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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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about Dirichlet test in convergence

whether this infinite sequence $\sum_{n=1}^\infty \frac{(-1)^ncos(nx)}{\sqrt n}$ converge on $\mathbb R$. I know that it would converge uniformly intuitivly on [$-\pi+\delta$, $\pi-\delta$] where $\...
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‎Is the functional sequence ‎$f_n(x)$ ‎uniformly ‎convergent‎ ‎on ‎$‎[1, 2]‎$‎?

Consider the functional sequence ‎‎\begin{align*}‎ ‎f_n(x) = ‎\sqrt{n+1} - \sqrt{n+2} + \sum_{k=1}^n ‎\frac{1}{2\sqrt{k+x+2}} - ‎‎‎‎\frac{1}{2\sqrt{k+x+1}}, (1\leq x\leq 2, n=1,2,3,...). ‎‎\end{align*}...
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Rearrangement of uniformly convergent power series is uniformly convergent?

Is rearrangement of uniformly convergent power series again uniformly convergent? Suppose we have $\sum a_nx^n$ converging unifomly on $[0,1]$. Is its rearrangement $\sum a_{\sigma(n)}x^{\sigma(n)}$ ...
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Probably dumb limit

I have a sequence of continuous functions $f_n : I^k \rightarrow I^k$ converging uniformly to a continuous function $f$. Then for each $n$ I choose a point $x_n$ and since they're chosen in $I^n$ ...
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What does the series $\sum\limits_{n=-\infty}^{\infty} \frac{1}{n}e^{i \pi n x}$ ($n\neq 0)$ converges to on $(-1,1)$?

What does the series $\sum\limits_{n=-\infty}^{\infty} \frac{1}{n}e^{i \pi n x}$ ($n\neq 0)$ converges to on $(-1,1)$? I tired plotting this series for different values of $n$ and it seemed it is zero ...
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Uniform convergence of Bessel series - question concerning proof in Watson 3.13

Good afternoon, I am currently studying on Bessel functions using Watson's Treatise on the Theory of Bessel functions. In chapter 3, the series defining the Bessel function of first kind for ...
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Is it possible for a sequence of matrices to have pointwise but not uniform convergence?

Is it possible for a sequence of matrices to have pointwise but no uniform convergence? The norm for the matrices is the operator norm.
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How to prove the non uniform convergence of sequence of functions.

The sequence of functions $\{f_n\}$, defined by $$f_n(x) = n\log \left(1+\frac{x^2}{n}\right)$$ is not uniformly convergent on $\mathbb{R}$. We have $\lim_\limits{n\to \infty}f_n(0)= 0$ and $$ \lim_{...
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How do I evaluate this limit: $\max \{\lim_{x,k \to \infty} G_k(x)\}$

Let $$G_k(x)=A(x)B_k(x)=\pi(x)^{\frac{1}{\log(x)}}\pi(k-x)^{\frac{1}{\log(k-x)}},$$ where $\pi(x)$ is the prime counting function. What is $$\max \{\lim_{x,k \to \infty} G_k(x)\}?$$ This is what I ...
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Condition on Uniform Convergence of Fourier Series

When I was reading a proof of some problem, it said that "Since $f$ and $f'$ are in $L^2([-\pi,\pi])$, the Fourier Series of $f$ converges to $f$ uniformly". My question is that, is this statement ...
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Detecting Uniform Convergence By Fourier Methods

A close snorbaki friend has posed a problem about uniform convergence of a sequence of functions, and I've noted it has a particularly well behaved sequence of Fourier transforms. Can I dazzle my ...
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1answer
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Does the Weierstrass M-test apply if a term of the functional series is undefined on the domain of definition?

I'm analysing the series of functions $$\sum_{n=1}^{\infty} \frac{1}{e^{x\cdot n^2}-1} \qquad \textrm{where } \space x \geq 0$$ for uniform convergence. I'm trying to use the Weierstrass M-test so I ...
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Convergence of images of a sequence of converging continuous functions

Suppose that $\phi_n$ are continuous functions from a compact set $\Omega\subset \mathbb R^n$ to $\mathbb R^n$, and that they converge uniformly to a continuous function $\phi:\Omega \to \mathbb R^n$. ...
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1answer
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Proving the uniform convergence of a the recursively sequence given as $P_{n+1}=P_{n}(x)+\frac{1}{2}(x^{2}-P_{n}(x)^{2})$.

