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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22 views

Non stationary solutions of the PDE $u_t + u_x = u_{xx}$

Problem. Consider the PDE $$ u_t + u_x = u_{xx}, \qquad (t,x) \in (0,+\infty) \times (0,1). $$ (i) Write the unique solution $\overline{u}=\overline{u}(x)$ which does not depend on time and ...
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0answers
49 views

If $f_n(t):=f(t^n)$ converges uniformly to continuous function then $f$ constant

Let $f \colon [0,1] \to \mathbb R$ be continuous and let $f_n$ be defined by $$ f_n(x):=f(x^n), \qquad x \in [0,1], \,\, n \in \mathbb N. $$ Suppose there exists a continuous function $g$ on ...
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0answers
49 views

Let $f_n(x)=x^n$ on $I=[0,1)$, does $\sum\limits_{n=1}^\infty f_n(x)$?

so I have a problem with 4 convergence questions. I have done the ones where I have to use other methods such as the ratio test and they were OK, but I cannot understand this one in particular. The ...
1
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1answer
33 views

Are the family of given nice functions $f\subset C^0(I,[0,1])$ equicontinuous?

The family of continuous functions $f\in\mathcal{F}$ are defined on a closed subset of real numbers $I\subset\mathbb{R}$ as follows: \begin{equation} f(y) = \begin{cases} 0, &l(y)<\rho \\ ...
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2answers
32 views

Uniform Convergence and Sup

Let $X \subset \mathbb{R}^d$ be open and let $(f_n)_{n \in \mathbb{N}} : X \to \mathbb{C}$ be a sequence of complex-valued functions. Suppose $f_n \to f$ uniformly on $X$. Fix $a \in X$. I would like ...
1
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0answers
32 views

Uniform convergence result in proof of second-derivative formula

This is a fairly basic analysis question. Consider a continuous function $f: \mathbb{R} \to \mathbb{R}$ which is twice differentiable at a point $x$. If necessary, also assume that $f \in ...
3
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1answer
39 views

Show that $\lim_{s \to \infty}F_s(t) = F(t)$ uniformly for $t \in (0,+\infty)$

Given the following functions: $$ F(t)= \int_0^\infty e^{-tx}\dfrac{\sin{x}}{x}\,dx, \quad t>0$$ $$ F_s(t)= \int_0^s e^{-tx}\dfrac{\sin{x}}{x}\,dx, \quad t \geq 0, s>0$$ Show that $\lim_{s \to ...
0
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1answer
35 views

A counter-example to Dini's Theorem (after removing a hypothesis)

Recall Dini's Theorem: Let $K$ be a compact metric space. Let $f: K\to\mathbb{R}$ be a continuous function and $f_{n}: K\to\mathbb{R}$, $n\in\mathbb{N}$ be a sequence of continuous functions. ...
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2answers
40 views

How to show that $f_n(x) = \frac{1}{1+nx^2}$ on $\mathbb{R}$ is not uniformly convergent

I need to understand pointwise and uniform convergence of function series and have several exercise examples. Most of them I was able to solve but I got problems showing that: $$ f_n(x) = ...
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2answers
37 views

It is a question about uniform convergence of a function.

I solved this problem . Is my answer a correct ? $$ x \in [0,\infty) ,\lim_{n \to \infty} \frac{nx}{1+n^2x^2}=0\ \ \ $$ Is $ \frac{nx}{1+n^2x^2} $ converged uniformly on $0$ ? My solution $$ ...
7
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2answers
58 views

Uniform convergence of $f^2_n$ when $f_n$ converges uniformly

Let $(f_n)$ be a sequence of functions that converge uniformly to $f$ on the interval $I$. Prove or disprove: $f^2_n \to f^2$ uniformly on I. I was almost certain this claim is false but was unable ...
3
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2answers
44 views

Show if the series $f(x)=\sum\limits_{k=1}^\infty \frac{1}{k} \sin(\frac{x}{k})$ converges uniformly or not.

