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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-1
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1answer
38 views

Checking if $(x^{2n}-1.5x^n+\frac12)^\infty_{n=1}$ converges uniformly on$ [\frac12,1]$ and on $[0,\frac12]$

Checking if $\left(x^{2n}-1.5x^n+\frac12\right)^\infty_{n=1}$ on $ [\frac12,1]$ and on $[0,\frac12]$. What tests/techniques do I use to prove uniform convergence?
0
votes
1answer
26 views

Test the uniform convergence of $x^n-x^{2n}$ in $[0,1]$

$$ f_n(x)=x^n-x^{2n} \\ f_n:[0,1]\rightarrow \mathbb R $$ I know that the function $x^n$ is not converging uniformally because the limiting function is not continuous (when x=1 there's a "step" in ...
5
votes
1answer
52 views

Prove the uniform convergence of $f_{1}(x)= \sqrt x , f_{n+1}(x)=\sqrt{x+f_n(x)}$ in $[0,\infty]$

As far as I understand most of these questions use the M-test, but I can't find a series that suffices.
0
votes
1answer
23 views

How to prove that a sequence of polygonal functions converges uniformly?

Suppose that $\{\varphi_{n}\}$ is a sequence of polygonal functions from [0,1] to $X$, where $X$ is a compact set. How to prove that $\{\varphi_{n}\}$ converges uniformly to a certain $f$, or $\{\...
1
vote
2answers
32 views

Show the uniform convergence of the partial derivatives

Let $f\in C_1\left(\mathbb{R}^n,\mathbb{R}\right)$, is it true that if $[a,b]$ is a closed interval and $\left(x_2,...,x_n\right)$ is fix, then for all $\varepsilon>0$ there exists $\delta$ such ...
1
vote
1answer
28 views

How can we fill in some missing details in this proof?

Let $(X,d)$ and $(Y,\rho)$ be metric spaces and $X$ is compact. Suppose $f:X\to Y$ and $f_n:X\to Y$ are continuous functions such that for every $x\in X$, $\rho(f_n(x),f(x))$ decreases to $0$ as $n\...
0
votes
3answers
50 views

Uniform convergence of $(1 − x^{n+1})/(1 − x)$ [closed]

Prove that the sequence $(1 − x^{n+1})/(1 − x)$ converges uniformly to $1/(1 − x)$ on each interval of the form $[−r, r]$ with $r < 1,$ but it does not converge uniformly on $(−1, 1).$
0
votes
0answers
21 views

convergence to a non-injective function

Let G be a simply connected domain, $G \not \neq \mathbb{C}$ and $z_0 \in G$, I got to show that for every $n \in \mathbb{N}$ there is a holomorphic and injective mapping: $f_n:G \rightarrow D_1(0)$, ...
0
votes
0answers
17 views

problem in convergence and uniform convergence of sequence of functions?

Let $f: \mathbb{R} \to [0, \infty)$ be a non- negative real-values continuous function. Let $$ \phi_n(x) = \begin{cases} n, \ \quad if \ f(x)\geq n\\ 0, \ \quad if \ f(x) < n\end{...
0
votes
0answers
29 views

Uniform convergence of a equicontinuous sequence of functions [duplicate]

Let $X$ be a compact metric space, and $(C(X),d_{\infty})$ the space of continuous functions. Let $D\subset{X}$ be a dense subset, and $\{{f_n\}}_{n \in N}$ a equicontinuous sequence of functions from ...
1
vote
3answers
87 views

Does the series $S(x) = \sum_{n=1}^\infty \frac{n^2x^2}{n^4+x^4}$ converge uniformly?

does the above series uniformly converges? If it does, how to find the interval of x in which it uniformly converges? $ {(n^2-x^2)}^2 \ge 0 \Rightarrow n^4 + x^4 - 2n^2x^2 \ge 0$ This gave me $\frac{...
0
votes
0answers
32 views