Given a sequence of algebraic polynomials $\lbrace P_{n}:[0,1] \to \mathbb{R}\rbrace_{n=0}^{\infty}$ defined recursively as $P_{0}(x)=1$ and for every $n \in \mathbb{N}$; $$P_{n+1}=P_{n}(x)+\frac{1}{...
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Show uniform convergence implies uniform convergence of squares/root

Let $f_n ,\,f\colon [0,1]\rightarrow [0,\infty)$ be given. I wanted to show that if $f_{n}{\underset {n\ }{\rightrightarrows }}f,$ then $f^{1/2}_{n}{\underset {n\ }{\rightrightarrows }}f^{1/2}$ ...
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Question about epsilon-representative in machine learning

I am working on the exercise of book Understanding machine learning:from theory to algorithm. Now I am stucked by this question: Sample S is epsilon-representative of H w.r.t distribution D, and S' ...
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how can we calculate the integral

let $(\lambda_n)_{n \in N}$ be a sequence creasing of positive reals going to infinity . $f(x)= \sum_{n=1}^{\infty}( -1)^ne^{- \lambda_n x}$ how can we calculate $\int_{0}^{\infty} f(t)dt$ for ...
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Convergence of an infinite series involving a continuous bijection

I am having some difficulty with a real analysis problem. Suppose $f:[0,\infty)\to[0,\infty)$ is a continuous bijection and consider the series $$\sum_{n=1}^{\infty}\frac{nf(x^2)}{1+n^3(f(x^2)^2)}$$ ...
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1answer
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Converging sequence of functions on a compact set.

I have this yes/no question regarding uniform convergence of real functions: "If a sequence of functions defined on a compact set converges uniformly, then the uniform limit is a bounded function." ...
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Exercise in measure theory about dominated convergence

I'd like a confirm that my solution is correct, thank you in advance. Suppose $(X, B, \mu)$ is a measure space, $(f_n)_n \subset L^1(\mu)$ and $f_n \to f$ uniformly for some $f:X \to \mathbb{R}$. ...
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1answer
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Weierstrass M-test and Sophomore's Dream.

I am trying to prove the Sophomore's Dream $\int_0^1 x^{-x} \ d x = \sum\limits_{n = 1}^\infty n^{-n}$. I get to this point, $\lim\limits_{a \to 0^+} \int_a ^1 \sum\limits_{n = 0}^\infty \frac{(-1)^n(...
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Convergence of suprema of sequence of functions

Let's say that I have a sequence of continuous, bounded functions $\lbrace f_n \rbrace$ which converge uniformly to some continuous, bounded function $f$. $f_n$, $f : X \to \mathbb{R}$, where $X$ is a ...
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1answer
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Criterion for determining when uniform convergence of multivariate convex functions is convex

Suppose we have a sequence of convex multivariate functions $f_a(\vec{v}): S \to \Bbb R$ defined on some open convex subset $S$ of $\Bbb R^n$, which converge uniformly to some limit function $f_\...
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1answer
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Show local uniform convergence for composition of continuous and affine map on compact set.

Let $D \subseteq \mathbb{R}^d$ be compact and nonempty. Furthermore, let $W^{(n)},W \in \mathbb{R}^{m \times d}$ be matrices and $b^{(n)}, b \in \mathbb{R}^m$ be vectors, such that $$ \max_{i,j} \...
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1answer
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Show that $\inf \{ \| f-P \|_{\infty}\mid P \in P_n \} \geq \delta_n$ for any decreasing sequence $\delta_n \to 0$

I'm trying to show that given any decreasing sequence $\delta_n \to 0$, we can find a continuous function $f: [-1,1] \to \mathbb{R}$ such that $$\inf\{\|f-P \|_{\infty}\mid P \text{ a polynomial of ...
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Behavior of coefficients of sequence of polynomials converging to the zero function

Consider a sequence of polynomial functions $\{f_n(x)\}$ defined on a closed interval $[a,b]$ with $0 < a < b < 1$. The function $f_n(x)$ is of the form $$ f_n(x) = a_{n1} x + a_{n2} x^2 + ....
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convergence of bounded, holomorphic functions on the disk

I want to show that a sequence of holomorphic, zero-free functions on the disk converges uniformly to zero on compact subsets of the disk if $|f_n| < 1$ and $\lim_{n \rightarrow \infty} f_n(0) = 0$....
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If a function $f: A \to \mathbb R$ is bounded and continuous on $A$, does its integral on said interval converge uniformly?

