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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Question about applying Dominated Convergence Theorem

Question: $\phi_n(x)=\int_{x_0}^xf(t,\phi_n(t))dt$ ,where $\phi_n(x)$ is continuous on $(a,b)$ and $f$ is continuous and bounded on $(a,b)\times(-\infty,+\infty)$. 1.If $\phi_n(x)$ converges uniformly ...
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Prove uniform convergence of the series $\frac{(-1)^n}{n+x}$ in $[0,\infty)$

Can it be proved without using dirichlet test for uniform convergence? Edit: Answer is Cauchy Criteria for Uniform Convergence
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Does this sequence converge uniformly?

We consider some sequence $(f^{(k)})_{k\in \mathbb{N}}\subset L^p(\mathbb{R}^3)$ such that $f^{(k)}\to f$ in $L^p(\mathbb{R}^3)$. Further assume there is some $R>0$ so that $\text{supp}(f^{(k)}),\...
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Rudin’s PMA, Theorem 11.20

This is the definition which we need for the theorem: (source) 11.19 $\; \;$ Definition $\; \;$Let $s$ be a real-valued function defined on $X$. If the range of $s$ is finite, we say that $s$ is a ...
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$\int_E fd\mu=\lim \limits_{n\to\infty}\int_E f_nd\mu$

Let $(X,\mathcal S, \mu)$ be a measure space, $E\in \mathcal S$, $\mu(E)<\infty$ and $f_n$ be a non negative sequence, measurable real-valued functions that converge uniformly to $f:X\rightarrow \...
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Show that $\left(V_{n}\right)_{n \geq 1}$ converges in $L_{1}$

Let consider the Galton-Watson process with immigration which is given by the following recursion $$ Z_{n+1}=\sum_{k=1}^{Z_{n}} \xi_{k}^{(n+1)}+\eta_{n+1} $$ where $\left(\xi_{k}^{(n)}\right)_{k \geq ...
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Investigate whether the sequence of function $f_n(x)=\sum_{k=0}^n (-1)^k \frac{x^{2k}}{(2k)!}$ is converges uniformly.

Let $(f_n)$ be a sequence of functions with $$f_n(x)=\sum_{k=0}^n (-1)^k \frac{x^{2k}}{(2k)!}.$$ Investigate whether $(f_n)$ converges uniformly to $f$ on $[0,1]$, where $f(x)=\cos x$. Attempt: Notice ...
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There exists such result for any convergence of sequence definiton

Assume that $L(x_n) $ stands for the limit of the sequence $(x_n) $ in some sense, not necessarily the usual limit. For example, could be the almost convergence limit or the statistical limit. I ...
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Find a function $f:[0,1] \to \Bbb R$ such that $f_n \to f$ on $[0,1]$ where $f_n(x)=\sum_{k=0}^n (-1)^k \frac{x^{2k}}{(2k)!}$.

Let $(f_n)$ be a sequence of functions with $$f_n(x)=\sum_{k=0}^n (-1)^k \frac{x^{2k}}{(2k)!}.$$ Find a function $f:[0,1] \to \Bbb R$ such that $f_n \to f$ on $[0,1]$. I know that this sequence of ...
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Questions in understanding proving this series of functions is $C^\infty.$

Define $f_n : [0,2\pi]\times (1,\infty)\to \mathbb C$ as $f_n(\theta,t)= e^{in\theta}e^{-n^2 t}$ for all $n\in \mathbb N$. Here is the proof of $f\in C^\infty ([0,2\pi]\times (1,\infty))$ written in ...
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Bounded increments of a martingale converges proof.

Prove or disprove: There exists a martingale $\left(M_{n}\right)_{n}$ with $\mathbb{P}\left(M_{0}=1\right)=\mathbb{P}\left(M_{0}=-1\right)=1 / 2$ and $\left|M_{n}-M_{n-1}\right| \leq n^{-3}$ for all $...
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How to prove that this series of functions converges uniformly?

