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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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Asymptotic Normality lemma (Serfling - 1980)

I'd like some assistant on the proof of the following Lemma: If $X_n$ is $AN(\mu,\sigma_n^2)$, then also $X_n$ is $AN(\overline\mu,\overline\sigma_n^2)$ if and only if $\frac{\overline\sigma_n}{\...
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Uniform convergence of a two variable real analytic function with respect two one variable.

Let $f(x,y)=\sum_{i,j} a_{ij}x^i y^j$ be a two variable real analytic function defined on $\mathbb{R}^2$ such that: 1- $\sum_{i,j} a_{ij}x^i y^j$ is absolutely convergent of all $x,y \in \mathbb{R}$....
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Uniform convergence of polynomial approximation on Schwartz space

I have a question regarding uniform convergence of basis expansion in Schwartz space. For $L^2(\mathbb{R},\lambda)$, $\lambda$ Lebesgue measure, the partial sums of basis expansion (Hermite functions) ...
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1answer
19 views

Uniform convergence about the function $h$

Consider the sequence of functions $$h_n(x)=\begin{cases}2nx&\text{if}\;x \in [0,\frac{1}{2n}]\\\\2-2nx&\text{if}\;x \in [\frac{1}{2n},\frac{1}{n}]\\\\0&\text{if}\;x\geq \frac{1}{n} \end{...
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1answer
33 views

Struggling with finding a potential counterexample for a convergent series.

This question comes with two parts. Part (a): Let $\{f_n(x)\}$ be a sequence of nonnegative functions for $x \in S \subseteq \mathbb{R}$ such that $f_1 \geq f_2 \geq \dots \geq 0$, and that $f_n \to ...
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Uniform convergence and integrals when domain is not a compact set.

Suppose sequence of continuous functions ${f_n}$ that converges uniformly to a continuous function $f$ on a closed interval $[a,b]$, then we have $$\lim_{n\to\infty}\int_a^b f_n(x) dx = \int_a^bf(x) ...
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Sequence of bounded variation functions that converge uniformly to a function with unbounded variation

I need help with this exercise: Prove that $\exists (f_{n})_{n=1}^{\infty}\subset BV([0,1])$ such that $f_{n}\to f$ uniformly, but $\|f_{n}-f\|_{BV}\not\to 0$. So I proposed the sequence $(f_{n})_{n=...
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Convergence of harmonic functions in $L^1$ implies uniform convergence on compact sets

Resorting to an analog of what's done here, I'm trying to prove the following statement: Let $u_m: \mathbb{R}^n \to \mathbb{R}$ be a sequence of harmonic functions and suppose there exists a ...
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1answer
64 views

Uniform convergence of $\sum_{n=1}^{\infty} \frac{(-1)^{n-1}x^2}{(1 + x^2)^n}$ on $\Bbb{R}$.

I was trying to prove that $\sum_{n=1}^{\infty} \frac{(-1)^{n-1}x^2}{(1 + x^2)^n}$ converges uniformly on $\Bbb{R}$. was trying to use $M$ test, in which I am trying to bound $|f_{n}(x)| < M_{n} $ ...
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Why is $\sum_{n=1}^{\infty} x^n-x^{n+1} $ not uniformly convergent in $[0,1)$?

So, I needed to determine whether $\sum_{n=1}^{\infty} x^n-x^{n+1} $ is uniformly convergent in $[0,1)$. This is what I tried to do: First, this a series of functions where $f_n(x)=x^n-x^{n+1}$ for ...
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1answer
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Is it possible to apply the Dirichlet's uniform convergence test

For a series of functions given by $$\sum_{k=1}^{\infty}\frac{1}{k}\sin\left(\frac{x}{k+1}\right)$$ on some bounded nonempty set $A$ in $\mathbb{R}$, is it possible to apply the Dirichlet's uniform ...
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1answer
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uncontinuous sequence function convergent uniforme [duplicate]

i wanted to know if there is some example of uncontinuous function converging uniformly to a continuous function. Does a function converge uniformly to a bounded function must be bounded too ?
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$\lim_{n \to \infty} n \int_{0}^{100}f(x)g(nx)dx=f(0)$

Let $g: \mathbb{R} \to \mathbb{R}$ be a continuous function with $g(y)=0$ for all $y \notin [0,1]$ and $\int_{0}^{1}g(y)dy=1$. Let $f: \mathbb{R} \to \mathbb{R}$ be a twice differentiable function. ...
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Is it true that $f_n(x)=\left(\frac{n}{n+x}\right)^n$ uniformly converges to $e^{-x}$

Is it true that: $$f_n(x)=\left(\frac{n}{n+x}\right)^n$$ uniformly converges to $e^{-x}$? I have tried to prove it using definition with supremum, but have not succeed. Can you give me a hint, please?...
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Is the dominated convergence theorem applicable whenever “THIS” theoem is applicable?

