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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].

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Qualifying Exam in Real Analysis question 2 [on hold]

Let $f_{n}(x)= \int_{1/2}^{x}arctan^2(t/n)dt$ where $n=1,2,...$ (a) Show that $\sum_{n=1}^{\infty} f'_{n}(x)$ is uniformly convergent on $[-1,1].$ (b) Show that $g(x)=\sum_{n=1}^{\infty} f_{n}(x)...
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Uniform convergence of $f_n(x) = \frac{nx}{(2+nx)(4+x^2)}$

I need to study the uniform convergence of $$f_n(x) = \frac{nx}{(2+nx)(4+x^2)}$$ on the interval $[2,+\infty)$ I've shown that on $[0,+\infty)$: at $x =0$ $f_n(0)=0 \xrightarrow{} 0$ at $x \neq 0$...
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1answer
21 views

Proof of equicontinuity via uniform convergence

I am studying for an entrance exam in August and I am trying to review and cover the real analysis section. I came across this problem on one of the past exams and I am having some trouble with it. ...
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1answer
47 views

Uniform Convergence of a subsequence over an arbitrary interval

I am getting ready for an entrance exam that is in August and I am trying to get a head start on the analysis section. I recently came across this problem and it is giving me some trouble. Consider ...
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A difficulty in understanding the proof of Dini`s theorem.

The question and its answer is given below: But I am not understanding the sequence of steps in the solution what are the ideas he using, for example: 1- In the second line why he is using a ...
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1answer
31 views

Determining whether the sequence ${f_{n}}$ converges uniformly on the set $A.$

Determining whether the sequence ${f_{n}}$ converges uniformly on the set $A$ \begin{equation*} f_{n}(x) = g(nx),\quad g(x)= \begin{cases} x & \text{if }0 \leq x < 1/2,\\ 1-x & \text{if } ...
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1answer
29 views

Convergence of an exponent function

Following the answer to this MSE question, it is claimed that we easily show the following convergence: $$ \text{exp}(-\sqrt{\lambda}it + \lambda (e^{it/\sqrt{\lambda}}-1)) \rightarrow_{\lambda \...
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1answer
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Techniques for proving non-uniform-convergence of improper integrals

I have been reviewing uniform convergence of series and improper integrals with parameters. Working through examples I found it easier to prove uniform convergence (using M-test, Dirichlet-Abel test, ...
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2answers
30 views

Does pointwise convergence to a continuous function in compact set imply uniform convergence? [duplicate]

Suppose $K$ is a compact set in $\mathbb{R}$, $f_n$ is a sequence of functions such that $f_n$ converges pointwise to a continuous function $f$, does it imply $f_n$ converges uniformly to $f$ in $K$ ? ...
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Given pointwise convergence of $f_n \rightarrow f$, does $\lim_{\epsilon \rightarrow 0} f(x+\epsilon) = f(x)$ implies uniform convergence?

I was wondering if provided that the partial sums $f_n$ converge to $f$ matching pointwise convergence, what this would imply: $$ \lim_{\varepsilon \rightarrow 0} f(x+\varepsilon) = f(x) $$ If that ...
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57 views

Prove that if $F: X \longrightarrow C(Y, Z)$ is continuous, then $f : X \times Y \longrightarrow Z$ is continuous. [closed]

The space $C(Y, Z)$ is supposed to be a topological space with topology of uniform convergence. I wanted to prove the fact using the definition of continuity, but I've failed. I really don't ...
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proof that a sequence $\{f_n\}$ in $(C(G,\Omega),\rho)$ converges to f if and only if convergence is uniform on compact subsets of G

I want to show that a sequence $\{f_n\}$ in $(C(G,\Omega),p)$ converges to $f$ if and only if $\{f_n\}$ converges uniformly on compact subsets of G. Preliminary definitions: Here $\rho$ is defined ...
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1answer
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Proving a sequence of functions converges, is differentiable, etc.

I have this question asking to prove that a sequence of functions is differentiable, pointwise and uniformly convergent, and something regarding the equality of limits of the function's derivative. ...
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Pointwise and uniform convergence $\sum_{n=1}^{\infty}\frac{(n+1)^n-n^n}{n!}x^{n^n}$

I am in deadlock studyibg the pointwise and uniform convergence of the following series: $$\sum_{n=1}^{\infty}\frac{(n+1)^n-n^n}{n!}x^{n^n}$$ Maybe should I handle it as a power series? But how? Any ...
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28 views

Show that there exists no sequence of complex polynomials which converges uniformly to $\frac{1}{z^2}$ on the annulus $1<|z|<2$.