So this is part of a different problem. The book and my professor say that the series $f(x)=\sum\limits_{k=1}^\infty \frac{1}{k} \sin(\frac{x}{k})$ converges uniformly on [0,1] by the Weierstrass ...
0
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2answers
40 views

$f_n:[0,1]\to [0,1]$ continuous, $f_n\to f$ uniformly, prove: ${\frac1n}\sum_{k=1}^{n}{f_k} \to f$ uniformly

I was able to prove, hopefully correctly, that the sum converges uniformly. But, I'm not sure how to show it converges uniformly specifically to $f$. The way I proved it converges uniformly was by ...
2
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1answer
42 views

Answer check on two series

I want to determine if these two are absolutely convergent, conditionally convergent or simply divergent. 1) $$\sum_{n=2}^\infty \left(\frac xn - \frac x{n-1}\right)$$ $$= \frac x2 - \frac x1 + ...
3
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1answer
37 views

The uniform limit of a sequence of functions

Which of the following sentences are not always correct? $A.$ The uniform limit of a sequence of differentiable functions is integrable. $B.$ The uniform limit of a sequence of integrable functions ...
2
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1answer
24 views

Prove that $f_n$ is uniformly bounded on $E$ and $f$ is a bounded function on $E$

I have the solution for the following problem, but I don't understand most of it. Question: A sequence of functions $f_n$ is said to be uniformly bounded on a set $E$ iff there exists $M>0$ such ...
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1answer
32 views

two question about uniformly convergence of sums

First question: I want to show that this series : $$\sum _{k=0}^{\infty }\:\frac{x^2}{\left(1+x^2\right)^k}$$ converges uniformly in $[a,\infty)$ for $a>0$. I thought about the M test and i ...
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1answer
27 views

Question about example i saw for uniform convergence

This is the sequence $f_n(x) = nxe^{−nx}$ on $(0,\infty)$. Well, i can see that $f_n$ tend to $0$ pointwise for every $x \in (0,\infty)$. But if i want to check if $f_n$ tend to $0$ uniformly in ...
0
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1answer
44 views

how to show a series is not uniformly converges?

Well , this is the function sequence:$$f_k\left(x\right)\:=\:\frac{1}{k+k^2x}$$ I want to prove that there is no uniform convergence for $\sum _{k=1}^{\infty }\:\frac{1}{k+k^2x}$ , in ($0$,$\infty $). ...
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0answers
19 views

Prove the sequence converges uniformly [duplicate]

I'm having trouble with this a bit. It is given that a sequence $f_{n} \rightarrow f$ is pointwise convergent. also that each function in the sequence is a Lipschitz with the same constant L meaning: ...
2
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3answers
201 views

Evaluate the limits

$$\lim_{p\rightarrow\infty}\int_0^1e^{-px}(\cos x)^2\text{d}x$$ I tried to prove the integrand is uniformly convergent so that the limit and integral can be exchanged. But I failed. Any ideas?
0
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1answer
53 views

The limit of $f(x,y)=\sum_{n=1}^\infty \frac{x}{x^2+yn^2}$ as $x\to \infty$.

Let $$f(x,y)=\sum_{n=1}^\infty \frac{x}{x^2+yn^2}, \ \ \ \ y\neq 0$$ how to show that for each $y\neq 0, g(y)=\lim_{x\rightarrow \infty} f(x,y)$ exists, evaluate $g(y)$. Then prove the convergence ...
1
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1answer
40 views

$\lim_{n\to \infty} \int^{\infty}_{-\infty} f_n(x) dx\, = \int^{\infty}_{-\infty} f(x) dx\,$

If $\{f_n(x)\}$ is sequence of continuous function on $\mathbb{R}$ converging uniformly to $f(x)$,then does $\lim_{n\to \infty} \int^{\infty}_{-\infty} f_n(x) dx\, = \int^{\infty}_{-\infty} f(x) ...
0
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3answers
43 views

If $f_n$ is a sequence of integrable functions that converges uniformly, is it's limit integrable?

Well, i want to prove that if $f_{n}$ is sequence of integrable functions on $[a,b]$, and $f_{n}\rightarrow f$ uniformly, then $f$ is also integrable on $[a,b]$. So far, i divided the proof for ...
0
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0answers
23 views

How to find out convergence of sequence of functions in short time.

I was studying about sequence of functions and about uniform and pointwise convergence of these series. To prove or disprove the convergence of sequence of functions we have several methods like by ...
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0answers
24 views

A question about uniform convergence in proof that $L^\infty$ is Banach.

