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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35 views

How to take the limit of the improper integral of a sequence of functions

Suppose $f_1, f_2, . . .$ are (Riemann) integrable functions. Then what is the $\epsilon$ definition of $$\lim_{n \rightarrow \infty} \lim_{M \rightarrow \infty} \int_{0}^{M} f_n(x) dx = L $$ for $L ...
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0answers
20 views

Is functional analysis a tool to prove uniform convergence? [on hold]

I am very new to functional analysis. Is functional analysis a tool to prove uniform convergence ?
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1answer
26 views

What kind of convergence is $\sum |f_n|$?

We know that $\sum |f_n|$ converges on $E\subseteq \mathbb R$ then $\sum f_n$ is said to converge absolutely on $E$. But in terms of pointwise or uniform convergence, I am willing to know what kind ...
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0answers
23 views

How many sequence of functions are there to converge pointwise to a given function on $E\subseteq \mathbb R$?

yesterday night, I was studying sequence of functions in $\mathbb R$ and then this question came to mind. When a sequence of real valued function is given, we can find out it's pointwise limit ...
2
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1answer
21 views

If $\{f_n\}$ and $\{g_n\}$ be uniformly convergent sequences of bounded functions on S, then $\{f_ng_n\}$ is uniformly convergent on S.

If $\{f_n\}$ converges uniformly to $f$ and $\{g_n\}$ converges uniformly to $g$, does it mean $\{f_ng_n\} $ will converge uniformly to $fg$? I am absolutely stuck on this. Please help.
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0answers
39 views

Darboux Integrable Functions vs. Uniform Closure of Simple Functions

Is there a Darboux integrable function $f:[0,1]\to E$ with values in a Banach space $E$ that is not the uniform limit of simple functions $s=\sum_\alpha\chi_{A_\alpha}$ taken over finite sums with ...
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0answers
166 views

Proof on why $0-1+2-3+4-\ldots\neq-1/4$

When reviewing my notes on series' convergence, I thought of applying a workaround on why $\sum_{n=0}^{\infty}(-1)^nn$ should or shouldn't be $-1/4$ (I recalled this page). I started by considering ...
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0answers
13 views

Uniform convergence with respect to a parameter

So this is just a notational question. Assume one has a sequence $f_n\to f$ uniformly, where $f_n,f:X\to Y$ for some metric/Banach spaces $X,Y$. Now suppose that $f_n$ and $f$ depend on a parameter ...
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1answer
30 views

$f$ is uniformly continuous on $(a, b]$ implies $\lim\limits_{x\to a^+} f(x)$ exists and finite

I need to prove: $f$ is uniformly continuous on $(a, b]$ implies $\lim\limits_{x\to a^+} f(x)$ exists and finite Now, I already have a sketch for the proof: Let $\{x_n\}$, a sequence such ...
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1answer
25 views

Uniform convergence of a sequence of polynomial logarithm

Let $P\in \Bbb{C}[X]$ of degree $d\ge 2$. For $n\in \Bbb{N}$ (include $O$). Denote by $P^n$ the $n$-th composition and $g_n: z\mapsto \frac{1}{d^n}\log(\max \{1,\vert P^n\vert\})$. Show that ...
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1answer
39 views

Complex Analysis Weierstrass M-Test

Prove that each of the following series converges uniformly on the corresponding subset of $\mathbb C$: $$\begin{align*} \text{(a)} \; & \sum_{n=1}^\infty \frac{1}{n^2 z^{2n}}, & & ...
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1answer
67 views

Prove the uniform convergence of the following function series

Prove that $$\sum_{k=0}^{\infty}\left(1+\frac{k}{x}\right)^{-x}$$ is uniformly convergent on $x\in\left[a,\infty\right).$ According to the equality, $$\frac{x}{1+x}<\ln(1+x)$$ we have, ...
2
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1answer
46 views

$f_n(x) = x(1-x)^n$ converges uniformly to $0$ on $[0, 1]$?

I am trying to show that the sequence of functions $f_n(x) = x(1-x)^n$ converges uniformly to $0$ on $[0, 1]$. Well at $0$ and $1$, $f_n(x) = 0$ for all $n$. So let $x \in (0, 1)$. $f_n(x)$ ...
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1answer
69 views

Is the space of continuous and bijective functions $f\colon [0,1] \to [0,1]$ complete?

