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
36 views

Does $(f_n)$ converge pointwise/uniformly on $I$?

Does $(f_n)$ converge pointwise/uniformly on $I$ if $$f_n(x) = \frac{x^n}{1+x^n} ~~~~~~ I=[0,1]$$ My attempt: If $x \in [0,1): \displaystyle \lim_{n \to \infty}f_n(x) = 0$ If $x=1: ...
0
votes
0answers
23 views

Sums convergent but not uniformly convergent on [0,1]

Show that both $\sum_{n=1}^{\infty} ({1-x}){x^n}$ and $\sum_{n=1}^{\infty} (-1)^n({1-x}){x^n}$ are convergent on [0,1] but only one converges uniformly. Which one? Why? I was playing around with the ...
3
votes
3answers
74 views

How to show that $\sum_{k=0}^{\infty} \frac{x^{k}}{k!}$ represents a continuous function

(This is a homework problem) I am trying to show that the series $\sum_{k=0}^{\infty} \frac{x^{k}}{k!}$ represents a continuous function on $\mathbb{R}$. My idea was to show that the functions ...
1
vote
2answers
62 views

How do I determine if $f_n\to f$ uniformly on $[0,\infty)$? [on hold]

For $x\in [0,\infty)$, let $f_n(x) = \frac{x^n}{n+x^n}$. Not sure if it converges or not, but can someone help explain it to me using the definition of uniform convergence?
0
votes
2answers
48 views

Uniformly cauchy sequences

A sequence of functions $f_n$ is said to be uniformly cauchy if $$\forall \varepsilon > 0 \ \exists N > 0 :\forall z , \forall r, s > N: |f_r(z) - f_s(z)| < \varepsilon$$ How can I show ...
0
votes
1answer
28 views

Uniform convergence of real function

Let $f:[a,b] \to \mathbb{C}$ where $0 < a < b < 2 \pi$ be defined by $f(x) = \sum_{k=1}^{\infty} \frac{exp(ikx)}{k}$. Show that $f$ converges uniformly in $[a,b]$. The problem is that I ...
1
vote
1answer
29 views

Explicit example of normal family

Suppose $\mathscr F \subset H(\Omega$) for some region (i.e. open connected) $\Omega$. ($H(\Omega)$ means the set of all holomorphic function in $\Omega$) We call $\mathscr F$a normal family if every ...
3
votes
1answer
34 views

Weak convergence in probability implies uniform convergence in distribution functions

This is a problem that I am totally stuck at. I know the fact that $F_n$ converges pointwise to $F$ in this question. Also, I looked through Google and found out that I have to show first that ...
2
votes
1answer
33 views

Uniform convergence for the sequence $\psi_n(x)=n^{-1}e^{-n^2/(n^2-x^2)}$

How can I prove that the sequence of functions: \begin{equation} \psi_n(x)=\begin{cases} n^{-1}e^{-n^2/(n^2-x^2)}, & |x|\leq n \\ 0, & |x|\geq n \end{cases} \end{equation} convergences ...
2
votes
1answer
45 views

How do I prove that $f_n$ converges to $f$ uniformly on the interval $[\frac{1}{3},1]$?

Let $f_n(x) = \frac{1}{n^2x}$ and let $f(x)=0$. You may take it as a given that $f_n$ converges to $f$ pointwise for all $x\neq 0$. (a) Prove that $f_n$ converges to $f$ uniformly on the interval ...
0
votes
5answers
67 views

How can I write $\lim_{n\to\infty} \frac{1}{n^6}{(1+2^5+…+n^5)}$ as an Integral?

How can I write this series as a definite integral? $\lim_{n\to\infty} \frac{1}{n^6}{(1+2^5+...+n^5)}$ We've just covered $\lim_{n\to\infty}\int_{a}^{b}f_n=\int_{a}^{b}\lim_{n\to\infty}f_n$ When ...
2
votes
1answer
19 views

Uniform convergence of $(f_n(x)) = \frac{nx}{n+x}$ on $I=[0,1]$

Test whether or not $(f_n(x)) = \frac{nx}{n+x}$ on $I=[0,1]$ converges uniformly on $I=[0,1]$. My attempt: $\displaystyle \lim_{n \to \infty}f_n(x) = \lim_{n \to \infty}\frac{nx}{n+x} = \lim_{n ...
4
votes
1answer
23 views

Does $(f_n(x))= (\frac{nx}{1+nx^2})$ converge pointwise/uniformly on $I= [0,1]$?