Show that there exists no sequence of complex polynomials which converges uniformly to $\frac{1}{z^2}$ on the annulus $1<|z|<2$. Suppose there exists a sequence of complex polynomials $f_n(z)=...
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1answer
24 views

Uniform convergence as $\epsilon\to 0^+$

Reading some lectures on Hamilton-Jacobi PDE theory I found some terminology that I really don't understand. Let $\Omega$ be an open subset of $\mathbb{R}^n$. Suppose that $u_\epsilon:\Omega\to \...
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75 views

Uniform convergence of this series?

Let us consider : $f_n : \mathbb{R}\ni x \mapsto \frac{x}{(x^2+n^2)\log(n)}\in\mathbb{R}$ for $n> 1$. I need to prove that $\sum _{n\ge 0}f_n$ is uniformly convergent. I've already proved that it ...
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34 views

Uniform convergence after transforming?

I face a problem now, it is known that $f_n(x)$ converge to $+\infty$ pointwise as $n\rightarrow+\infty$, but I want to prove $f_n(x)$ converge to $+\infty$ uniformly. I can prove $\arctan f_n(x)$ ...
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Uniformly convergent on each ccmpact set of $\mathbb R$ but not on $\mathbb R$

As the title says, I am looking for a sequence of function which is uniformly convergent on all compact sets of $\mathbb R$ but not on $\mathbb R$. I thought $f_n(x) = x/n$ is such a function since ...
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What is the main difference between pointwise and uniform convergence as defined here?

I have a little confusion here. I have seen the following several times and seem to be a bit confused as to differentiating them. Let $E$ be a non-empty subset of $\Bbb{R}$. A sequence of functions $\...
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1answer
36 views

Uniform convergence of $\frac{y/(2N)}{\sin(y/(2N))}$ towards 1

I can't come up with a proof, why $f_N(y) := \frac{\frac{y}{2N}}{\sin\left(\frac{y}{2N}\right)}$ converges uniformly against $1$ for $y\in(0,\pi),\ N\to\infty$. I would be thankful for any advice.
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absolut and uniform convergence of complex power series on an open disk imply absolut and local uniform convergence on the whole disk of convergence [duplicate]

I'm stuck in the proof of the following theorem of complex analysis: There is a $R\in\mathbb{R}_{0}^{+}\cup\{\infty\}$ s.t.: (i) $\sum_{k=0}^{\infty} a_k (z-z_0)^k$ converges absolutely and ...
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61 views

If $\int_{a}^{b} f(x) g(x) dx=0$ with $f(x)=\sum_{i=0}^{\infty} a_n x^n$, can I integrate term by term?

Suppose $\int_{a}^{b} f(x,l) g(x) dx=0...(1)$ with Taylor series of $f(x)=\sum_{i=0}^{\infty} a_n x^n$, where $a_n$ depends upon parameter $l$. I want to construct non-zero bounded integrable function ...
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1answer
10 views

Uniform Cauchy and convergence clarification.

I have a doubt in the definition of uniform Cauchy sequence of functions in the domain $A$. We say that the sequence $(f_n)_{n\in\mathbb{N}}$ is uniformly Cauchy when for any positive $\epsilon$ we ...
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1answer
43 views

Prove that $\lim\limits_{n\to\infty} \int^{b}_{a}\left(\sum^{n}_{k=1} (f_{k}(x))^{n}\right)^{1/n}dx=\int^{b}_{a}f(x) dx$

I found this question interesting from "Introduction to Analysis by William R. Wade" and decided to share its proof and also like to learn your approach to proving it. Supposing that $f_k:[a,b]\to \...
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2answers
56 views

Prove that $ B_{n}(x)=\sum^{\infty}_{k=0} \dfrac{(-1)^k}{k!(n+k)!} \left(\dfrac{x}{2}\right)^{n+2k}$ converges uniformly.