Let $z_0 \in \mathbb{C} \setminus \{0\}$ be a fixed complex number and let $z \in \mathbb{C}$. Let $K \subset \mathbb{C}$ be a compact set. I'm trying to use the Weiertrass $M$ - test to show that the ...
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Uniform convergence integration question

EDIT: Possibly major detail I forgot to mention: $f$ is $C^1$. The question: Show that there exist constants $c_n>0$ (independent of $f$) such that the sequence of functions $$g_n(x)=c_n \int_{-\pi/...
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Sequence of holomorphic functions converges uniformly, but sequence of $k^{th}$ derivatives doesn't

Is there a sequence of holomorphic functions $(f_{n})$ on the closed unit disc such that $(f_{n})$ converges uniformly to a holomorphic function $f$ on the closed unit disc, but $(f_{n}^{(k)})$ does ...
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Uniform convergence of $f_n(x) = \frac{x^n}{1+x^n}$ [duplicate]

This is problem 9.3.1 from Bruckner's 'Elementary real Analysis': I've found previously that it is uniform convergent in $(0,1)$, but it seems to me that for an appropriate $a>0$, this sequence is ...
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Uniform continuity of sequence of function

Is $\dfrac{x^{1.9}}{(x^2 + n^4)}$ uniformly convergent to 0 given that $x>0$ To prove it’s indeed uniformly convergence, how do I find its maximum? i only got x^1.9/(x^2+n^4) <= x^1.9/n^4. have ...
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An attempt at proving Dini's theorem.

By Dini's theorem, I mean $E$ is a compact metric space. $f: E \to \mathbb{R}$ and $f_n: E \to \mathbb{R} $ are all continuous functions. Moreover $f_n \to f$ pointwise. Suppose for each $p \in E$ ...
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Rudin: Prove that if $f(x)=\sum_{n=0}^{\infty}c_nx^n$ converges for $|x|<R$, then $f'(x)=\sum_{n=1}^{\infty}nc_nx^{n-1}$ in $(-R, R)$.

The Problem: If $f(x)=\sum_{n=0}^{\infty}c_nx^n$ converges for $|x|<R$, then $f'(x)=\sum_{n=1}^{\infty}nc_nx^{n-1}$ in $(-R, R)$. The problem is a theorem from Walter Rudin's PMA: Here is part of ...
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Uniform convergence under matrix inverse

Let $A_n\in\mathbb{R}^{d\times d}$ be a sequence of positive definite matrices such that $\sigma_{\text{min}}(A_n) = 1/n$ and $\sigma_{\text{max}}(A_n) = 1$ for all $n\in\mathbb{N}$. Let $X\in\mathbb{...
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Uniform convergence via Arzela-Ascoli

I want to show that $u_\epsilon = -\epsilon \log\left(\frac{ e^{\frac{x}{\epsilon}} + e^{-\frac{x}{\epsilon}}}{e^{\frac{1}{\epsilon}} + e^{-\frac{1}{\epsilon}}} \right)$ converges uniformly to $1-|x|$ ...
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5 votes
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Proving $f_n \rightrightarrows f$, provided that $f_n \rightarrow f$, each $f_n$ is increasing, and $f$ is continuous.

Clarification: this is not Dini's theorem. The title is highly succinct but here is the task in full detail: Let $[a,b]\in \mathbb{R}$ where $a<b$ are reals. Each $f_n: [a,b] \to \mathbb{R}$ is an ...
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How can I integrate $\int_0^1 \bigl(1-x^n\bigr)^{2^n} \mathrm{d}x$? [closed]

How can I integrate $$ \int_0^1 \bigl( 1 - x^n \bigr)^{2^n}\, \mathrm{d}x? $$ I need to find the limit of the integral as $n \to \infty$, but cannot switch the integral and the limit as the function ...
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Uniform convergence over $[0,1]$ preserves zeroes proof verification

Our teacher gave this problem in class and said that the hint to solve this was by using triangle inequality. Let $\{f_n\}$ be a sequence of continuous real-valued functions on $[0,1]$ converging ...
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Uniform Convergence of Difference Quotient to Derivative using Compactness

I have a question about a proof to a question in another discussion. This is regarding the uniform convergence of the difference quotient to the derivative of a function. Here is a link to the ...
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Convergence of derivatives.