Let $X$ be the space of continuous and bijective functions $f$, such that $$ f\colon [0,1] \to [0,1] \quad , \quad f(0)=0 \quad , \quad f(1)=1 \, .$$ Is $X$ complete (under the supremum norm $ ...
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0answers
38 views

Uniform convergence of $f_n(x)=\frac{n}{2}\int_{x-\frac{1}{n}}^{x+\frac{1}{n}}f(t)\ dt$

Let $f:\mathbb{R}\longrightarrow\mathbb{R}$ be a continuous function. I need to prove that a sequence $(f_n)$ defined as $$f_n(x)=\frac{n}{2}\int_{x-\frac{1}{n}}^{x+\frac{1}{n}}f(t)\ dt$$ is uniformly ...
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1answer
60 views

Convergence in $C(X)$ is uniform convergence.

I read this the convergence in $C(X)$ is uniform convergence. Where $X$ is compact hausdorff topological space and $$C(X)=\{f:X\to\mathbb{C}\;\mid \; f\ \text{is continuous}\}$$ And ...
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1answer
32 views

$F_n \overset{w}{\to} F$, and $F$ is continous. Show that $F_n$ converges to $F$ uniformly on $\mathbb{R}$

$\{F_n\}$ and $F$ are distribution functions, and $F$ is continuous on $\mathbb{R}$. If $F_n$ converges weakly to $F$, show that $$ \sup_x | F_n(x) - F(x) | \to 0, n \to \infty $$ I know that ...
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2answers
27 views

Uniform convergence on compact sets allows switching the limit and the integral.

Why does uniform convergence on compact sets allows switching the limit and the integral?
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1answer
71 views

Show that there is sequence of homeomorphism polynomials on [0,1] that converge uniformly to homeomorphism

Let $f:[0,1]\rightarrow [0,1]$ be a homeomorphism. Show that , there exists a sequence of polynomials $$(P_n(x))_n$$ such that $P_n(x)$ converge uniformly to $f$ on $[0,1]$ and every $P_n(x)$ is a ...
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2answers
32 views

Prove if $f_n(x)=n^2 x^n(1-x)^2$ converges pointwise and/or uniformly on $I=[0,a]$, where $a<1$

I have the following problem and I am kind of stuck in the second part of it. So the problem says... "Find the pointwise limit of the given sequence and determine whether or not the convergence is ...
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0answers
38 views

Non stationary solutions of the PDE $u_t + u_x = u_{xx}$

Problem. Consider the PDE $$ u_t + u_x = u_{xx}, \qquad (t,x) \in (0,+\infty) \times (0,1). $$ (i) Write the unique solution $\overline{u}=\overline{u}(x)$ which does not depend on time and ...
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0answers
55 views

If $f_n(t):=f(t^n)$ converges uniformly to continuous function then $f$ constant

Let $f \colon [0,1] \to \mathbb R$ be continuous and let $f_n$ be defined by $$ f_n(x):=f(x^n), \qquad x \in [0,1], \,\, n \in \mathbb N. $$ Suppose there exists a continuous function $g$ on ...
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0answers
55 views

Let $f_n(x)=x^n$ on $I=[0,1)$, does $\sum\limits_{n=1}^\infty f_n(x)$?

so I have a problem with 4 convergence questions. I have done the ones where I have to use other methods such as the ratio test and they were OK, but I cannot understand this one in particular. The ...
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1answer
35 views

Are the family of given nice functions $f\subset C^0(I,[0,1])$ equicontinuous?

