For questions about uniform convergence of a sequence or a series of functions on a set. Also used with the tag [convergence].

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1answer
16 views

Infinite products of scaled indicator variables: almost sure convergence vs. uniform convergence of the sample mean

Let $\frac{X_i}{2}\sim Ber(0.5) \implies E[X_i]=1$, and let $Y_n=\prod\limits_{i=1}^n X_i$. Since the $X_i$ are iid, $E[Y_n]=1,\;\forall n<\infty$. However, something weird appears to be happening ...
0
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1answer
17 views

uniform convergence of diff'ble functions with derivatives converging in $L^1$

Suppose we have a sequence of differentiable functions $f_n$ from a closed interval $I\subseteq\mathbb{R}$ to $\mathbb{R}$ with the following properties: $f_n$ converges uniformly to some function ...
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3answers
86 views

What does it mean for a topology on a mapping space to correspond to a type of convergence?

I've been asked to prove that three different topologies on $Y^X = \{f : X \to Y | f \text{ is continuous} \}$ correspond to three different types of convergence, but I don't understand exactly what ...
0
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1answer
38 views

Iterating average

If $f$ is a continuous function $[0,1]\to \mathbb R$, we define a linear application $T$ as follows $$T(f)(x)=\begin{cases} f(0) & \mathrm{if }~ x=0 \\[0.2cm] \displaystyle ...
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1answer
24 views

Uniform convergence of series and continuity of $f$

This is from Ross's Elementary Analysis Textbook: The series $(2^{-n})(x^n)$ from $n=1$ to $n= \infty$ represents a continuous function on $(-2,2)$, but the convergence isn't uniform. He points out ...
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1answer
31 views

Weierstrass M-test help

I am supposed to use M-test on this one $$\sum \frac {n\ln (1+nx)}{x^n}$$ on $$1<x< \infty$$ But I face problems finding an appropriate $M_n$, thanks for help
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1answer
56 views

Determine whether $\sum\limits_{n=1}^\infty \frac{1}{n^x}$ converges uniformly on $(1,\infty)$

Detemine whether $\sum\limits_{n=1}^\infty \frac{1}{n^x}$ converges uniformly on $(1,\infty)$. My attempt: Upon attempting to use the Weierstrauss M-test I get ...
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2answers
39 views

limit and uniform convergence of sequence of function

I have a sequence of function $g_n$: $$ g_n(x) = xe^{-nx}, \qquad \text{for } x \in [0,\infty ) $$ I need to find the limit, and determine the uniform convergnece of the sequence. I think the ...
3
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1answer
22 views

uniform convergence of product of two uniformly convergent sequences of function

Given 2 sequences of functions $f_n$ and $g_n$ on an interval $[a, b]$. $f_n$ is uniformly convergent to $f$. $g_n$ is uniformly convergent to $g$. And there exists 2 real numbers $M_1$ and $M_2$ ...
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0answers
32 views

Application of Weistrass M- Test

Suppose that the series $\sum_{1}^{\infty}n|b_n|$ converges. Show that the series $\sum_{1}^{\infty}b_n \sin{nx}$ converges uniformly on R, and that it can be intergrated and differentiated term by ...
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2answers
35 views

uniform convergence of sequence of function

I have a sequence of function $f_n$: $$ f_n(x) = \sqrt{x^2 + \frac1n} \qquad \text{on the interval } [-1,1] $$ and $$f(x) = |x| $$ I need to prove that the sequence of functions $f_n$ is uniformly ...
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2answers
24 views

limit of sequences of functions and uniform convergence

I have two sequences of functions: $$ h_n(x) = (x-\frac1n)^2 \qquad \text{for } x \in [0,1] $$ $$k_n(x) = (x-\frac1n)^2 \qquad \text{for } x \in \mathbb R $$ I need to find their limits and ...
0
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1answer
23 views

uniform convergence of a function (continuous or differentiable or both?)

