=1}^{\infty}{\frac{nx}{1+n^2x^2}}$ for uniform convergence [duplicate]

Test the series of function $$\sum_{n=1}^{\infty}{\frac{nx}{1+n^2x^2}}$$ for uniform convergence. I tried various methods like Weierstrass M test. Also tried to find $S_n(x)$ and then do. But ...
0
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0answers
56 views

Convergence of $\sum_{n=1}\frac{1}{1-x^n} $

Consider $$ f(x)=\sum_{n=1}^\infty\frac{1}{1-x^n} $$ I am trying to find all $x\in\mathbb{R}$ at which the above series converges, absolutely converges, and all intervals of $\mathbb{R}$ on which the ...
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1answer
71 views

Check proof that $\int _{-\pi}^\pi\sum\limits_{n=2}^\infty\tan(x/2^n)\mathrm dx=\sum\limits_{n=2}^\infty\int_{-\pi}^\pi\tan(x/2^n)\mathrm dx$

Prove that $$( A) \quad \int _{-\pi}^{\pi} \sum_{n=2}^{\infty} \tan \frac{x}{2^n} \mathrm{d}x=\sum_{n=2}^{\infty}\int _{-\pi}^{\pi} \tan \frac{x}{2^n}\mathrm{d}x$$ We have $$ \forall n\ge 2, \quad \...
4
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5answers
264 views

Understanding uniform convergence of sequence

Given $f_n(x)=(x-1)^{3n}$ on $(0,2]$ we can show point-wise convergence to the function $$ f(x)=\begin{cases} 0&x\in(0,2)\\1&x\in\{2\}\end{cases} $$ But how do I show that uniform convergence ...
1
vote
0answers
83 views

Uniform convergence and series of expansion

Assume the following sum: $$ s(x) = \sum_{n=1}^{\infty} f_n(x) = \sum_{n=1}^{\infty}\frac{1}{n \sqrt{n^2+x^2} \left(\sqrt{n^2+x^2}+n\right)}, $$ where $x \in \mathbb{R}$. Since $f_n(x) = f_n(-x)$, it ...
0
votes
1answer
39 views

Can we apply Egorov's theorem?

We have the following example : for all $n\in \mathbb{N}$ and $x\in [0,1[$ we define : $f_n(x)=x^n$. We know that $\forall x \in [0,1[$, $f_n$ is pointwise convergent to $f\equiv 0$. However its not ...
0
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1answer
31 views

Show uniform convergence of series with |x|

Let $f:\mathbb{R}\to\mathbb{R}$ be a $2\pi$ periodic function where $$ f(x)=|x|= \begin{cases} -x&x\in[-\pi,0[\\ x&x\in[0,\pi] \end{cases} $$ Show that $\sum_{n=0}^{\infty} f(2^nx)/2^n$ ...
4
votes
3answers
103 views

Why $\sum_{n=1}^{\infty}\frac{n\left(\sin x\right)^{n}}{2+n^{2}}$ is not uniformly convergent on $[0,\;\frac{\pi}{2})$?

Why $\sum_{n=1}^{\infty}\frac{n\left(\sin x\right)^{n}}{2+n^{2}}$ is not uniformly convergent on $[0,\;\frac{\pi}{2})$? I was thinking that we need to show partial sums \begin{equation} \left|S_{2n}\...
2
votes
2answers
39 views

Is $f_{n}\left(x\right)=\sin\left(x+\frac{x^{2}}{n}\right)$ uniformly convergent on $\left[0,\:2\pi\right]$

Is $f_{n}\left(x\right)$ uniformly convergent on $\left[0,\:2\pi\right]$? \begin{equation} f_{n}\left(x\right)=\sin\left(x+\frac{x^{2}}{n}\right) \end{equation} We can see that $f_{n}\left(x\right)$ ...
0
votes
1answer
20 views

Exchange of limit and sum for a uniformly convergent sequence

Let $\{x_n^k\}$ be a set of nonnegative numbers with $n$ being in the integers, and $k$ being natural numbers, such that the $x_n^k \rightarrow 0$ uniformly over $n$ as $k$ tends to infinity and $\...
0
votes
1answer
37 views

Convergence of $h_{n}(x) = x^{ 1 +\frac{1}{2n-1} }$ defined on [-1,1]

