For questions about uniform convergence of a sequence or a series of functions on a set. Also used with the tag [convergence].

learn more… | top users | synonyms

1
vote
2answers
69 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 ...
7
votes
0answers
41 views
+50

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 ...
0
votes
2answers
24 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 ...
1
vote
1answer
33 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 ...
2
votes
0answers
45 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 ...
3
votes
1answer
43 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 ...
0
votes
3answers
44 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} ...
1
vote
4answers
38 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 $f$ ...
3
votes
1answer
32 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?
0
votes
0answers
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 ...
2
votes
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 ...
2
votes
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 ...
1
vote
0answers
26 views

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), ...
1
vote
0answers
22 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: ...
0
votes
2answers
62 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 [$0,1$] ...
3
votes
2answers
70 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 ...
0
votes
0answers
51 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 ...
2
votes
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} ...
0
votes
0answers
28 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 ...
0
votes
3answers
64 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 ...
0
votes
2answers
56 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 $$ ...
0
votes
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
votes
1answer
45 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
vote
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 ...
2
votes
2answers
31 views

Uniform Convergence of $\sum_{n=0}^\infty (1-x^2)^2x^n$ on $[0,1]$; subsequent integral

Let $a_n = (1-x^2)^2 x^n$. Show that $\sum_{n=0}^\infty a_n$ converges uniformly on $[0,1]$ and deduce that $\int_0^1 \frac{(1-x^2)^2}{1-x} dx = \sum_{n=1}^\infty \frac{8}{n(n+2)(n+4)}$. Attempt: ...
0
votes
1answer
35 views

$f_n ^{'}$ is uniformly convergent but the seq $f_n$ is not uniformly convergent.

Let $f_n(x) = n + x/n , x \in \mathbb R$. Then the seq $f_n ^{'}$ is uniformly convergent but the seq $f_n$ is not uniformly convergent. We see that $f_n ^{'} = 1/n$ which is trivially uniformly ...
0
votes
3answers
47 views

Uniform convergence and integration: $\lim_{n \to \infty} \int_{0}^{1} \frac{1}{1+x^2+\frac{x^4}{n}} dx=?$

For f$_n$(x)=$\frac{1}{1+x^2+\frac{x^4}{n}}$, we need to calculate $\lim_{n \to \infty} \int_{0}^{1} f_n(x) dx$ . I want to prove f$_n$ is Riemann integrable and f$_n$ uniformly converges to f, then ...
0
votes
0answers
37 views

help with investigating the uniformly convergence of a function sequence.

I have to check the uniform convergence of the below mentioned function sequence: $f_n(x) = \frac{1-\ln x}{nx}$ while $0<x<1$ at the answers, it's told that the sequence doesn't converge ...
0
votes
2answers
55 views

Uniform convergence of improper integrals

I'm having trouble w/ the following result, which is Ch 6, Thm 15, in Buck's Advanced Calculus: If $f(x,t)$ is continuous on $x\geq b$,$t\geq c$, $\int_c^\infty f(x,t)\;dt=F(x)$, uniformly on $x\geq ...
1
vote
1answer
21 views

Sequence uniform convergence but the derivatives are not.

Give a sequence $(f_n)_{n\in \mathbb{N}}$ of differentiable functions which uniformly converge to $0$, but for which the seqeunce $(f_n')_{n\in \mathbb{N}}$ of the derivatives isn't even pointwise ...
1
vote
2answers
48 views

Is $\sum_{n=1}^\infty a_n\sin(nx)$ converges on $[\varepsilon, 2\pi-\varepsilon]$?

Let $a_n$, a sequence monotonically decreasing to $0$. Consider $$\sum_{n=1}^\infty a_n\sin(nx)$$ Is the series converges uniformly on $[\varepsilon, 2\pi-\varepsilon]$? ($\varepsilon ...
3
votes
2answers
212 views

How to prove the non-existence of a polynomial series that uniformly converges to this function?

I was asked to prove the following. There exists a function $p(x)\in C(a,b)$ $(-\infty<a<b<+\infty)$, such that there does not exists a polynomial series which uniformly converges to ...
1
vote
1answer
31 views

Uniform convergence on an interval.

Let $f_n(x)=\frac{x^n}{1+x^n},~x\geq 0, ~n\in \mathbb{N}$ Show that there is no uniform convergence on $[1,+\infty[$. I found this particular part of an exercise in my textbook and ...
1
vote
0answers
40 views

Approximating functions such that the left- and right limit exists everywhere

Every continuous function $f : \mathbb R \to \mathbb R$ could be uniformly approximated by step functions. For a proof consider an interval $[a,b]$, then $f$ is bounded on this compact interval, i.e. ...
0
votes
1answer
39 views

Finding a metric such that $\Phi$ becomes a contraction

Let $$ f:\:[0,1]\rightarrow\mathbb{R};\\ \Phi: \mathcal{F}([0,1], \mathbb{R}) \rightarrow \mathcal{F}([0,1], \mathbb{R}),\\ f\mapsto\Phi(f):=Φf := \left(x\mapsto\left\{ ...
0
votes
2answers
41 views

How can I show uniform convergence?