I have a function $S$: $$ S(x) = \sum_{n=1}^\infty \frac1{x+n^2} \\ \text{for} \ x \ge 0 $$ I need to determine if $S$ is continuous or differentiable or both.
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2answers
27 views

limits of sequences of functions and uniform convergence

I have a sequence of functions $(f_n)$ defined by $$ f_n(x) = \begin{cases} 0, & x=0,\\ x, & 0< x<1/n, \\ x^2, & 1/n <x. \end{cases} $$ I need to ...
0
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1answer
34 views

Show the sequence converges uniformly?

I was trying out some problems in the textbook and I came across this one which is bit tricky for me. Would anyone please be kind enough to help me out? Show that the sequence $\sum_{n=1}^\infty ...
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2answers
23 views

Convergence and uniform convergence of $f_n(x)$

The given sequence of functions is defined as $f_n(x) = \frac{x^n}{n+x^n}$ for $x\ge 0$ and $n = 1,2,\ldots$; let $f = \begin{cases} 0 &:0\le x \le 1\\ 1 &:x>1 \end{cases} $ My hypothesis ...
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3answers
33 views

Prove that function is not uniformly convergent

I am in a trouble of showing that the sequence of function $$f_n(x) = \frac{e^{-nx}}{3n^2x^2+1}, \ \ n = 1, 2, \cdots$$ does not converge uniformly on $[0,\infty)$. I tried to show that $f_n(x)$ ...
0
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2answers
29 views

Prove $\{h_n\}_{n=1}^\infty$ converges uniformly on $[a,1]$

Let $h_n: [a,1] \to R$ defined by: $h_n(x) = n^2x$ for $a \le x < \frac 1n$ and $h_n(x) = \frac 1x$ for $ \frac 1n \le x \le 1$. Show that $\{h_n\}_{n=1}^\infty$ converges uniformly on $[a,1]$ ...
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1answer
43 views

Proving uniform convergence of an integral-defined function on compact sets

If $f$ is a compactly supported smooth (infinitely differentiable) function into $[0, 1]$ such that $\int f(x)dx = 1$, $g$ is a continuous function, and $f_\epsilon(x) = ...
0
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1answer
33 views

$k_n(x) = \frac x{1+nx^2}$ uniform convergence

Prove that the sequence $\{k_n\}_{n=1}^\infty$ defined by $$k_n(x) = \frac x{1+nx^2}$$ for all $x \in R$ and each positive integer $n$, converges uniformly on R. $$$$ I know the definition of uniform ...
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1answer
24 views

Smooth Compactly Supported: Uniform Convergence?

Can you give me a hint how to check uniform convergence here: $$\varphi\in\mathcal{C}^\infty_0(\mathbb{R}):\quad\left|\frac{1}{h}(\varphi(t+h)-\varphi(t))-\varphi'(t)\right|\to0$$ Pointwise is clear ...
0
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1answer
50 views

Does the sequence$ f_n(x)=\dfrac{x}{1+nx^2}$ converge uniformly on $\mathbb{R}$?

I have to figure out if the sequence $f_n(x)=\dfrac{x}{1+nx^2}$ converges uniformly on $\mathbb{R}$
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3answers
31 views

Let $f_n(x)=xne^{-nx}$ for all $x \ge 0$ and $n \ge 1$. Show that $(f_n)$ converges to zero on $[0, \infty)$ pointwise but not uniformly

Let $f_n(x)=xne^{-nx}$ for all $x \ge 0$ and $n \ge 1$. Show that $(f_n)$ converges to zero on $[0, \infty)$ pointwise but not uniformly. I know we have to evaluate at $f_n(0)$ and where $x$ does ...
0
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3answers
14 views

Disprove uniform convergence of $x^n$ where $ x\in[0,1)$

So I'm looking at $f_n(x)=x^n ~~~x\in[0,1)$. It is obvious that it converges pointwise to $f(x)=0$. Is it now possible to disprove uniform convergence using the Uniform norm? This would mean I have to ...
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1answer
44 views

Let $f_n(x)=nx(1-x^2)^n$ on $[0,1]$ for $n\ge1$. Find $f(x)= \lim f_n(x)$. Is this a convergence uniform?