Function sequence and some properties

Consider functions $f_n$, $f : \mathbb{R} \rightarrow \mathbb{R}$ such that the sequence $\{f_n\}$ is uniformly convergent to $f$ and every $f_n$ has property $W$. Determine whether $f$ must have $W$ ...
0
votes
1answer
43 views

Show that the mapping $(C^1, ||.||_{C^1}) \rightarrow (C, ||.||_\infty) $ is continuous

Assume $D:(C^1, ||.||_{C^1}) \rightarrow (C, ||.||_{\infty}),$ $$D(f) = f',$$ is a mapping with $$||f||_{C^1} := ||f||_{\infty} + ||f'||_{\infty},$$ $$||f||_{\infty} := sup_{x \in [a, b]} f(x).$...
1
vote
2answers
48 views

Uniform convergence of n-fold composition using Schwarz lemma

Let $f$ be an analytic function mapping the unit disk $\mathbb D$ to itself with $f(0) = 0$ and $|f'(0)| < 1$. Let $f^{n} = f \circ f \circ \dots \circ f$ be the function obtained by composing $f$ ...
0
votes
1answer
29 views

Asymptotically equivalent series for uniform convergence

I have to find sets of uniform convergence of $$\sum_{n=1}^{\infty}n^2 \sin \frac{x}{n^4}$$ what if I study this series passing to the asymptotically equivalent $$\sum_{n=1}^{\infty}n^2 \frac{x}{n^4}$$...
1
vote
1answer
23 views

Convergence of minimum of a sequence of functions

Consider functions $f_n, f$ from $\mathbb{R}$ to $\mathbb{R}$. Suppose $f_n(y)$ converges pointwise to $f(y)$ for all $y$ as $n \rightarrow \infty$. I would like to know under what conditions is the ...
1
vote
3answers
58 views

Find the values of $x$ where $\sum_n \frac{x^{2n}}{1+x^{2n}}$ converges, and show that we have a continuous function on this set

Find the values of $x$ where $\sum_n \frac{x^{2n}}{1+x^{2n}}$ converges, and show that we have a continuous function on this set I can see that $$\frac{x^{2n}}{1+x^{2n}}<x^{2n},\forall x\in(-1,1)...
1
vote
0answers
36 views

Show if these statements about uniform convergence of series are true or false

a. If $\sum_{n\ge 1}g_n$ converges uniformly then $(g_n)$ converges uniformly to zero Im not sure because if $|\sum_{n=m}^{m+k}g_n|<\varepsilon$ comes from an alternating series then $|g_m-g_{m+k}...
0
votes
2answers
91 views

Find $\lim_{x\to0}\sum_{n=1}^{\infty}\frac{\sin x}{4+n^2x^2}$

I have problem with finding the following limit. I suspect that it should be easy, but really I don't have a clue. $$ \lim_{x\to0}\sum_{n=1}^{\infty}\frac{\sin x}{4+n^2x^2} $$
2
votes
0answers
23 views

Convergence of the minima of functionals

Let $\mathcal{H} \subset \mathbb{R}^3$ denote a compact subspace. Suppose we have a sequence of functionals $(Q_n)_{n\geq 1}$ and a functional $Q$ from $C(\mathcal{H},\mathbb{R}^3)$ (which is the ...
3
votes
2answers
133 views

Prove this series does not uniformly converge

$$\sum_{n=1}^{\infty}n^2 \sin \frac{x}{n^4}$$ It is easy to show that it absolutely converges. But what about uniform convergence? With M-test: $$|| f_n|| = \sup (| n^2 \sin \frac{x}{n^4}|) \leq \...
0
votes
1answer
26 views

$f'_n(x)$ is bounded and $f_n(x) \to 0$ for each x, then $f_n(x) \to 0$ uniformly

I want to show the question If $f_n(x)$ is differentiable on [a.b] with $|f'_n(x)|<10$ for all n and if $f_n(x) \to 0$ at each x, then $f_n(x) \to 0$ uniformly. I think I should use triangle ...
5
votes
3answers
43 views

How to show that $f_{n}(x)=\frac{1}{1+x^{n}}$ converges uniformly to $1$ on $[0,a]$ with $0<a<1$?