Let $n\geq 0,$ be a fixed. The Bessel function of order $n$ is the function defined by \begin{align} B_{n}(x):=\sum^{\infty}_{k=0} \dfrac{(-1)^k}{k!(n+k)!} \left(\dfrac{x}{2}\right)^{n+2k} .\end{...
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1answer
39 views

Uniform convergence of power sequence

I'd like to prove that the power sequence $f_n(x) = x^n$ doesn't converges uniformly on $[0,1]$, but it does on $[0,a]$ if $a < 1$. My textbook states that a sequence of functions converges ...
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Different ways of proving that $|\sum^{\infty}_{k=1}(1-\cos(1/k))|\leq 2 $

I've found two ways of proving that \begin{align} \left|\sum^{\infty}_{k=1}\left[1-\cos\left(\dfrac{1}{k}\right)\right]\right|&\leq 2 \end{align} Are there any other ways out there, for proving ...
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82 views

Uniform convergence of $ \zeta'(x) = \sum \frac{\ln(k)}{k^x}$

As a part of a proof that the $\zeta$ function is differentiable I want to check that the series: $$-\sum_{k=1}^\infty \frac{\ln(k)}{k^x}$$ Converges uniformly for $x \in (a, \infty)$ whenever $a>1$...
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1answer
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Proving the uniform convergence of the average sequence of $f_n(x)=\sin(nx)$

I was asked to prove that: 1) $f_n(x)=\sin(nx)$ does not converge pointwise. 2) The average sequence of $f_n(x)=\sin(nx)$ is uniformly convergent. I secceed to prove the first part but I cannot ...
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38 views

Proving $f_{s}(t) = \frac{e^{-st}}{t}$ is uniformly convergent

I am new to proving uniform convergence and am trying to show that $$f_{s}(t) = \frac{e^{-st}}{t}$$ is uniformly convergent $\forall$ $s\geq 0$. I am pretty sure that $t\epsilon \mathbb{R}$. I first ...
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Let $f : \mathbb{R} \to \mathbb{R}$ be approximated arbitrarily well by polynomials of bounded degree. Prove $f$ is a polynomial.

I am trying to prove that if a function $f : \mathbb{R} \to \mathbb{R}$ can be approximated arbitrarily well by polynomials of bounded degree, then $f$ itself must be a polynomial. For starters, let $...
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2answers
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Hint requested for: If $\sum_{n=0}^{\infty} a_n x^n$ converges for some $x_0$, then it converges uniformly and absolutely on $[-a, a]$ with $a<|x_0|$?

I would like to prove If $\displaystyle\sum_{n=0}^{\infty} a_n x^n$ converges for some $x_0$, then it converges uniformly and absolutely on $[-a, a]$ with $0<a<|x_0|$. (Sorry not enough ...
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In functional analysis, is there a commonly accepted short-hand notation for specific types of convergence?

In math literature on functional analysis I found various short-hand notations for specific types of convergence, e.g. a single right arrow for pointwise convergence $$f_n(x) \underset{n \to \infty}{\...
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1answer
26 views

An extension of a corollary of the Arzela-Ascoli theorem for smooth functions

I'm trying generalize this corollary for the case which the sequence of functions $\{ f_n \}_{n \in \mathbb{N}}$ are defined on a bounded domain (open and connected) $U \subset \mathbb{R}^m$ ($m \geq ...
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36 views

Uniform convergence of $f_n(x)=\frac{\sqrt{1+{(nx)}^2}}{n}$ and ${f_n}^\prime$

Discuss the uniform convergence of $f_n(x)=\frac{\sqrt{1+{(nx)}^2}}{n}$ and its first derivative on real line. I think both $f_n$ and ${f_n}^\prime$ do not converge uniformly. If we put $x=0,1$ in $...
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Uniform convergence of $f_n(x) = \sqrt{x^2 + \frac{1}{n^2}}$, solution verification

Is my reasoning right? I have $f_n(x) = \sqrt{x^2 + \frac{1}{n^2}}$ for $x \in \mathbb{R}$, so I conclude that it's pointwise convergent $f_n \to |x|$, and moreover it's uniformly convergent to $|x|$, ...
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1answer
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Uniform Continuity of sum of a series of functions

Let $f(x)=\sum_{n=1}^{\infty}f_n(x)$, where $$f_n(x)=\begin{cases}n(x-n+\frac1{n}),\ \ x\in[n-\frac1{n},n]\\n(n+\frac1{n}-x),\ \ x\in[n,n+\frac1{n}]\\0,\ \ \text{otherwise}\end{cases}$$ Then, is $f(x)...
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Uniform convergence of the exponential series on a bounded interval

