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
24 views

uniformly convergent sequence of differentiable functions, series of derivative of terms not convergent

I am attempting to come up with a uniformly convergent sequence of differentiable functions $g_{n}:(0,1)\to \mathbb R$ such that the sequence $\{g_{n}'\}$ does not converge. I was thinking that ...
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3answers
40 views

Proving uniform convergence on disk within radius of convergence

Needham's Visual Complex Analysis 2.III.2 states that a power series $S_k=\sum{C_k z^k}$ with RoC $R$ converges uniformly on any disk $r<R$. He leaves the proof as an exercise to the reader. But ...
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0answers
40 views

Is this expression for $x\pmod n$ interesting; nontrivial?

For example, we would get several interesting results if we had a formula for $x\pmod n$ that was uniformly convergent, however, according to Wikipedia (Floor and Ceiling Functions) these formulas do ...
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0answers
26 views

Check pointwise convergence and uniform convergence of $f_n(x) = n^3x\exp(-nx^2)$ on $[0,1]$

Check pointwise convergence and uniform convergence of $f_n(x) = n^3x\exp(-nx^2)$ on $[0,1]$ Pointwise convergence: $$ \lim_{n\rightarrow\infty} f_n(x) = \lim_{n\rightarrow\infty} ...
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1answer
20 views

Conditions for convergence of derivatives from pointwise convergence

Let $\{f_n\}_n$ be a sequence of functions $\mathbb R \rightarrow \mathbb R$ which converges pointwise to $f$, ie: $$f_n(x) \rightarrow f(x) \hspace{10pt}\hbox{for all $x$}.$$ What additional ...
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1answer
45 views

How to prove that this series converges uniformly?

I have a series $$ -\frac{\pi}{12} + \sum_{k=1}^\infty \frac{\left(3k\pi^2-16\right)\sin{\frac{k\pi}{2}} + 8\pi\cos{\frac{k\pi}{2}}}{\pi^2k^3}\cos{kt} $$ And I have to use Weierstass test to prove ...
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2answers
22 views

Calculating values of integrals using Fourier series and uniform convergence

I have a problem that I don't know how to begin solving. I have f(t) $$ f(t) = \sum_{k=1}^\infty\frac{1}{k^2+1}\sin{kt} $$ First I had to show that this series converges uniformly, I've done that ...
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1answer
46 views

Proving if $\frac{nx}{e^{nx}}$ converges uniformly

I have this expression $$ \frac{nx}{e^{nx}} \qquad x\geq 0 $$ and I had to test if it's pointwise and uniformly convergent(separately). Pointwise convergence could be easily proved by just evaluating ...
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0answers
23 views

A question uniform convergence

Let $X$ be a compact Hausdorff space, $a$ a continuous real-valued function on $X$, and for $t\in\mathbb{R}$ let $f_t(x)=\exp(ia(x))$ such that the function $t\mapsto f_t$ is continuous (where we use ...
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1answer
44 views

Geometry of Analysis

I am a recently graduated student and doing Post Graduation now. I often come across uniform convergence, uniform continuity etc. As we all know that we check continuity and convergence easily by just ...
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2answers
44 views

Limits of Sequences of functions and uniform convergence 3

I have the sequence of functions: $s_n(x)= \displaystyle\sum_{k=0}^n \frac{k^2x}{1+k^4x^2}$ where $x \in [0,1]$ Consider$ [\delta, 1]$ for any $\delta \in (0,1]$ Now $0 \leq ...
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2answers
100 views

How to show that $\sum\limits^\infty_{n=0}a_n(\frac{2x}{1+x})^n$ is continuous?

$(a_n)$ is a bounded sequence and $x\in (-1,1)$. I have shown point wise convergence. I have tried using Weierstrass but failed, and the only other thing I can think of is using Abel's Theorem for ...
3
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3answers
45 views

Given $f_n:= e^{-n(nx-1)^2} $, any suggested approaches for showing that $\lim_{n \to \infty}\int^1_0 f_n(x)dx = 0$?