Let $f_n(x)=\frac{x^n}{1+x^n},~x\geq 0, ~n\in\mathbb{N}$. 1.1.: Determine the pointwise limi of $(f_n)$, $x\geq 0$. 1.2.: Show that the sequence $(f_n)$ is uniformly convergent on ...
0
votes
0answers
41 views

Checking if $f_n(x)=\sqrt[n]{1+x^n}$ uniformly converges.

$f_n(x)=\sqrt[n]{1+x^n}$. First of all, I have to find its different limits so that I can find $f(x)$. This is where I have a problem: if $x> 1$, then $x\le f_n(x)=f(x)\le 2x$, but I can't get the ...
1
vote
0answers
25 views

Converse of uniform convergence theorem

Let $f(x) = \sum_{i=1}^\infty f_i(x)$. Suppose that $f$ is continuous, and each $f_i$ is continuous. Does it follow that the series converges uniformly to $f$?
1
vote
1answer
34 views

For which values of $x \in I$ is $(f_n)$ differentiable term by term?

Let $f_n(x) = \frac{1}{n}e^{-nx}$ on $x\in[0,1] =I$. Discuss the pointwise and uniform convergence of $(f_n)$ on $I$. For which values of $x \in I$ is $(f_n)$ differentiable term by term? ...
2
votes
1answer
39 views

Uniform convergence in the endpoints of an interval

Study the pointwise and uniform convergence of the series $$\sum_{n=1}^\infty\dfrac{4^n}{n^2}\dfrac1{(1+x^2)^n}$$ I'm doing this exercise and I'm not sure about the following: What I've done ...
0
votes
0answers
40 views

A Simple Question About Directional Derivatives

I am stuck with this one question in our worksheets. The question is : Let $f:\mathbb{R}^n\mapsto \mathbb{R}$ and $x\in\mathbb{R}^n$. For all $v∈\mathbb{R}^n$ the directional derivative exists and ...
0
votes
1answer
48 views

How to show uniform convergence of series

Let $$f(t) = \sum_{k=0}^\infty ke^{-t\sqrt{k}}u_k$$ for $t \in (0,\infty)$, where the $u_k$ is such that $\sum \sqrt{k}u_k$ converges, but we know nothing about the convergence of $\sum ku_k$. How do ...
3
votes
1answer
70 views

If $f_n \to f$ uniformly and $f$ is continuous, does that imply $f_n$ is continuous?

I have a theorem in my book which says if $(f_n)$ is a sequence of functions uniformly converging on $A$ to $f$, and is continuous at some point $c \in A$, then $f$ is also continuous at this point ...
1
vote
2answers
61 views

Does the Taylor series of $\ln (1-x)$ converge uniformly on $[0, 1)$?

We know that: $$\ln (1-x) = -\sum_{n=1}^{\infty} \frac{x^n}{n}$$ Does the Taylor series converge uniformly on $[0, 1)$? I guess the answer is yes. What I have tried to do is that, after I showed ...
0
votes
1answer
30 views

Help with convergence of series of functions

We have this series of functions $\sum\limits_{n=1}^{\infty}nx^n(1-x)$ for $x$ ranging from $[0, 1]$. How do we determine whether it's uniformly convergent or not? Thanks in advance.
2
votes
2answers
24 views

Uniform convergence method?

I am self-studying analysis and have reached the topic of uniform convergence. When it comes to series, it seems that Weierstrass' M-test is a powerful tool to determine uniform convergence. However, ...
0
votes
1answer
30 views

Convergence in distribution and weak convergence implies convergence of expectations

I'm preparing myself for the final exam of my graduate Probability Theory course and was stuck with one of the exercises our professor gave us. Let $x_n, n=1,2,\cdots$, and $X$ be random variables ...
2
votes
1answer
55 views

Prove $f_n(x)$ converges pointwise to some $f(x)$ but the convergence is not uniform

EDIT: Given $f_n(x)=\frac{x}{1+x^2}-\frac{(x^2+1)x}{1+(n+1)^{2}x^{2}}$ Show that $f_n(x)$ converges pointwise to some $f(x)$ but that the convergence is not uniform. EDIT: I tried finding ...
1
vote
1answer
20 views

Uniform convergence for sequence

For a series, we can use Weierstrass to show uniform convergence. What test does one have for a sequence? Here's the one I am working on right now: $$\frac{ln(x^n)} {1 + x^n} \ , \ x \in [a, ...
3
votes
2answers
63 views

Uniform convergence in distribution

Consider a sequence of stochastic processes, $X_n(x)$ and a limiting process $X(x)$. For a fixed $x$, if $\mathbb{P}(X_n(x) \leq y)$ converges to $\mathbb{P}(X(x) \leq y)$ for continuity points of ...