The family of continuous functions $f\in\mathcal{F}$ are defined on a closed subset of real numbers $I\subset\mathbb{R}$ as follows: \begin{equation} f(y) = \begin{cases} 0, &l(y)<\rho \\ ...
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2answers
34 views

Uniform Convergence and Sup

Let $X \subset \mathbb{R}^d$ be open and let $(f_n)_{n \in \mathbb{N}} : X \to \mathbb{C}$ be a sequence of complex-valued functions. Suppose $f_n \to f$ uniformly on $X$. Fix $a \in X$. I would like ...
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0answers
33 views

Uniform convergence result in proof of second-derivative formula

This is a fairly basic analysis question. Consider a continuous function $f: \mathbb{R} \to \mathbb{R}$ which is twice differentiable at a point $x$. If necessary, also assume that $f \in ...
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1answer
41 views

Show that $\lim_{s \to \infty}F_s(t) = F(t)$ uniformly for $t \in (0,+\infty)$

Given the following functions: $$ F(t)= \int_0^\infty e^{-tx}\dfrac{\sin{x}}{x}\,dx, \quad t>0$$ $$ F_s(t)= \int_0^s e^{-tx}\dfrac{\sin{x}}{x}\,dx, \quad t \geq 0, s>0$$ Show that $\lim_{s \to ...
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1answer
41 views

A counter-example to Dini's Theorem (after removing a hypothesis)

Recall Dini's Theorem: Let $K$ be a compact metric space. Let $f: K\to\mathbb{R}$ be a continuous function and $f_{n}: K\to\mathbb{R}$, $n\in\mathbb{N}$ be a sequence of continuous functions. ...
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2answers
45 views

How to show that $f_n(x) = \frac{1}{1+nx^2}$ on $\mathbb{R}$ is not uniformly convergent

I need to understand pointwise and uniform convergence of function series and have several exercise examples. Most of them I was able to solve but I got problems showing that: $$ f_n(x) = ...
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2answers
37 views

It is a question about uniform convergence of a function.

I solved this problem . Is my answer a correct ? $$ x \in [0,\infty) ,\lim_{n \to \infty} \frac{nx}{1+n^2x^2}=0\ \ \ $$ Is $ \frac{nx}{1+n^2x^2} $ converged uniformly on $0$ ? My solution $$ ...
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2answers
62 views

Uniform convergence of $f^2_n$ when $f_n$ converges uniformly

Let $(f_n)$ be a sequence of functions that converge uniformly to $f$ on the interval $I$. Prove or disprove: $f^2_n \to f^2$ uniformly on I. I was almost certain this claim is false but was unable ...
3
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2answers
51 views

Show if the series $f(x)=\sum\limits_{k=1}^\infty \frac{1}{k} \sin(\frac{x}{k})$ converges uniformly or not.

So this is part of a different problem. The book and my professor say that the series $f(x)=\sum\limits_{k=1}^\infty \frac{1}{k} \sin(\frac{x}{k})$ converges uniformly on [0,1] by the Weierstrass ...
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2answers
45 views

$f_n:[0,1]\to [0,1]$ continuous, $f_n\to f$ uniformly, prove: ${\frac1n}\sum_{k=1}^{n}{f_k} \to f$ uniformly

I was able to prove, hopefully correctly, that the sum converges uniformly. But, I'm not sure how to show it converges uniformly specifically to $f$. The way I proved it converges uniformly was by ...
2
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1answer
50 views

Answer check on two series

I want to determine if these two are absolutely convergent, conditionally convergent or simply divergent. 1) $$\sum_{n=2}^\infty \left(\frac xn - \frac x{n-1}\right)$$ $$= \frac x2 - \frac x1 + ...
3
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1answer
39 views

The uniform limit of a sequence of functions

Which of the following sentences are not always correct? $A.$ The uniform limit of a sequence of differentiable functions is integrable. $B.$ The uniform limit of a sequence of integrable functions ...
2
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1answer
24 views

Prove that $f_n$ is uniformly bounded on $E$ and $f$ is a bounded function on $E$

I have the solution for the following problem, but I don't understand most of it. Question: A sequence of functions $f_n$ is said to be uniformly bounded on a set $E$ iff there exists $M>0$ such ...
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1answer
32 views

two question about uniformly convergence of sums

First question: I want to show that this series : $$\sum _{k=0}^{\infty }\:\frac{x^2}{\left(1+x^2\right)^k}$$ converges uniformly in $[a,\infty)$ for $a>0$. I thought about the M test and i ...
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1answer
27 views

Question about example i saw for uniform convergence

This is the sequence $f_n(x) = nxe^{−nx}$ on $(0,\infty)$. Well, i can see that $f_n$ tend to $0$ pointwise for every $x \in (0,\infty)$. But if i want to check if $f_n$ tend to $0$ uniformly in ...
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1answer
45 views

how to show a series is not uniformly converges?