The problem is as follows. I have a secuence $f_{n}(x)=\frac{1}{1+x^{n}}$ and I have to show why $f_{n}$ converges uniformly on $[0,a]$, $0<a<1$ and why does not converge uniformly on $[0,1]$. ...
2
votes
1answer
37 views

Necessary and sufficient condition for uniform convergence.

Let $a_n$ and $b_n$ be real sequences and let $f_n(x) = a_nx + b_nx^2$ be a sequence of polynomials. What should be the necessary and sufficient conditions on the sequences $a_n$ and $b_n$ so that the ...
2
votes
2answers
31 views

Uniform convergence for particular values of $p$

The functions $f_n$ on $[0,1]$ are given by $$f_n(x) = \frac{nx}{1+n^2x^p} \ \ (p >0).$$ For what values of $p$ does the sequence converge uniformly to its pointwise limit $f$? Consider that \...
1
vote
1answer
48 views

$\sum \frac{x}{n^{a}(1+nx^{2})}$ converge uniformly in $\mathbb{R}$.

Let $a>1/2.$ Determine if a series converge uniformly in $\mathbb{R}:$ $$\sum_{n=1}^{\infty} \frac{x}{n^{a}(1+nx^{2})}$$ If $x>0$ ... it is easy to prove that $$\left |\frac{x}{n^{a}(1+...
0
votes
1answer
25 views

Derivative and uniform convergence

$f_n(x) = \dfrac{\arctan (n^{1/4} x^2)}{n^{3/2}}$ I need to calculate first derivative of it and then tell if first derivative is uniformly convergent. I calculated it but I got now idea how to bound ...
0
votes
0answers
19 views

Determine if a series converge uniformly

can you help me with this, I don't know from where to start: I need to determine if the series $\sum_{n=2}^\infty (-1)^{n-1}\frac{x^{2n}}{3^{2n}ln(n)}$ converge uniformly in the range of it converge. ...
3
votes
2answers
47 views

if $f_n(x)$ converges uniformly to a function $f(x)$ does $f_n'(x)$ converge uniformly to $f'(x)$?

Let [a,b] denote a finite interval and consider a sequence $\{f_n(x)\}_{n=0}^\infty$ in $C^1([a,b])$. if $f_n(x)$ converges uniformly to a function $f(x)$ on $[a,b]$, does $\{f_n'(x)\}$ converge ...
2
votes
1answer
34 views

Proving that $\frac{nx}{2+n+x}$ converges uniformly on $0 \le x \le 1$

Proving that $f_n(x)=\frac{nx}{2+n+x}$ converges uniformly on $0 \le x \le 1$ Now I know I have to use the infinity metric, but I can't understand the solution given for this question. The next ...
1
vote
1answer
49 views

Improper Integral of $f_n$ of a Uniformly Convergent Sequence

Let $(f_n)$ be a sequence of functions defined in $[a,\infty)$, which uniformly converges to $f$ in every interval $I_b$ of the form $[a,b]$. Assume every function in the sequence is integrable in $...
0
votes
0answers
26 views

Uniform convergence of the function sequence $f_n(x)=n(f(x+\frac{1}{n})-f(x))$

I'm new to uniform converge in sequence function, so I have: Let $f(x)$ be a continuous differentiable function in $R$. $f_n(x)=n(f(x+\frac{1}{n})-f(x))$. I need to find $\lim_{n\rightarrow \infty}...
0
votes
1answer
17 views

Uniform convergence of a sequence of functions given as product and convolution.

Suppose we have, for an open bounded set $\Omega \subset \mathbb{R}^n$: A function $u \in L^p(\mathbb{R}^n) \cap C(\mathbb{R}^n)$. A sequence of mollifiers $(\rho_n) \subset C_c^{\infty}(\mathbb{R}^...
2
votes
1answer
24 views

Uniform convergence of $\sum (-1)^nf_n(x)$ on $[0,1]$ where $f_n(x)=x^n(1-x)$.