Show: The function series $$\sum _ {k=0} ^\infty \frac{x^ k} {k!} $$ converges uniformly on each bounded interval in $\mathbb{R}$. Discussion I think a good approach will be to deploy the Cauchy ...
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2answers
23 views

Uniform convergence exercise

This was a question asked to me in an exam which I couldn't answer: Let $g:[0,1]\to \Bbb R$ be a continuous function, such that $0<g(x)<1$ for all $x\in[0,1]$. Let $f_n :[0,1] \to \Bbb R$ be a ...
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1answer
25 views

If a series of functions satisfies the Cauchy criterion uniformly, then the series converges uniformly on $S$

Problem: If a series $\sum_{k=0}^\infty g_k$ of functions satisfies the Cauchy criterion uniformly on a set $S$, then the series converges uniformly on $S$. If the series satisfies the Cauchy ...
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1answer
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Using proof by contradiction to prove Dini's theorem

Does this proof sound right? Thanks Dini's Theorem: If $f$ and $f_n$ are continuous functions on $[a,b]$ such that $f_n \leq f_{n+1} \forall n \geq 1$ and $(f_n)$ converges to $f$ pointwise, then $(...
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2answers
58 views

Is $\sum _{n=1} ^ \infty \frac{1}{(x+\pi)^2 \cdot n^2 }$ uniformly convergent on $(-\pi , \pi)$?

Is this series uniformly convergent on $(-\pi , \pi)$: $$\sum _{n=1} ^ \infty \frac{1}{(x+\pi)^2 \cdot n^2 }\,?$$ My Attempt: If the series were convergent we would have got a natural number $k$ ...
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2answers
29 views

Is $f_{n}$ is analytic on $(a, b)$ and $f_{n} \rightarrow f$ uniformly on $(a, b)$ then is $f$ analytic on $(a, b)$?

Is $f_{n}$ is analytic on $(a, b)$ and $f_{n} \rightarrow f$ uniformly on $(a, b)$ then is $f$ analytic on $(a, b)$? Intuitively, I think that the answer is no. I know that the statement holds for ...
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1answer
42 views

Suppose we have continuous $f_{n} : D \rightarrow \mathbb{R}$ converging to $f : D \rightarrow \mathbb{R}$ uniformly on $D$. Then, is $f$ continuous? [duplicate]

Suppose we have continuous $f_{n} : D \rightarrow \mathbb{R}$ converging to $f : D \rightarrow \mathbb{R}$ uniformly on $D$. Then, is $f$ continuous? I know that if we have $f_{n}$ converging to $...
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0answers
20 views

Determining uniform convergence of a sequence of exponential functions

I am given the sequence of functions $f_n(x)=e^{-nx^2}$ on $[-10,10]$. I must find the pointwise limit function $f(x)$ and decide whether convergence is uniform. If it is, I must find a $B_n$ such ...
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1answer
51 views

Show that $f_n(x)=\frac{x^{2n}}{1+x^{2n}}$ is not uniformly convergent

Show that $f_n(x)=\frac{x^{2n}}{1+x^{2n}}$ is not uniformly convergent. For $f_n$ to be uniform convergent, $\sup_{x\in\mathbb{R}}\{|f_n(x)-f(x)|\}\lt\epsilon$. I know that $f_n$ converges pointwise ...
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1answer
45 views

Uniform convergence of iterated improper integrals on $(0,\infty)$

I'm trying to get a better understanding of when it is permissible to swtich conditionally convergent improper integrals (when Fubini inapplicable) and I looked at a case where it works: $$\int_0^\...
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2answers
40 views

Continuous function as pointwise limit but not as uniform limit of a sequence of continuous functions on $[0,1]$

I recentaly find an article where it is said that there is a sequence of continuous functions $\{f_n:[0,1]\rightarrow\Bbb R\}_{n\in \Bbb N}$ that converges pointqise almost everywhere to zero function ...
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1answer
11 views

Uniform convergence of a sequence on limited t range

Hi I know the following sequence converges pointwise to $t$ from the limit but I am not sure if it converges uniformly for $t \in [0,1]$ and $n≥1$ $$f_n(t)= \frac {nt}{n+t}$$ I have found $|f_n(t)-f(...