I was reading this post Understanding proof of completeness of $L^{\infty}$ and it is mentioned that the sequence $(f_n)_{n\in\mathbb N}$ converges uniformly on a conegligible set $N^C$. Could someone ...
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0answers
21 views

Interchange of limits in computation with Gamma

On page 164 of Stein and Shakarchi's Complex Analysis there is the following computation: (here $0 < s < 1$, and line 1 -> 2 is a preceding lemma) \begin{align*} \Gamma(1 - s)\Gamma(s) &= ...
4
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2answers
63 views

Limit of $f_n(x) = n\log(\frac{nx+1}{nx-1})$ and its uniform covergence

Let $f_n(x) = n\log(\frac{nx+1}{nx-1})$. What is the pointwise limit of $f_n$ for $x \in[1,\infty]$? Is it uniformly convergent? $$\lim_{n\longrightarrow ...
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0answers
16 views

analyticity of an integral

Studying $$f(z)=\int_0^1 g(z,x)\ dx $$ where $g(z,x)$ is analytic in the open unit disk $D$ for all $x\in [0,1]$ and continuous for $|z| <1$ and $0\leq x \leq1$.Now, $$\lim_{n\to\infty} [1/n ...
1
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1answer
18 views

Find its pointwise limit and determine if its uniform

$f_k:[0,1]\to\Bbb R,$ $f_k(x)=\begin{cases}kx,x\in[0,\frac 1k]\\1,x\in[\frac1k,1]\end{cases}$ Find the pointwise limit of $(f_k(x)) $. And is this convergence uniform? for x=0 $f_k(0)=0$ and x=1 ...
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1answer
62 views

Uniform convergence of $\sum\limits_{n=1}^\infty \sin \left(\frac{x}{n^2}\right)$

Can someone please verify my answers? Consider the series $$\sum\limits_{n=1}^\infty \sin \left(\frac{x}{n^2}\right)$$ Prove that the series converges uniformly on the bounded interval $[-M, ...
1
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3answers
33 views

Uniform convergence and pointwise limit of $f_n(x)=\frac 1{1+x^n}$

let $f_n(x)=\frac 1{1+x^n}$ a)Find the pointwise lim. $f(x)=lim_{n\to\infty}f_n(x),n\in[0,\infty)$ b)is $f_n$ uniformly convergent on the interval $[0,\frac 12]$? for b) $x=0\to ...
1
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1answer
44 views

Is $f_n(x)=\frac {2nx}{1+n^2x^2}$ uniformly convergent?

$f_n(x)=\frac {2nx}{1+n^2x^2}, 0\leq x \leq1 $ we know $f_n(0)\to 0$ and $f_n(1)\to 0$ and $f'_n(x)=0\to x=\pm\frac1n$ since for x<1/n $f'_n(x)>0$ x=1/n is max. point. it seems ...
1
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1answer
55 views

Convergence of tricky series

I've stumbled upon a particularly unpleasant series, and I can't quite seem to crack it. $\sum\limits_{n=1}^{\infty}\dfrac{\ln(1+nx)}{n^{2}} $ I need to show uniform convergence on any interval of ...
1
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1answer
43 views

Which of the following functions on R are uniformly continous?

$a)\frac {1}{x^2+1} $ $b)\cos^3x$ $c)\frac {x^2}{x^2+2} $ $d)x\sin x$ $a)|\frac{1}{x^2+1}-\frac{1}{y^2+1}|\leq|\frac{|x-y|(|x|+|y|)}{(1+x^2)(1+y^2)}|\leq ...
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2answers
62 views

How to show $\sqrt x$ is uniformly continuous at [0,1] and $[1,\infty )$

From the definition if we choose $\delta=\epsilon^2$ $|\sqrt x−\sqrt y|^2≤|\sqrt x−\sqrt y||\sqrt x+\sqrt y|=|x−y|<ϵ^2⟹|\sqrt x−\sqrt y|<ϵ.$ does this suffice both interval [0,1] and ...
3
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2answers
67 views

$\int _0^{\pi }\:\sum _{n=0}^{\infty \:}\frac{n\cdot \sin \left(nx\right)}{e^n}dx=\frac{2e}{e^2-1}$

It is asked to prove: $\int _0^{\pi }\:\sum _{n=0}^{\infty \:}\frac{n\cdot \sin \left(nx\right)}{e^n}dx=\frac{2e}{e^2-1}$ I have tried to search for convergence and it gave me 0 so i can't solve it. ...
0
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1answer
31 views

Uniform convergence on functions

For a $f(x)=x^n, x\in[0,0.5]$ I know that $d_\infty (x^n,0)=sup|x^n-0|=\frac{1}{x^n} \rightarrow 0 $, when $n\rightarrow \infty$ and $x\in[0,0.5]$. For $\{x^n-x^{n+1}\}, x\in[0,1]$ this isn't the ...
0
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2answers
52 views

Taylor series of $\sin(x)$ converges uniformly on $[-\pi,\pi]$?