We have a course on complex analysis in the current semester.Our professor introduced a theorem in class which is as follows: Theorem Let $f_n:\Omega\to \mathbb C$ be a sequence of holomorphic ...
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Uniform converges of $f_n(x)=x-\frac{x^n}{n!} , x\in [-a,a] ,a>0$

I have 2 questions : 1.with the given following function series : $f_n(x)=x-\frac{x^n}{n!}$ I need to show that for $x\in [-a,a]$ $ ,a>0 $ there is $P,Q>0$ such that : $|f_n(x)-f_{n-1}(x)|\leq ...
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2 votes
2 answers
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uniform convergence of $\sum_{n=1}^\infty (1-e^{-\frac xn})\sin(nx)$

I have this question that I'm struggling to do in my calc textbook Note that I have not been exposed to complex numbers yet The question is: let $a$ be a positive real number such that $0 < a < \...
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Uniform convergence of $\sum_{n=1}^\infty x^\alpha e^{-nx^2}$ in the set $X=(0;+\infty)$ depending on value of the $\alpha$

I'm trying to explore uniform convergence of series $\sum_{n=1}^\infty x^\alpha e^{-nx^2}$ in the set $X=(0;+\infty)$ depending on value of the $\alpha$. I tried to use theorem that series are ...
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Can we show that this series is uniformly convergent using Weierstrass-M test?

Let $D$ be an open connected set. How can I show that the series $$f(z) = \sum_{k=1}^{\infty}kz^k,$$ where $|z|<1$ for all $z\in D$ is uniformly convergent on $D$? I wanted to use Weierstrass-M ...
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2 votes
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Do we have the following uniform convergence?

We are given a sequence $f_N$ such that $f_N(x) \xrightarrow{N\to\infty} f(x)$ uniformly in $x$. $f$ is continuous, but the $f_N$'s are not necessarily continuous. Moreover we know that $y_N = \frac{\...
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Is this enough for uniform convergence?

Let $f_n$ be a sequence of continues functions from $(-a,a)$ to $\mathbb{C}$ such that $\lim_{n \to \infty} f_n(x)=0$ does this mean $f_n(x)$ is uniformly convergent on $[-\epsilon,\epsilon]$ whenever ...
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1 vote
1 answer
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Equicontinuity, compactness and uniform convergence.

Let $X$ be a compact topological space and let $K\subset \mathbb{R}^n$ be compact. Let $F = \{f_n\}_{n \ge 1}$ be a sequence of continuous functions where $f_n : X \to K$ that converges pointwise to $...
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A question about convergence and uniform convergence

Suppose that $f_{m,n}$ is continuous alomst everywhere, and $f_{m,n}$ converges to $g_{m}$ alomst everywhere as $n \to \infty$, $g_{m}$ converges uniformly to $g$ almost everywhere as $m \to \infty$. ...
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A problem about convergence of measures

I am trying to understand the proof sketch of the following. Let $\kappa$ and $(\kappa_n)_{n\geq 1}$ be a measure and a sequence of finite measures defined on the borelians of $\mathbb{R}^p$ such that ...
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A uniformly convergent sequence of real valued functions need not to be term by term Integrable?

We know that if $\{f_n\}_{n=1}^\infty$ be a sequence of real valued functions converging uniformly to a function $f$ in a bounded interval $[a, b]$, then sequence is term by term integrable, i. e., $$...
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the limit of the infimum of a sequence of bounded functions that converge uniformly is equal to the infimum of the limiting function

prove $ \lim_{n\to\infty}inf [f_n(x)|x\in E]=inf[f(x)|x\in E]$ where $f_n$ are bounded functions of a set $E\subset R$ that converge uniformly to a function $f$. I've looked at similar proofs on the ...
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Rudin's PMA Theorem 7.9

Here is rudin's statement Theorem 7.9 suppose $$\lim_{n\to\infty}f_n(x)=f(x)(x\in E).$$ Put $$M_n=\sup_{x\in E}|f_n(x)-f(x)|.$$ Then $f_n \to f$ uniformly on $E$ if and only if $M_n \to 0$ as $n \to \...
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Uniform convergence through Supremum [duplicate]