Let the bounded interval $A$ be a subset of the reals and let $f : A\to\mathbb R$ be a bounded and continuous function. Since the sequence $(f_n): f_n = f/n$ has the limit $\displaystyle\lim_{n\to\...
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2answers
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Uniform convergence of a specific sequence of function

i have trouble proving that $f_n(x)= sin(\frac{nx}{n+1})$ is not uniformly convergent on [1,$\infty)$. $x \in \mathbb{C} $. I know that it converges pointwise to sin(x) but i don't know how to proceed....
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Semi-locally uniform convergence

I am looking for a topology on a set of functions which is somehow between uniform convergence and locally uniform convergence. For simplicity reasons I will explain my idea for the space of functions ...
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1answer
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Non-uniform convergence of improper integrals II

I'm looking at approaches for proving or disproving uniform convergence of improper integrals. I asked this question: Techniques for proving non-uniform-convergence of improper integrals but I ...
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1answer
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Proof Verification for Uniform Convergence on Sequence of Functions

just looking for a verification on a proof. Thanks in Advance Let $f_n$ be a sequence of functions such that $f_n=\frac{x^{2n}}{1+x^{2n}}$ defined on $[-2,2]$. Prove or Disprove Uniform Convergence ...
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Uniform convergence of the series of functions $\sum\limits_{n=1}^{+\infty} {n^n} {\arctan({x}^{n^3})}$

I have pointwise convergence of the sum of $f_n$ in $(-1,1)$.In fact: 1) if $|x|>1$ $f_n(x)$ doesn't converge to zero 2) if $|x|<1$ $f_n(x) \sim n^n x^{n^3}$ for $n\rightarrow +\infty$ and $\...
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Pointwise and uniform convergence of $\sum\limits_{n=1}^{+\infty}(-1)^n\frac{x}{x+e^{-nx}}$ $\quad$ $x\in R$ [closed]

How can I prove pointwise and uniform convergence of $\sum\limits_{n=1}^{+\infty}(-1)^n\frac{x}{x+e^{-nx}}$ $\quad$ for $x\in \mathbb{R}$? uniform convergence: the pointwise convergence is on $E=(-\...
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1answer
52 views

Reciprocal of the Weierstrass $M$ test.

I need help with this problem: Suppose that $f_n$ are continous non-negative fucntions bounded on A and let $M_n=\sup f_n$. If $\sum_{n=1}^\infty f_n$ converges uniformly on A, it follows that $\...
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2answers
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Uniform convergence of $\sum\limits_{n=1}^{+\infty} \left(\sin{{1} \over {n}}\right) x^n$

$f_n(x)=\left(\sin{{1} \over {n}}\right) x^n$ pointwise convergence: $|f_n(x)|=\left(\sin{{1} \over {n}}\right) |x|^n \sim {{|x|^n}\over{n}}$ for $n \rightarrow +\infty$ $\sum\limits_{n=1}^{+\infty}{...
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1answer
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Does $f_n\to f$ in $L^1(0,1)$ imply $\int_0^tf_n\to \int_0^tf$ uniformly? [closed]

I am trying to find a reference that provides an answer to this question: Does $f_n\to f$ in $L^1(0,1)$ imply $\int_0^tf_n\to \int_0^tf$ uniformly?
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Show that $\sum_{n=1}^\infty \ \frac{x}{n(1+nx^2)}$ converges uniformly on $R$.