THIS theorem: Let $I =[a,b]$ be a closed and bounded interval and $\forall n\in \mathbb{N}$, $f_n:I \to \mathbb{R}$ be Riemann integrable on $I$. If the sequence $(f_n)$ converges uniformly to a ...
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1answer
22 views

Uniform convergence- Point-wise convergence. Doubt regarding the difference

I know the definition of "pointwise convergence" and "uniform convergence", nevertheless I have some difficulties understanding the difference between those two concepts. My book defines Uniform ...
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1answer
41 views

Is $S{(x)}$ will uniformly convergent on $[-1,1]$ ? True/false .

Is $S(x) = \sum_{n=1}^{\infty} x^2 (1-x^2)^{n-1}$ will uniformly convergent on $[-1,1]$ ? True/false . My attempt : Yes , $S_n$ will uniformly convergent on $[-1,1]$ if i put $x = \frac{1}{\...
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1answer
60 views

Is $ S_n$ is uniformly convergent ? Yes/NO

Is $\displaystyle S_n = \sum_{n=1}^{\infty} 2^n \sin\frac{1}{3^nx}$ uniformly convergent on the interval $[1, \infty)$ ? True /false My attempt : NO, the given series will not uniformly ...
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1answer
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Uniform Cauchyness

Assume the sum of $|f_n|$ is uniformly cauchy. Does this imply that the sum of $(f_n)$ is uniformly cauchy? My reasoning is yes, since every $f_n$ is bounded by $|f_n|$.
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If a sequence of functions $f_n$ converges uniformly to $f$, does this imply $f_n$ is continuous? How to prove?

I know I can find an $f_n$ that converges to f uniformly, but can we use this to argue $f_n$ is continuous? Is there a theorem I can state?
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Suppose $\hat F_n(y)\overset{p}\to F(y)$, can we obtain $\sup_y|\hat F_n(y)-F(y)|\overset{p}\to0$?

If we have (i) $\hat F_n(y)\overset{p}\to F(y)$; (ii) $|F(y)|\le1$; (iii) $F(y)$ is a monotonically increasing function with respect to $y$. Can we obtain the following result directly $$\sup_y|\...
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2answers
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Uniform convergence of $x^n$ on interval $[0,b]$ for $0 \leq b < 1$.

Suppose we have $f_n (x) = x^n$ for $x \in [0,b]$ for $0 \leq b < 1$. If $\epsilon > 1$, any value of $K$ will do because every $x^n$ for any $x$ or any $n$ is less than $1$. If $0 < \...
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1answer
60 views

How to compute the $n^{th}$ partial sum of a series?

Compute explicitly $S_n(x)$, the $n^{th}$ partial sum of the series $$\sum_{k=1}^∞ \frac{x\left[-1+4k(k+1)x^2\right]}{(1+4k^2x^2)(1+4(k+1)^2x^2)}$$ then compute the sum $S(x)$ of the infnite series, ...
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0answers
64 views

Calculate $\lim_{n \to \infty} \int_0^1nx^nf(x)dx$. [duplicate]

Consider the function $f$ which is continuous. Calculate $\lim_{n \to \infty} \int_0^1nx^nf(x)dx$. Here first I attempted to prove $f_n=nx^n$ is uniformly convergent using sup-norm limit but ...
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1answer
42 views

Supremum of uniformly converges functions goes to the supremum of the limiting function

Let ${ f_n\left(x\right):[0,1] \to \mathbb{R} }$ uniformly convergent to function $f(x)$ which is bounded on the interval $[0,1]$. Prove the following: ${ \lim_{n\to\infty} sup_{[0,1]} f_n\left(x\...
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58 views