I came up with a solution to a simple problem that I was working on but it seemed too simplistic so I doubt that my solution is correct which is why I want some help. $f_n(x)$ is a sequence of complex ...
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1 answer
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How do I prove that $\sum_ {k=0}^ {\infty} \frac{\sin (kx)^2}{ (1+k^ 2x^2)}$ is not continuous at zero? [closed]

How do I prove that $\sum_ {k=0}^ {\infty} \frac{\sin (kx)^2}{ (1+k^ 2x^2)}$ is not continuous at zero? This is a question on uniform convergence I tried to solve it I just don't have any clue how I ...
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2 votes
1 answer
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Does uniform convergence in $L^1$ imply uniform convergence in $L^2$?

I had to keep the title reasonably long, so below are the full details. This is a question that came to my mind while working on a distantly related problem; since I do not know whether the answer is ...
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Uniform convergence of sequence of functions that converges pointwise to an unbounded function

I have the following sequence of functions before me: $f_n(x)=\dfrac{1-x^n}{1+x},-1<x<1$ I have to determine whether above sequence of functions is uniformly convergent or not. First of all, I ...
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2 votes
2 answers
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Check uniform convergence of $\sum_{n=1}^{\infty}\frac{x\sqrt{n}}{1+x^2n^2}$

I need to check if $\sum_{n=1}^{\infty}\frac{x\sqrt{n}}{1+x^2n^2}$ for $x>0$ converges uniformly. I solved easier version of this problem which was to check if $\sum_{n=1}^{\infty}\frac{\sqrt{n}}{1+...
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Convergence a.e. but not in $L_{p}$ norm Corollary

Let $p>1$. I have considered the following example of a function in $L_{p}(\mathbb{R})$ that converges to $0$ almost everywhere but doesn't converge to $0$ in the $L_{p}$ norm: for each $n$, let $...
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1 vote
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Sequences of Functions, continuity and Uniform convergence.

I need help verifying my solution to a problem, it seemed really simple but that's often when I make mistakes so I'd appreciate some help. Let $D \subset \mathbb{C}$ and $f_n:D\rightarrow \mathbb{C},n\...
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Proving $f_n(x)=\frac{1+\cos^2(nx)}{\sqrt{n}}$ converges uniformly to $0$ on $\mathbb{R}$

We wish to prove that the sequence of functions $f_n(x)=\frac{1+\cos^2(nx)}{\sqrt{n}}$ converges uniformly to $0$ on $\mathbb{R}$. Scratch work: $|f_n(x)|=|\frac{1+\cos^2(nx)}{\sqrt{n}}|\le|\frac{2}{\...
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(Locally) Uniform convergence of the given series

Determine the locally uniform convergence of the series $\sum_{n=1}^{\infty} f_n(z)$ on $V=\mathbb{C}\setminus\{-n : n\in\mathbb{N}\}$, where $$f_{n}(z) = (-1)^n \frac{1}{z+n}.$$ Is $f_n$ also uniform ...
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Is uniform convergence required for a continuous limit function?

Consider the sequence $(f_{n})_{n=1}^{\infty}$ of continuous functions on $I = [0, \infty)$ defined recursively by $f_{1}(x)=x, f_{n}(x)=x+\int_{0}^{x}f_{n-1}(t)\sin(x-t) dt, \forall n\geq 2$. This ...
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Convergence in $L^2$ doesn't imply uniform convergence

I am working on implications between different types of convergences in series of functions. I think that $L^2$ convergence doesn't imply uniform convergence, however I cannot find a counter-example. ...
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Issues with theorem 7.15 from Rudin's PMA.

Some prerequisite definitions and theorems. Given a metric space $X$, let $$ \mathscr{C}(X) = \{f: X \rightarrow \mathbb{C} \; | \; f \text{ bounded and continuous.}\}. $$ Next we have the familiar ...
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2 votes
1 answer
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Proving $f_n(x) = x^n$ for $x \in [0,1]$ is not a uniformly Cauchy sequence

I know this can be argued more succinctly by citing theorems on uniform convergence. I have also seen answers to this question on this page, but I made a somewhat different (I think) attempt of my own....
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