Does $\displaystyle(f_n(x))= \bigg(\frac{nx}{1+nx^2}\bigg)$ converge pointwise/uniformly on $I= [0,1]$? My attempt: Pointwise: $\displaystyle \lim_{n \to \infty}f_n(x) = \lim_{n \to \infty} ...
1
vote
1answer
25 views

Pointwise but not uniform convergence of continuous functions on [0,1]

As I was going over the definitions of pointwise and uniform convergence I came to the following problem: since the canonical example for continuous functions on $[0,1)$ which are pointwise but bot ...
0
votes
1answer
31 views

$\sum_{n=1}^{\infty} \frac {\sin(nx)}{\sqrt{n}}$ uniform convergence in $[0,2\pi]$

Let $\sum_{n=1}^{\infty} \frac {\sin(nx)}{\sqrt {n}}$ be a series of functions. How can I show that it is not uniformly convergent in $[0,2\pi]$? I thought about the series $\sum_{n=1}^{\infty} \frac ...
1
vote
1answer
19 views

Uniform convergence in closed segment

Let $f_n(x)=\frac{2nx+1}{n+nx^2}$. I want to prove that it is uniform converging in $[0,3]$. The pointwise limit function is $f(x)=\frac{2x}{1+x^2}$. I was able to find a supremun to $|f_n(x)-f(x)|$ ...
1
vote
1answer
20 views

Understanding uniform convergence and the M-test

Consider $$\sum_{n=1}^\infty \frac{1}{n^2(1 + x/n)}$$ where $x \in (-1,\infty)$. At $n = 1$, we have $\frac{1}{1 + x}$, and surely this can be made as close to infinity as we want. Is it then not ...
1
vote
2answers
38 views

Uniform convergence of function series

Let $f_n(x)=n(\sqrt{x^2+\frac{1}{n}}-x)$. I want to prove that $f_n(x)$ is uniformly converging in $[1,\infty)$. I found that the pointwise limit function is $f(x)=\frac{1}{2x}$, and looked for the ...
0
votes
2answers
24 views

Determine if the convergence of $f_n(x) = \sqrt{x^2 + \frac{1}{n^2}}$ is uniform.

$f_n(x) = \sqrt{x^2 + \frac{1}{n^2}}$ converges pointwise in a set $E = [0, \infty)$ to $f(x) = x$. This problem reminds me a lot of how $\frac{x}{n}$ fails to converge uniformly to $f=0$ on ...
2
votes
0answers
38 views

Unifrom Convergence of series of product of two sequences

Suppose {$f_n$}, {$g_n$} defined on $E$ and, (a) $\Sigma f_n$ has uniformly bounded partial sums; (b) $g_n \to 0$ uniformly on $E$ (c) $g_1(x)\geq g_2(x)\geq g_3(x)\geq ...$ for every $x \in E$. ...
0
votes
0answers
29 views

Uniform convergence to a differentiable function

Let $(a_n)_{n\in \mathbb{N}}$ be a bounded sequence. Prove that the series $\displaystyle \sum_{n=1}^{\infty} \frac{a_n}{n^{2x}}$ converges absolutely and uniformly on $(1, +\infty)$ to a ...
0
votes
2answers
33 views

Uniform convergence of sequence of functions with infinite roots to a limit with finite roots

Consider a sequence of continuous functions $(f_n)$ defined over $[0,1]$ such that, for all $n$, the set: $$A_n = \{x\in [0,1] : f_n(x) = 0\}$$ is infinite in cardinality. Can $(f_n)$ uniformly ...
1
vote
1answer
40 views

If K $\subset \mathbb{R}$ is compact prove that ${f_n}$ converges uniformly to f on K.

Suppose that we have a sequence of functions $\{f_{n}\}$ that converges uniformly to a function $f$ on any $(a,b)\subset \mathbb{R}$. If $K\subset \mathbb{R}$ is compact prove that $\{f_{n}\}$ ...
3
votes
2answers
37 views

Pointwise convergence and the convergence of a distribution sequence.