Let $f_n(x)=nx(1-x^2)^n$ on $[0,1]$ for $n\ge1$. Find $f(x)= \lim f_n(x)$. Is this a convergence uniform? I have to use the hint that $\lim_{n\to \infty}(1-\frac hn)^n=e^{-h}$. Do we show $|f(x)- ...
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2answers
18 views

Pointwise/Uniform Convergence of a function

Let $f_n(x)$=$nx ,\space \space x\in[0,\dfrac{1}{n}]\brace 0, \space\space x\in(\dfrac{1}{n},1]$. Find the pointwise limit of $(f_n)$ and check if $(f_n)$ is uniformly convergent.
0
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1answer
13 views

uniform convergence in sequence particularly infinite matrix

I am trying to understand uniform convergence and I almost understand the concept from the series of functions. But the I came up with the question that I do not know is it logical or does it have a ...
3
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2answers
51 views

Uniform convergence of $\sum f(x)^n$

Let $f:X\to\mathbb{R}$ be such that $\sup\{|f(x)|:x\in X\}<1.$ Show that $\sum_{n=1}^{\infty} f(x)^n$ converges and compute the sum.. Every value given by $f$ is less than one, then if ...
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1answer
42 views

Convergence of the sequence $f_n(x)=\frac{1}{1+nx^2}$

I'm trying to find the convergence of $f_n$ and $f_n'$ where $f_n(x)=\frac{1}{1+nx^2}$. From the function if I derivate the result is $f'n= -\frac{2nx}{(1+nx^2)^2}$. To determine $f$ I have to take ...
0
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1answer
22 views

Convergence of the sequence of maxima of a function sequence

Suppose we have a compact set $K \subset \mathbb{R}$ and a sequence of continuous functions $f_n: K \rightarrow \mathbb{R}$. Let $f$ be the uniform (and hence continuous) limit of $(f_n)_n$. Assume ...
3
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1answer
23 views

Convergence of monotone $f_n:[0, \infty) \rightarrow [0,1]$ to continuous, monotone $g$ is uniform

Let $g: [0, \infty) \rightarrow [0,1]$ be a continuous, monotone increasing function where $g(0)=0$ and $g(x)\rightarrow 1$ as $x \rightarrow \infty$. Also let $f_n:[0, \infty) \rightarrow [0,1]$ be a ...
2
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1answer
21 views

Prove that if $\sum g_k$ converges uniformly on a set $S$ and if $h$ is a bounded function on $S$, then $\sum hg_k$ converges uniformly on $S$

Prove that if $\sum g_k$ converges uniformly on a set $S$ and if $h$ is a bounded function on $S$, then $\sum hg_k$ converges uniformly on $S$ A little confused about this question, would love to ...
2
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1answer
34 views

Uniform convergence to exponential exercise

Yesterday I encountered the following exercise in a tutorial sheet from the University of Lyon : define a sequence of functions $(f_n)$ (with $f_n:[0,\infty) \to {\mathbb R}$) by ...
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0answers
25 views

Expansion of integrand before integration?

I have the following integral as part of a calculation $$\int_{-A}^{A} \int_{-A}^{A} \frac{1}{(z^2 + d^2)^3} dz dx, $$ where $A$ is a constant. I am given the condition $d \gg z$ so I am wondering if ...
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0answers
23 views

Condition for uniform convergence of Fourier series

Let $f$ be a Lebesgue summable periodic function on $[-T/2,T/2]$. I read in Kolmogorov-Fomin's (p.414 here) that if $f$ is bounded on a set $E\subset[-T/2,T/2]$ and for any $\varepsilon>0$ there is ...
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1answer
33 views

Is the uniform limit of continuous functions continuous for topological spaces?

Question: [For notations see the context given below.] If $E$ is a topological space and $(f_n)_{n\geq1}$ is a Cauchy sequence in $\mathcal{C}\mathcal{B}(E)$, then we know $(f_n)_{n\geq1}$ ...
2
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1answer
57 views

How do I prove that $\sum f_n g_n$ converges uniformly?

Let $E$ be a subset of $\mathbb{R}$. Let $\{f_n\}$ be a sequence of functions on $E$ such that $\sum f_n$ converges uniformly on $E$. Let $\{g_n\}$ be a sequence of functions on $E$ such that ...
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1answer
34 views

Proving that a series is not uniform convergent but convergent.