I have a question on infinite products

The question: Suppose $F(z) = \prod_{j = 1}^{\infty} [1 + f_{j}(z)]$ converges uniformily on compact subsets of a open set $U \subseteq \mathbb{C}$ to a limit that could be zero or non-zero. Suppose ...
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1answer
28 views

Defining a subset of uniform convergence of a succession of functions

I am new to convergence analysis of successions of functions, but I'm quite stuck on this particular exercise. $$f_n(x)=-\arctan\left(\frac{nx}{n+x}\right)$$ is a succession of functions. Study ...
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28 views

Computing $S_n(x)$, the partial sum of a series explicitly

Compute explicitly $S_n(x)$, the $n^{th}$ partial sum of the series $$\sum_{k=1}^∞ \frac{x\left[-1+4k(k+1)x^2\right]}{(1+4k^2x^2)(1+4(k+1)^2x^2)}$$ then compute the sum $S(x)$ of the infnite series, ...
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2answers
22 views

Uniform convergence of a seriesobtained by dominating it with another series

Let $\{f_n\}$ be a sequence of complex valued functions on the real line and $n$ be integers. If there is a sequence of nonnegative numbers $\{a_n\}$ such that $\sum _{n} \mid f_n(x) \mid \leq \sum_{...
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31 views

Prove that the sum of a series is differentiable

Prove that the series $$\sum_{k=2}^∞ \sin (kx)/k\ln^2(k)$$ is absolutely and uniformly convergent on $\mathbb{R}$. If the sum of the series is denoted by $f(x)$ prove that $f$ is differentiable at ...
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2answers
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Uniform convergence of a sequence of functions 4

Prove that the sequence $\left((nx)/(1+4n^2x^2)\right)_{n\in\mathbb N}$ is not uniformly convergent on $(-a,a)$, where $a > 0$ My attempt: $\lim_{n\to\infty}(nx)/(1+4n^2x^2) = 0 = f(x)$ Now, ...
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1answer
44 views

Prove that the sequence is pointwise convergent.

Let $f_n:\Bbb R\to \Bbb R$ be a continuously differentiable sequence of functions and assume that the sequence $f_n^{'}$ converges uniformly on $\Bbb R$. Also Assume that the sequence $f_n(0)$ ...
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1answer
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Puntual and uniform convergence of function

I am facing the following problem: Let $H:[0,1] \to R$ a function such thath $H(0) = 0 = H(1)$, H continous in $x = 0$ and $H(x) \neq 0$ for some $x$. Let $H_n:[0,1] \to R$ be such that $H_n (x) = H(...
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2answers
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Uniform or pointwise convergence of a sequence of functions

My problem is that I find it kind of hard to contrast between uniform and pointwise convergence. For example with this proof I'm not quite sure whether I have proven uniform or poitwise convergence: $...
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1answer
21 views

Uniform convergence NBHM(2019) [closed]

Check the uniform convergence of following : 1) $f_{n}(x)=n\log(1+ \frac{x^{2}}{n})$ on $\mathbb{R}$ 2) The series $\sum_{1}^{\infty}2^{n}$$\sin(\frac{1}{3^{n}x})$ on $[1,\infty)$ For the ...
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1answer
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Sequence of polynomials $f_n [0, 1] \to \mathbb{R}$ of the same degree converges uniformly to a polynomial

Prove that if $f_n : [0, 1] \to \mathbb{R}$ is a sequence of polynomials of the same degree and $f_n$ converges uniformly to $f$ then $f$ is a polynomial. I've seen similar questions on MSE, but with ...
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1answer
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Is $f_n (x)$ is uniformly convergent on $[0, \pi /2]$?