Given $f_n:= e^{-n(nx-1)^2} $, any suggested approaches for showing that $\displaystyle \lim_{n \to \infty}\int^1_0 f_n(x)dx = 0$? I have already shown that $f_n \to 0$ pointwise (and not ...
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2answers
68 views

Finding a particular type of sequence of functions

For every bounded function $f:[a,b] \to \mathbb R$ on a closed bounded interval $[a,b]$ , which is dis-continuous at at most countably many points of its domain ; can we find a sequence of ...
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0answers
31 views

Sequence of functions that converge uniformly on a compact metric space.

I'm getting stuck with this problem: Let $\mathbb{X}$ be a compact metric space and $f_n, g_n: \mathbb{X} \rightarrow \mathbb{R}$ be functions converge uniformly to $f, g: \mathbb{X} \rightarrow ...
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1answer
94 views

$ \sum_{n=0}^\infty a_ne^{-\alpha_nx} $ converges for $x_0$ prove it uniformly converges in $[x_0,\infty]$

consider the folloing sum $$ \sum_{n=0}^\infty a_ne^{-\alpha_nx} $$ for every n $ 0 < \alpha_n < \alpha_{n+1}$ it is also given that for $x_0 \in \Bbb{R}$ the sum converges prove that the ...
2
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1answer
117 views

if $ L = \lim_{x\to1^-} \sum_{n=1}^\infty a_nx^n$ then $\sum_{n=1}^\infty a_n = L$

let ${a_n}$ be a sequence of Real non negative numbers. assume the following limit exists and is finite: $$ L = \lim_{x\to1^-} \sum_{n=1}^\infty a_nx^n$$ prove that $\sum_{n=1}^\infty a_n$ ...
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2answers
39 views

Uniform Convergence and limit $(n+1)\int_0^1 x^nf(x) \; dx$ [duplicate]

If $f$ is a continuous real-valued function, show that $$ f(1)=\lim_{n\to \infty} \int_0^1 (n+1)\,x^n \,f(x) \; dx $$ I am looking for a general hint or steps to proceed but I want to fill them in. ...
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0answers
13 views

Proving that an asymptotic series is uniform for parameter belonging to a compact subset of the parameter space.

Let $U$ be a disk centered at $z=1$ of radius $\delta$. Given the function $h(z)=\frac{1+i(z^2-1)^{1/2}\sin(\alpha/2)}{1-i(z^2-1)^{1/2}\sin(\alpha/2)}$, define the function $$w(z)=\frac{1}{2}\ln h(z) ...
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1answer
28 views

Proving uniform convergence of a Series

I need a hint to solve this little problem: Let $f_n:]0,1] \to \Bbb R$ defined by $f_n(x)=x^n (\ln x)^2$. Prove that $$\sum_{n=1}^{\infty} f_n(x)$$ converges uniformly in $]0,1]$. I started proving ...
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2answers
25 views

Pointwise/Uniform Convergence of Sequence of Functions (and continuity!)

Define a sequence of functions on $\mathbb{R}$ by: $$ f_n(x)=\begin{cases}1, & \text{if $x=1$, $1\over 2$ ,$1\over 3$,...,$1\over n$} \\ 0, & \text{otherwise} \end{cases} $$ and let $f$ ...
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1answer
45 views

Stationary points of a family of functions which converge uniformly.

Here's the motivation of my question: Consider the family of functions $\mathrm{f}_n(x) := \frac{1}{n}\sin(nx)$. It is easy to show that $\mathrm{f}_n$ converges uniformly to the zero function. To ...
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1answer
43 views

Uniform Convergence Proof in Spivak's Calculus

I have done everything in this problem except "conclude that the series does not converge uniformly on $\mathbb{R}$". I know something happens at $x=0$ and that $$ \lim_{N \to \infty} f \bigg( ...
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1answer
31 views

Pointwise Convergence to 0 Implies Uniform Convergence to 0

I have seen some related posts on Dini's Theorem, and am actually working a problem related to it, but I have come across some troubling logic unrelated to the theorem. I believe my question to be ...
3
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2answers
72 views

Is the uniform limit of piecewise continuous functions $f:[a,b]\rightarrow\mathbb{R}$ piecewise continuous?