How would I show that: $$\sum_{n=1}^{\infty} \frac{y^2}{(1+y^2)^n}$$ converges for all $|y| \leq 1$ but that this does not converge uniformly? I wrote out the series which is: $$\frac{y^2}{(1+y^2)} ...
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0answers
29 views

In what sense does the Maclaurin series for $\ln(x+1)$ converge on $[-1,1]$? [duplicate]

This is regarding Modes of Convergence. The question is as follows. Given $f(x) = \ln(x+1)=\sum_{n=1}^{\infty}(-1)^{n-1}\frac{x^n}{n}$. Let $f_N(x)=\sum_{n=1}^{N}(-1)^{n-1}\frac{x^n}{n}$ ...
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0answers
5 views

Stationary distribution for given Markov Chain

Would any body mind to tell me, whether MC shown in the above figure is time reversal or not, and how to calculate the stationary distribution of the following MC.
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2answers
53 views

Doubt on Arzela-Ascoli theorem

Consider a sequence of equicontinuous and uniformly bounded functions on a compact set. Under which condition I can say that it has a unique uniformly convergent subsequence ? Or, atleast uniform ...
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0answers
38 views

A question on convergence of derivative of power series

This is a question from Fourier Analysis with Applications by Folland. First we write Fourier series for $$e^{\theta}=\sum c_ne^{in\theta}$$ We differentiate this series term by term to obtain ...
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2answers
36 views

Necessary Condition for uniform convergence of series of functions

I would like to make sure that the follwing is a necessary condition for uniform convergence of series of functions: :Let $$ \sum _{n=1}^{\infty }f_{n}(x) $$ be a series of functions, than a ...
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3answers
72 views

Does $\{(1-x)x^k\}$ converge uniformly on $[0,1]?$

The convergence is clearly pointwise to the function $f(x) = 0$, but I'm not sure if this convergence is uniform. I wanted to prove that it wasn't uniform because I had a feeling it wasn't, but I'm ...
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0answers
47 views

Contraction Mapping Principle for System of Equations

Show that the system of equations: $x_1 = \frac{1}{4}x_1 - \frac{1}{4}x_2 + \frac{2}{15}x_3 +3 $ $x_2 = \frac{1}{4}x_1 + \frac{1}{5}x_2 + \frac{1}{2}x_3 -1 $ $x_3 = -\frac{1}{4}x_1 + ...
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votes
2answers
22 views

Uniform convergence to $0$ of the $n$-th iterate of $f$

Let $a>0$ and $f:[-a,a]\to[-a,a]$, continuous, such that $$\forall x\in[-a,a]-\{0\}:\left\vert f(x)\right\vert <\left\vert x \right\vert.$$ Denote $f^n$ the $n$-th composition of $f$. ...
0
votes
1answer
36 views

composition of uniformly convergence sequence with continuous function, is uniformly convergence?

Let $(f_n)$ be a series of functions in $C[0,1]$ that uniformly converge to a continuous function $f\in C[0,1]$. a. Let $g: [0,1]\to [0,1]$ be a continuous function. Is it true that $f_n\circ g$ ...
3
votes
0answers
31 views

Fourier transform of $\frac{1}{r}$ (Coulomb potential)

When calculating the Fourier transform of a function of the form $f(\vec{r}) = \frac{1}{4 \pi \left|\vec{r}\right|}$, one encounters the problem that the resulting integral does not converge, i.e. ...
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0answers
20 views

Convergence of Fourier Sine Series for Gerneral Continuous Function

This is my question: How do I should that, for $f \in C[0,\pi]$ with $f(0) = f(\pi) = 0$, the Fourier sine series $$\tilde f_n = \sum_{r=0}^n b_r \sin(r s)$$ converges uniformly to $f$ on ...
0
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0answers
12 views

Uniform convergence and Riemann integration (including improper integrals)

Let, for each $n\in\mathbb{N}$: $f_n:S\to \mathbb{R}$ a Riemann-integrable function over $S$ where $S\subseteq \mathbb{R}^N$ is bounded. Suppose that $f_n$ converges uniformly to a function $f:S\to ...