The sequence of functions $f_n(x)=\{tanh(nx)\}_{n=0}^{\infty}$ does not converge uniformly to $f(x)=sgn(x)$ but only pointwise. Is it then, still possible that the sequence of distributions ...
0
votes
1answer
31 views

Uniform convergence of $\{\tanh(nx)\}_{n=0}^{\infty}$

Quick question. How can I prove that the sequence of functions: \begin{equation} f_n(x)=\{\tanh(nx)\}_{n=0}^{\infty} \end{equation} converges uniformly to: \begin{equation} f(x)=\begin{cases} -1, ...
1
vote
1answer
46 views

Prove that $f_n(x)=\frac{1-(x/b)^n}{1+(a/x)^n}$ is uniformly convergent on the interval $[a+\epsilon ; b - \epsilon]$

Title says it all; I have to prove that the function sequence $f_n(x)=\dfrac{1-(x/b)^n}{1+(a/x)^n}$ is uniformly convergent on the interval $[a+\epsilon ; b - \epsilon]$, with ...
1
vote
1answer
28 views

Show that $f_n(x)=\dfrac{1-(x/b)^n}{1+(a/x)^n}$ with $0<a<b,$ and $x \in [a,b]$ is not uniformly convergent

So, I have to show that $f_n(x)=\dfrac{1-(x/b)^n}{1+(a/x)^n}$ with $0<a<b$ and $x \in [a,b]$ is pointwise convergent, but not uniformly convergent. The pointwise convergence is pretty straight ...
2
votes
1answer
51 views

Convergence of the sequence, $f_{n}(x)$.

Correct me if I am missing something or show me the better way. Let $0<a<b$ and consider the sequence of functions $$f_{n}(x)=\frac{1-(x/b)^{n}}{1+(a/x)^{n}}$$ for $n\in \mathbb{N}$. ...
0
votes
2answers
32 views

Uniform convergence, and how to show it?

We've just been introduced to uniform convergence, and the method presented is to take the supremum of the absolute value of the difference between the limit function and the function in the sequence, ...
0
votes
0answers
13 views

Uniform convergence of polynomials (including first and second derivatives)

I am searching for a proof of the following statement: Given a twice continuously differentiable (real-valued) function on $\mathbb{R}^n$ and a compact set $K$, one can find a sequence of polynomials ...
1
vote
1answer
40 views

Proving uniform convergence of sequence of functions

Let $f$ be continuous on $[0,1]$ with $f(1)=0.$ How can I show that the sequence $\{x^nf(x)\}$ converges uniformly on $ [0,1]$? Can I deduce that $f(x)$ is bounded? Then I can get $lim_{n \to \infty} ...
1
vote
0answers
45 views

Non-uniform convergence example

For the past couple of days, I've been trying to come up with a example for a problem in which $\sum_{k=1}^{\infty} |f_{k}(x)|$ does not converge uniformly but $\sum_{k=1}^{\infty} |f_{k}(x)|$ ...
6
votes
2answers
61 views

Showing that $P_r(x)=\frac{1-r^2}{1-2r\cos x+r^2}\rightarrow 0$ uniformly on $[-\pi,-\delta]\cup[\delta,\pi]$ as $r\uparrow 1$

Let $0<r<1$ and consider the series $$s = \sum_{n=-\infty}^\infty r^{|n|}e^{inx}.$$ I have shown that the series converges uniformely to $$P_r(x)=\frac{1-r^2}{1-2r\cos x+r^2}$$ on all of ...
5
votes
5answers
90 views

Why is $f_n(x) = x^n$ not uniformly convergent on $(0, 1)$?

Definition of uniform convergence: For all $\epsilon > 0$, there exists an $N \in \mathbb{N}$ such that $d(f_n(x), f(x)) < \epsilon$ for all $n > N \in \mathbb{N}$ and all $x \in (0,1)$. ...
1
vote
1answer
41 views

How to prove $\frac{nx^3}{1 + n^2 x^2}$ converges uniformly on $[1, \infty)$

I know this sequence of functions converges to $0$ pointwise, so I have to show that for all $\epsilon > 0$, there exists an $N$ such that for all $n > N$, $d(\frac{nx^3}{1 + n^2 x^2}) < ...
0
votes
1answer
24 views

Uniform convergence on compact sets implies interchange of summation and integration

Suppose there is a non-bounded set $S$ in the complex plain and a sequence of analytic functions $f_n$ defined over $S$. The series $$f(s)=\sum_{n=1}^{\infty} f_n(s)$$ converges uniformly on compact ...
0
votes
0answers
32 views