Let $(f_n)$ be a sequence of functions $f_n:[a,b]\to\mathbb{R}$, all of which are piecewise continuous. Does $f_n\rightrightarrows f$ imply that $f$ is piecewise continuous? EDIT: I wrote a proof ...
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0answers
57 views

Additional assumptions on function to ensure uniform convergence

Given a sequence $u=(u_n)_{n\geq1}$ converging to $1$, I would like to prove uniform convergence of the sequence of functions $f_n$ defined by $f_n(x)=f(u_n x)$ for $x\in\mathbb{R}_+$ to the function ...
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2answers
87 views

On the sequence of function $f_n(x)=n^2x(1-x^2)^n $

Is the sequence of functions $f_n(x)=n^2x(1-x^2)^n $ uniformly convergent over $[0,1]$ ? Is the series $\sum_{n=1}^{\infty} f_n(x)$ convergent with $0<x<1$ ? Is the series uniformly convergent ...
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1answer
79 views

When does interchangibility of limit and Riemann integral imply uniform convergence?

Let $\{f_n\}$ be a sequence of real-valued functions defined on an interval $[a,b]$ such that each $f_n$ is Riemann integrable, $\{f_n\}$ converges point-wise to $f$, $f$ is Riemann integrable and ...
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2answers
26 views

Convergence when the derivative is uniformly continuous

Let $f: \Bbb R \to \Bbb R$ be a derivable function. $f'$ is uniformly continuous in $\Bbb R$ Prove that $[n(f(x+1/n)-f(x))]$ converges uniformly to $f'(x)$ I'm having a hard time seeing why does ...
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1answer
34 views

Does $f_n(x)=\cos^n(x)(1-\cos^n(x))$ converge uniformly for $x$ in $[π/4 , π/2]$?

Does $f_n(x)=\cos^n(x)(1-\cos^n(x))$ converge uniformly for $x$ in $[π/4 , π/2]$? Its clear to see that the point-wise convergence is to $0$. By finding the derivative I obtained that the maximum of ...
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0answers
47 views

Question on Egoroff-like theorem

Hi all I was tackled by this question from Folland's real analysis second edition in the second chapter, it looks like a modified Egoroff theorem but I cannot really tackle it, it is question 41 of ...
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1answer
48 views

Does this sequence of functions converge uniformly?

So the questions says, let $a_n$ be a sequences of real numbers such that $\limsup |a_n| = 0$. Let $X = [0, 1]$ and for each $n \in \mathbb{N}$ the function $\space$ $f_n :$ $X \mapsto ...
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3answers
46 views

Does the following series converge uniformly?

I know how to show that the following series will converge absolutely. But am unsure how to show it will or will not converge uniformly for $z\in (0,1).$ $\displaystyle \sum_{n \mathop = 1}^{\infty} ...
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4answers
42 views

Pointwise convergence and uniform convergence of $f_n(x) = x^n(1-x)$

Ok, I am new to this pointwise and uniform convergence so don't mind if I make mistakes here. Let: $f_n(x) = x^n(1-x), x \in [0,1]$ $f(x) = 0, x \in [0,1].$ Prove that $f_n$ converges to $f$ ...
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1answer
33 views

Convergence that preserves smoothness

One of the advantages of uniform convergence is that it preserves continuity (among other properties). Unfortunately, it does not preserve derivability. Is there a convergence mode preserving it?
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42 views

Show that the series does not converge uniformly on $\mathbb{R}$

My Work: Here, according to the given facts $f(0)=0$ and $f$ is strictly increasing. I proved part (a) and (b) but failed to prove (c). I was going to use the definition (actually wanted to show ...
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1answer
43 views

$f_n(x)$ converges uniformly to a function $f(x)$ then does it follow that the limit function $f(x)$ is also uniformly continuous.

If a sequence of functions $f_n(x)$ converges uniformly to a function $f(x)$, and if each $f_n(x)$ is uniformly continuous, then does it follow that the limit function $f(x)$ is also uniformly ...
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1answer
28 views

$f_n → f$ uniformly on $S$ and each $f_n$ is cont on $S$. Let $(x_n)$ be a sequence of points in $S$ converging to $x \in S$. Then $f_n(x_n) → f(x)$.

Assume that $f_n → f$ uniformly on $S$ and each $f_n$ is continuous on $S$. Let $(x_n)$ be a sequence of points in $S$ converging to $x \in S$. Then $f_n(x_n) → f(x)$. I'm stuck in thinking about it ...
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0answers
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the series $\sum_{1}^{\infty} \frac{\cos(nx)}{\{\log(n+1)\}^x}$ is uniformly convergent on any closed interval $[a,b]$ lying within $(0,2\pi)$.