Is $f_n(x) = \cos^n x(1-\cos^nx)$ is uniformly convergent on $[0, \pi/2]$? My attempt : I thinks yes. here Sup $\{ |f_n(x) | : x \in [0, \pi/2]\}= 0$ so $f_n(x) = \cos^n x(1-\cos^nx)$ converge ...
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1answer
128 views

Convergence of $\sum_{n=1}^\infty \frac{z^n}{2^n(1-z^n)}$

I'm solving the following problem: Find the maximal open set, $\Omega,$ where the following series converges: $$\sum_{n=1}^\infty \dfrac{z^n}{2^n(1-z^n)}.$$ Extra: Prove that the series ...
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1answer
42 views

Lim Supremum fn(x) = sup f(x)

We have a sequence of function converging uniformly to f(x) which is bound to [0;1] I want to proove that the lim of the sup fn(x) = sup f(x) . Can someone help me ?
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0answers
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How to prove $\sup_y\left|\frac1n\sum_{i=1}^n F(y|X_i)-E(F(y|X))\right|\overset{p}\to0?$

Suppose we have two random variable $X$ and $Y$. The conditional CDF of $Y|X$ is defined as $F(y|X)=P(Y\le y|X)$. Let $X_1,X_2,\cdots, X_n$ be random sample of $X$. Show that $$\sup_y\left|\frac1n\...
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1answer
26 views

Uniform convergence but limit not continuous [closed]

i wanted to know if someone have an example of sequence of functions converging uniformly but such that the limit function (f) is not continuous .
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2answers
48 views

Uniform convergence of $\sum_{n=0} ^\infty \frac{(-1)^n}{2n+1} \left(\frac{2z}{1-z^2}\right)^{2n+1}$

I am given: $$ f(z)=\sum_{n=0} ^\infty \frac{(-1)^n}{2n+1} \left(\frac{2z}{1-z^2}\right)^{2n+1}$$ And asked to show that this complex function series converges uniformly for $|z|\leq \frac{1}{3}$. ...
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1answer
59 views

$f_n(x)=n(f(x+\frac{1}{n})-f(x))$ converges uniformly to derivative

Let $f$ be a continuous function with continuous derivative on $(a,b)\subseteq\mathbb{R}$. Define $f_n(x)=n(f(x+\frac{1}{n})-f(x))$. Prove that $f_n$ converges uniformly to $f'$ on any ...
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1answer
39 views

Prove that $(f_n)$, $f_n =x^n$, $x \in (0,1)$ is not uniformly convergent on $(0,1)$

Question 1. Prove that $f_n:(0,1) \to \mathbb{R}$ is not uniformly convergent on $(0,1)$, where $f_n = x^n , n\in \mathbb{N}$ . Proof: We need to show that, $\forall \ k \in \mathbb{N} $, $\exists \...
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1answer
67 views

If $f_n\to f$ uniformly, then $\frac{1}{n}\log(f_n)\to 0$ uniformly

Let $X$ be a compact subset of $\mathbb{R}$. Let $f_n:X\to (0,\infty)$ be a sequence of function converging uniformly to a function $f:X\to(0,\infty)$. Show that the sequence of functions $\left(\frac{...
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1answer
26 views

Uniform Convergence and Improper Integration

Let $g:(0,\infty)\to\mathbb{R}$ and $f_n:(0,\infty)\to\mathbb{R}, n\ge 1$ be integrable functions over $[a,b]$ for any $0<a<b<\infty$. Also suppose that (1) $|f_n(x)| < g(x)$ for all $...
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1answer
36 views

Uniform convergence of a sequence of functions given by integrals

If $f_n:(0,\infty)\to \mathbb{R}$ is a sequence of functions converging uniformly to a function $f:(0,\infty)\to\mathbb{R}$, and also if each $f_n$ is Riemann-integrable over $[a,b]$ for any $0<a&...
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2answers
29 views

Uniform convergence to f(x) = 0

I want to prove that $$f_n(x) = \frac{\sin(n^2 x)}{n}$$ converges to $f(x) = 0$. It's easy to see that $$\lim_{n\to\infty} \frac{\sin(n^2 x)}{n}= 0$$ but how can I prove the uniforme convergence ?
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4answers
64 views

Uniform Convergence of $\sum_{k=0}^\infty\frac{kx}{1+k^4x^2}$

This problem arose in trying to establish that $$f(x)=\sum_{k=0}^\infty\frac{kx}{1+k^4x^2}$$ is uniformly convergent on $[a,\infty), a>0$, and I thought I could do this, as I was able to show that ...
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2answers
26 views

Uniform convergence of $f_n(x)$ defined on unit interval

Consider a function $f$ defined on $[0,1] $ to the reals and define $$f_n(x)=\frac{\lfloor nf(x)\rfloor}{n}$$ Show that $f_n$ converges uniformly to $f.$ I'm having trouble dealing with the floor ...