1. $$ h_{n}(x) = x^{ 1 + \frac{1}{2n-1} } = x^{ \frac{2n-1+1}{2n-1} } = x^{\frac{2n}{2n-1}} = (x^2)^{\frac{n}{2n-1}} = (x^2 )^{ \frac{1}{2-\frac{1}{n}} } $$ then $$ \lim_{n\rightarrow \infty} h_{n}(...
3
votes
1answer
38 views

Analytic and bounded implies uniform continuity

Let $f$ be analytic and bounded in $\{z\in\mathbb{C}\mid Re(z)>0\}$. Prove that $f$ is uniformly continuous in $\{z\in\mathbb{C}\mid Re(z)>C\}=:D$ for every $C>0.$ For uniform continuity, I ...
0
votes
2answers
87 views

uniform convergence $\sum _{n=0} ^{\infty} \frac{\log (1+nx)}{nx^n}$ on $x \in (1,\infty)$

How could I prove that $\sum _{n=0} ^{\infty} \frac{\log (1+nx)}{nx^n}$ is not uniformly convergent on an interval $I=(1,+\infty)$? So far I have been thinking of proving that $f_n(x)=\frac{\log (1+...
0
votes
1answer
32 views

Is $f\left(x\right)=nx^{n}\left(1-x\right)$ uniformly convergent on $x\in\left[0,\:1\right]$ [duplicate]

Is $f_{n}\left(x\right)=nx^{n}\left(1-x\right)$ uniformly convergent on $x\in\left[0,\:1\right]$ We can see $f_{n}\left(x\right)\rightarrow f\left(x\right)=0$ point-wise, but $f_{n}\left(x\right)\geq ...
3
votes
1answer
66 views

Sequence of polynomials converging to $\frac{1}{z}$ [duplicate]

Is there a sequence of polynomials converging uniformly to $\frac{1}{z}$ in $K:=\{z\in\mathbb{C}\mid 1<|z|<2\}$? My first attempt was to use the theorem of Runge which would apply if $K$ would ...
1
vote
1answer
53 views

If $e^{2 \pi i f_n(x)}$ converges to $e^{2 \pi i f(x)}$ then $f_n$ converges to $f$ uniformly

Let $f_n$ be a sequence of smooth functions on $I=[0,1]$ such that $f_n(0)=0$. Now it is given that for given $\varepsilon>0,\ \exists n_0\in\mathbb{N}$ such that$ \|e^{2 \pi i f_n(x)}-e^{2 \pi i ...
3
votes
5answers
54 views

$f_{n}(x) = \frac{1}{n}$ if $x=n$ and $f_{n}(x) = 0$ if $x\neq n$. Is $\sum f_{n}(x)$ uniformly convergent?

For each $n \in \mathbb{N}$ and $x \in \mathbb{R}$ define $$f_{n}(x) = \frac{1}{n}\;\mathrm{if}\;x=n\quad\mathrm{and}\quad f_{n}(x) = 0\;\mathrm{if}\;x\neq n.$$ The series $\sum f_{n}(x)$ is ...
3
votes
1answer
60 views

Counterexample: Interchange Limit and Integral

Given continuous functions $f_n:[0,1]\to\mathbb R$ uniformly converging to 0 and $p>1$ such that $t^{-p}f_n(t)\in L_1[0,1]$, is then \begin{align*} \lim_{n\to\infty}\int_0^1\frac{f_n(t)}{t^p}dt=0 \...
2
votes
1answer
35 views

Does convergence in $W^{1,1}([0,1])$ imply uniform convergence?

Let $I=[0,1]$. Let $f_n \in W^{1,1}(I)$ be continuous and suppose that $f_n$ converge in $W^{1,1}(I)$ to a smooth function $f$. Does there exist a uniformly convergent subsequence of $f_n$? (There ...
3
votes
1answer
98 views

Is it true that $\lim_{n\to\infty}\int_{0}^{1}f_n(x)\,dx = \int_{0}^{1}f(x)\,dx$ in general and if $|f_n(x)|\le 2017$?

Let $f_n(x)$ and $f(x)$ be continuous functions on $[0, 1]$ such that $\lim_{n\to\infty} f_n(x) = f(x)$ for all $x \in [0, 1]$. Answer each of the following questions. If your answer is “yes”, then ...