To show that the series $\sum_{1}^{\infty} \frac{\cos(nx)}{\{\log(n+1)\}^x}$ is uniformly convergent on any closed interval $[a,b]$ lying within $(0,2\pi)$. My Try: Let us consider $u_n(x) =\cos(nx), ...
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0answers
23 views

topology of uniform convergence on compacts and strong operator topology

I am trying to understand the proof for some lemma in a book, but the part the authors label as trivial is not trivial to me at all. Any help is appreciated. Here is the trivial part of the lemma: ...
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2answers
63 views

measure theory problems and step functions

I have several questions that I haven't worked out. Any hints or solutions will be appreciated. Suppose that {$f_n$} is a sequence of real valued continuously differentiable functions on [$0,1$] ...
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2answers
73 views

$\int_a^b f(x)g(x)dx = \sum \int_a^b f_n(x)g(x)dx.$

Let $\sum f_n(x) $ be uniformly convergent to $f(x)$ on $[a,b]$ where each $f_n$ is continuous on $[a,b]$. If $g: [a,b] \to \mathbb R$ be integrable on $[a,b]$, then $$\int_a^b f(x)g(x)dx = \sum ...
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0answers
52 views

If $\sum a_n$ is a convergent series of real numbers then the following series are convergent.

If $\sum a_n$ is a convergent series of real numbers prove that the series: 1) $\sum a_n e^{-nx}$ is uniformly convergent on $[0,\infty)$; 2) $\sum \frac{a_n}{n^x}$ is uniformly convergent on ...
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0answers
43 views

Uniform convergence of functions involving normal CDF

Consider two sequences of continuous functions $(f_n)$ and $(g_n)$ for $n \geq 0$ defined by $$ f_n (x) := \int_0 ^t \Phi\left(\frac{x\Phi ^{-1}(\alpha(s) + \beta_n(s))+\Phi^{-1} ...
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0answers
30 views

Normal convergence of complex series

I have troubles with this task: Let $\mathbb{R}\_$ be the set of non-positive real numbers and $U = \mathbb{C}\backslash \mathbb{R}\_$ For $n \ge 0$, consider a function $f_n$$:U \rightarrow ...
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3answers
65 views

The sequences $\{f_n\},\{f_n'\}$ both are uniformly convergent on $[0,1]$.

Let $f_n(x) = \frac{\ln(1 + n^2x^2)}{n^2}, x \in [0,1]$. Then the sequences $\{f_n\},\{f_n'\}$ both are uniformly convergent on $[0,1]$. Here, $f_n'(x) = \frac{2x}{1 + n^2x^2}$. Both limits of ...
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2answers
58 views

How to prove this limit of derivative

Here is a question that I need help to prove it. Let $f:\mathbb{R}\to\mathbb{R} \in C^{\infty}$ be periodic of period $1$ and nonnegative. Show that $$ ...
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1answer
21 views

The sequence $f_n '$ converge to a function $g$ on $[0,1]$ and $f'(x) = g(x), x \in (0,1), f'(1) \neq g'(1)$.

Let $f_n(x) = \frac{x^n}{n}, 0 \leq x \leq1$. Then $1) $ the sequence $f_n$ converge uniformly to a function $f$ on $[0,1]$. $2)$ the sequence $f_n '$ converge to a function $g$ on $[0,1]$ and ...
0
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1answer
50 views

Apply Stone- Weierstrass Theorem [duplicate]

Suppose $f: [0,1]\to \mathbb R$ is continuous and $$\int_{0}^{1} f(x)e^{nx} \mathsf dx=0$$ for every $n$. Prove that $f(x)=0$ for all $x \in[0,1]$. Since $f$ is continuous on $[0,1]$, by ...
1
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2answers
45 views

$f_n$ defined on $[-1,1]$ by $f_n(x) = |x|^{1+\frac1n}, \ x \in [-1,1]$ converges uniformly .

To show that the sequence of function $f_n$ defined on $[-1,1]$ by $f_n(x) = |x|^{1+\frac1n}, \ x \in [-1,1]$ converges uniformly to a function $f(x) = |x|, \ x \in [-1,1]$ and the sequence of ...