Pointwise and uniform convergence of increasing functions

Let $a< b$ and assume $f_n : [a,b] \to \Bbb R$ are increasing functions, $ n = 1,2,\dots.$ Prove that if $f_n \to f$ pointwise on $[a,b]$, then (i) $f$ is increasing, and (ii) if $f$ is continuous ...
1
vote
1answer
24 views

Sequence of functions and Uniform convergence

Question: Let $f_n$ be a sequence of bounded functions on a set $S\subset\mathbb R$. Suppose $f_n\to f$ uniformly on $S$. Prove $f_n$ is uniformly Cauchy on $S$. Attempt: I proved this easily by ...
0
votes
0answers
14 views

continuity and convergence [duplicate]

If we have a continuous function that converges on a compact subset of a metric space does it imply that it converges uniformly in general, or is this only in the case if f is monotonic (Dini's ...
0
votes
0answers
26 views

uniform limits and asymptotic equivalence

I am told that $\frac{f(\lambda r)}{f(r)}$ tends to 1 uniformly in $\lambda$. I also know that $x(t)$ is asymptotically equivalent to $ct$, so $x(t)\sim ct$. How can I show that $\frac{f(x(\lambda ...
1
vote
1answer
26 views

Prob. 6, Sec. 21 in Munkres' TOPOLOGY, 2nd ed: How to show directly that this sequence of functions does not converge uniformly?

For each $n = 1, 2, 3, \ldots$, let $f_n \colon [0,1] \to \mathbb{R}$ be defined by $$f_n(x) \colon= x^n \ \ \ \mbox{ for all } \ x \in [0,1].$$ Then $$ \lim_{n \to \infty} f_n(x) = \begin{cases} ...
1
vote
2answers
49 views

How to show this infinite sum converges uniformly?

Let $f_k$ be a real numbers such that $\sum_{k=1}^\infty f_k < \infty$. For each $R > 0$, define the convergent sum $$v(R) = \sum_{k=1}^\infty f_k(b_k(R)e^{-ky} - c_k(R)e^{ky})$$ where $0 \leq y ...
0
votes
1answer
28 views

Uniform convergence when no limit function is specified.

What does it mean when someone asks you to show that a series of functions converges uniformly without specifying to what function? In the past when I've delt with uniform convergence, I've always ...
0
votes
0answers
40 views

Prove locally uniformly convergence of a sequence

Let for $n = 0,1,2,...$ , $f_n : [0,1] \rightarrow \mathbb{R}$ defined by $f_n (x) = x^n$. 1) Is the convergence of {$f_n$}$_{n=0} ^\infty$ to $f$ locally uniformly on the interval $[0.1]$? 2) And ...
1
vote
2answers
27 views

uniform convergence of subsequences

I have the following question. Given a sequence of functions $f_n(x)$ for which there is a subsequence of functions uniformly converging to some $\overline{f}(x)$. What can we say about $f_n(x)$? ...
0
votes
1answer
39 views

Euler method uniform convergence

I have a question for you guys. Given a differential equation $$\dot{x}=f(x)\qquad x\in\mathbb{R}^n$$ on a compact interval $[0,T]$. If one considers for every $k\in\mathbb{N}$, the Euler's ...
0
votes
1answer
48 views

Uniform convergence of $\sum^n_{k=-n} \frac{1}{z+k}$

Let $D=\mathbb C \setminus \mathbb Z$ and define $$f_n(z)=\sum^n_{k=-n}\frac{1}{z+k}$$ I have to prove that $\{f_n\}^\infty_{n=0}$ is locally convergent on D. We are given the hint to write $f_n$ as a ...
2
votes
1answer
37 views

Pointwise convergence implies uniform convergence under concavity?

Suppose that $X$ is a subset of a Euclidean space and $f_n:X\to\mathbb{R}$ converges pointwise to $f:X\to\mathbb{R}$. A book I'm reading seems to say that: if $X$ is convex and compact and each ...
3
votes
1answer
47 views

When does $f_n(x) = a_n \times (1 - nx)$ converge uniformly?

The sequence of functions $\{f_n\}_n$ is defined on $[0,1]$ by: $$f_n(x) = a_n \times (1 - nx),\ {\rm\ if}\ x \in ]0,\frac{1}{n}],$$ and $f_n(x) = 0$ otherwise, where $(a_n)_n$ is a positive ...
1
vote
1answer
29 views

Can a series of polynomials converge non-uniformly?

Is there an example of a series of polynomials, say, the degree equals the index and converges non-uniformly? In other words, does point-wise convergence of a polynomial series imply uniform ...