I need help with this problem: Show that $$\sum_{n=1}^\infty \ \frac{x}{n(1+nx^2)}$$ converges uniformly on $R$. So, I tried using the Weierstrass $M$-test. This is what I did: $f_n(x)=\frac{x}{n(...
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1answer
31 views

Uniform convergence of series of functions given convergence of coefficients

Prove that if $\sum{a_n}$ converges then the series $\sum{a_nx^n}$ converges uniformly on [0,1] I believe that I must use the Weierstrass M-test to show this convergence, but this requires that the ...
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2answers
40 views

Uniform and pointwise convergence of $ f_n(x)= \begin{cases} 0,\ x\leq n \\ x-n, \ x\geq n \end{cases} $ and $f_n(x)=x^n-x^{2n}$

I need help with this problem: For the next sequence {$f_n$}, determine the pointwise limit of {$f_n$} on the interval and indicate if {$f_n$} converges uniformly to that function. $ f_n(x)= \...
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1answer
81 views

Why $f_n(x)=\sqrt[n]{x}$ on $[0,1]$ doesn't converge uniformly?

I was solving this problem: For the following {$f_n$} sequence, determine the pointwise limit of {$f_n$} (if it exists) on the invterval, and indicate if {$f_n$} converges uniformly towards this ...
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2answers
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Proof of uniform convergence of $f_n(x) = \dfrac{x^n}{1+x^n}$ on every interval $[b, \infty[$ with $b>1$

I can only proof this for $b>2$. Then, you can find $N\in\mathbb{N}$ so that $\left|\dfrac{x^n}{1+x^n} -1\right| = \dfrac{1}{1+x^n} \leq \dfrac{1}{n}< \varepsilon$ for all $n\geq N$ and for all ...
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1answer
17 views

Uniform Convergence of $f_n^k$ and polynomial

Suppose $f_n$ are continuous functions converging uniformly to $f$ on the compact interval $[a,b]$. a) Show that for any natural power $k$, the sequence $f_n^k$ converges uniformly to $f^k$ on [...
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0answers
54 views

Proving discontinuity at $x=0$ for $f(x)=\sum_{n=0}^{\infty} \sin^{2}(nx)/ (1+n^{2} x^{2})$

Define $$f(x)=\sum_{n=0}^{\infty} \sin^{2}(nx)/ (1+n^{2} x^{2})$$ It was previously shown that $f$ is uniformly continuous on $\Bbb R_{\ne 0}$, $f$ converges (absolutely) for all real $x$ and that $...
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1answer
22 views

Uniform convergence of $f_n (x)=x^{2} - \frac{x}{n}$ on $[0, 1]$.

I need help with this problem: Consider the sequences of functions $(f_n)$ defined by: $$f_n (x)=x^{2} - \frac{x}{n}$$. Show that $(f_n)$ converges uniformly to $f$ on$[0, 1]$. I've already found ...
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2answers
47 views

Uniform convergence of $f_n(x)$

Given a sequence of functions $f_n(x)={{(x-1)^n}\over{1+x^n}} \arctan({n^{x-1}}).$ I have studied its pointwise convergence and I found as set of convergence $E=[0,+\infty)$ and as limit function the ...
2
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1answer
52 views

Exchanging limits when functions may not converge uniformly

Let $X$ be a compact metric space and for each $i \geq 1$, let $f_i \colon X \to \mathbb{N}$ be continuous functions satisfying: $f_{i+1}(x) \geq f_i(x)$ and; for each $n \in \mathbb{N}$ ...
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0answers
33 views

How can I prove that the following series does not converge uniformly. [duplicate]

How can I prove that the following series does not converge uniformly for $|y|<1$? could anyone give me a hint please? $$\sum_{n=1}^{\infty} \frac{y^n}{2n-1},$$
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2answers
31 views

Limit of Sequence of Differentiable Functions and Sequence of Derivatives

I have a two Cauchy sequences: $$ \{u_{n} \} \ \text{and} \ \{u'_{n}\} $$ Where $u_{n}$ is a continuously differentiable and bounded function on $\mathbb{R}$, and the sequences are Cauchy with respect ...
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2answers
45 views

Abel Means of an Integrable Function is Uniformly Convergent

I am studying Stein Shakarchi's book on Fourier Analysis. This page mentioned that the Abel means are absolutely convergent and uniformly convergent, just because $f$ is integrable. I did figure out ...