According to my notes, the Taylor series of $\sin(x)$ converges uniformly on $[-\pi,\pi]$. I know that the remainder term needs to converge uniformly to $0$ for this to be the case. But I really ...
1
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1answer
34 views

If $f_n$ converges uniformly then $\cos(t)f_n$ converges uniformly?

In my Fourier Series course, it seems the following result is used: If a series of function $\sum a_n(t)$ converges uniformly then the sequence of functions $\cos(t)\sum a_n(t)$ converges ...
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1answer
33 views

Question on regulated functions

Suppose that $f:[a,b]→\mathbb{R}$ is a regulated function. Then the integral $\int_a^bf(x)dx$ is defined as the limit $\lim_{n→∞}\int_a^bs_ndx$, where $(s_n)_{n∈\mathbb{N}}$ is a sequence of step ...
0
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2answers
67 views

Prove $f(x)=\sum_{k=1}^\infty \frac{1}{k!\sqrt{k}}x^k $ defines a continuous function on $\mathbb{R}$.

Prove $$f(x)=\sum_{k=1}^\infty \frac{1}{k!\sqrt{k}}x^k$$ defines a continuous function on $\mathbb{R}$. I think we can show that if $\sum_{k=1}^\infty \frac{1}{k!\sqrt{k}}x^k $ is uniformly ...
0
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2answers
27 views

help with pointwise convergence and uniform convergence

Consider D a finite set, and $f_{n\:}$ pointwise converge for every $d\in D$. How to prove that $f_n$ converges uniformly in D?
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2answers
44 views

Proving that $f_n(x)=\sin^{n}(x)$ is uniformly convergent.

Consider $$f_n\left(x\right)\:=\:\sin ^n\left(x\right)$$ How to prove that this sequence is uniformly convergent in $\left[0,b\right]\: \text{for}\:\frac{\pi }{2}>b>0$?
1
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0answers
22 views

Uniform Convergence of integrals of sequences of functions

Just looking for feedback on if I am thinking about this correctly: Let {$f_n$} be a sequence in $\mathscr{R}[a,b]$ (the set of Riemann integrable functions on $[a,b]$, not sure if this is standard ...
1
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2answers
65 views

Is the series uniform convergent in $(0,\infty)$?

For $$f(x)=\sum_{n=1}^{\infty}\frac{1}{1+n^2x}$$ And is it bounded in $(0,\infty)$?
1
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3answers
35 views

Show that the series is Uniformly convergent

Show that the series $$\sum {x\over1+n^2x}$$ is uniformly convergent in $[\delta,1]$ for any $\delta>0$ but not uniformly convergent in $[0,1]$ I am unable to find a suitable $M_n$ for use in the ...
0
votes
1answer
25 views

Show that the given series in Uniformly convergent

Show that the series $$1+{e^{-2x}\over2^2-1}-{e^{-4x}\over4^2-1}+{e^{-6x}\over6^2-1}-\cdot\cdot\cdot$$ is uniformly convergent for all real $x\ge0.$ I tried applying the Dirichlet's theorem to the ...
1
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0answers
40 views

How to show a sequence of functions does not converge uniformly

Let $f$ be a continuous function on $[0, \infty)$ such that $0\leq f \leq Cx^{-1-\rho}$, where $C$ and $\rho$ are positive constants. Let $f_k(x)=kf(kx)$. $\textbf{Question}$: Show that $f_k$ does ...
0
votes
2answers
22 views

Show uniform convergence of the sequence $\langle x-x^n/n\rangle$

Show uniform convergence of the sequence $\langle x-x^n/n\rangle$ on $[0,1]$ To start with, i am not even able to see what the point was limit of this sequence should be. Little confused. like for ...