Well , this is the function sequence:$$f_k\left(x\right)\:=\:\frac{1}{k+k^2x}$$ I want to prove that there is no uniform convergence for $\sum _{k=1}^{\infty }\:\frac{1}{k+k^2x}$ , in ($0$,$\infty $). ...
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3answers
201 views

Evaluate the limits

$$\lim_{p\rightarrow\infty}\int_0^1e^{-px}(\cos x)^2\text{d}x$$ I tried to prove the integrand is uniformly convergent so that the limit and integral can be exchanged. But I failed. Any ideas?
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1answer
53 views

The limit of $f(x,y)=\sum_{n=1}^\infty \frac{x}{x^2+yn^2}$ as $x\to \infty$.

Let $$f(x,y)=\sum_{n=1}^\infty \frac{x}{x^2+yn^2}, \ \ \ \ y\neq 0$$ how to show that for each $y\neq 0, g(y)=\lim_{x\rightarrow \infty} f(x,y)$ exists, evaluate $g(y)$. Then prove the convergence ...
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1answer
40 views

$\lim_{n\to \infty} \int^{\infty}_{-\infty} f_n(x) dx\, = \int^{\infty}_{-\infty} f(x) dx\,$

If $\{f_n(x)\}$ is sequence of continuous function on $\mathbb{R}$ converging uniformly to $f(x)$,then does $\lim_{n\to \infty} \int^{\infty}_{-\infty} f_n(x) dx\, = \int^{\infty}_{-\infty} f(x) ...
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3answers
44 views

If $f_n$ is a sequence of integrable functions that converges uniformly, is it's limit integrable?

Well, i want to prove that if $f_{n}$ is sequence of integrable functions on $[a,b]$, and $f_{n}\rightarrow f$ uniformly, then $f$ is also integrable on $[a,b]$. So far, i divided the proof for ...
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0answers
24 views

How to find out convergence of sequence of functions in short time.

I was studying about sequence of functions and about uniform and pointwise convergence of these series. To prove or disprove the convergence of sequence of functions we have several methods like by ...
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0answers
27 views

A question about uniform convergence in proof that $L^\infty$ is Banach.

I was reading this post Understanding proof of completeness of $L^{\infty}$ and it is mentioned that the sequence $(f_n)_{n\in\mathbb N}$ converges uniformly on a conegligible set $N^C$. Could someone ...
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0answers
21 views

Interchange of limits in computation with Gamma

On page 164 of Stein and Shakarchi's Complex Analysis there is the following computation: (here $0 < s < 1$, and line 1 -> 2 is a preceding lemma) \begin{align*} \Gamma(1 - s)\Gamma(s) &= ...
4
votes
2answers
82 views

Limit of $f_n(x) = n\log(\frac{nx+1}{nx-1})$ and its uniform covergence

Let $f_n(x) = n\log(\frac{nx+1}{nx-1})$. What is the pointwise limit of $f_n$ for $x \in[1,\infty]$? Is it uniformly convergent? $$\lim_{n\longrightarrow ...
0
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0answers
18 views

analyticity of an integral

Studying $$f(z)=\int_0^1 g(z,x)\ dx $$ where $g(z,x)$ is analytic in the open unit disk $D$ for all $x\in [0,1]$ and continuous for $|z| <1$ and $0\leq x \leq1$.Now, $$\lim_{n\to\infty} [1/n ...
1
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1answer
19 views

Find its pointwise limit and determine if its uniform

$f_k:[0,1]\to\Bbb R,$ $f_k(x)=\begin{cases}kx,x\in[0,\frac 1k]\\1,x\in[\frac1k,1]\end{cases}$ Find the pointwise limit of $(f_k(x)) $. And is this convergence uniform? for x=0 $f_k(0)=0$ and x=1 ...
5
votes
1answer
63 views

Uniform convergence of $\sum\limits_{n=1}^\infty \sin \left(\frac{x}{n^2}\right)$

Can someone please verify my answers? Consider the series $$\sum\limits_{n=1}^\infty \sin \left(\frac{x}{n^2}\right)$$ Prove that the series converges uniformly on the bounded interval $[-M, ...