Let $f_n(x) = x^n(1-x)$ and $\sum (-1)^nf_n(x)$. I showed that this series point wise. Case1) $x=1$ $$ f_n(1)=0 \, , \sum{(-1)^nf_n(1)} = 0 $$ Case2) $x\neq 1$ $$|f_n(x)| \leq |x|^n$$ since $$\sum |...
0
votes
1answer
24 views

Convergence of product of sequences of functions

Suppose we have two sequences of functions $(f^1_n),(f^2_n) $ where $f_n^1,f_n^2: \mathbb{R}^n \to \mathbb{R}$. These sequences verify ($\overset{u}{\rightarrow}$ means uniform convergence): $$f_n^1 ...
1
vote
1answer
18 views

Trouble with uniform convergence in Hilbert space.

Let $f:[a,b]\to H$ for an arbitrary real-valued Hilbert space $H$ be continuous on the real interval $[a,b]$. We are given that $B=\{b_k:k\in \mathbb{N}\}$ is an orthonormal basis for $H$. I am trying ...
2
votes
1answer
57 views

Find $\lim_{n \to \infty} \int_0^1 \frac{\cos\left (\frac{x^2}{n}\right)}{1+x^2} dx$

Find $$\lim_{n \to \infty} \int_0^1 \frac{\cos\left (\frac{x^2}{n}\right)}{1+x^2} dx$$ What I've done is use a theorem I found online, Dini's Theorem. The interval $[0,1]$ is compact, and since $...
1
vote
2answers
52 views

Sequence of functions that converges to Gaussian

Define $f_n:[0,\infty)\to\mathbb{R}$ as follows: $$ f_n(x) = \begin{cases} \left(1-\frac{x^2}{n}\right)^n & 0\leq x\leq \sqrt{n} \\ 0 & \text{otherwise}\end{cases} $$ I need to show that $f_n$ ...
-2
votes
1answer
24 views

Test the series for uniform convergence [closed]

$$\sum_{n = 1}^\infty \frac{nx}{1+n^2x^2}$$ How to proceed? Can't we do it using partial sum method?
0
votes
1answer
28 views

Integral approximation of functions not defined for x=0

I have to approximate with an error less than 0.1 this integral: $$ \int_1^2 \exp\left(-\frac{1}{x^2}\right)\,dx $$ I understood I have to use the Taylor series, then prove that is uniforme ...
0
votes
1answer
17 views

What happens when there is uniform convergence on an unspecified set?

Let $\{f_n\}$ define a sequence functions between metric spaces. I know what it means to say that "$f_n$ converges uniformly on some set $U$.". However, what if no set is specified, what does the ...
2
votes
2answers
71 views

Prove or disprove: $\sum_{n=0}^\infty e^{-|x-n|}$ converges uniformly in $(-\infty,\infty)$

I'm trying to find out if $\sum_{n=0}^\infty e^{-|x-n|}$ converges uniformly in $(\infty,\infty)$. Here's my attempt at an answer: If $ x\in(-\infty,0]:$ $$e^{-|x-n|}=e^{(-(x-n))}=e^{x-n}\le e^{-n}$$ ...
0
votes
1answer
39 views

Check series for uniform convergence on real numbers

$$ \sum_{n=1}^{\infty} \frac{n^2}{1 + n} \frac{x^2 \sin x}{1 + n^5x^4}, E = \mathbb{R} $$ I tried to determine convergent subseries and something limited to use Abel - Dirichlet test, I can't find ...
0
votes
2answers
42 views

How do I prove that $f_n(x)=\frac{1-x^n}{1-x}$ converges pointwise and uniformly?

I'm trying to check if the function sequence $f_n(x)=\frac{1-x^n}{1-x}$, $x \in (-1,1)$ converges pointwise and uniformly. If I had the sequence of $g_n(x)= \frac{1-x^n}{1-x^n}$ and $x \in (-1,1)$ i ...
1
vote
0answers
25 views

About the uniform convergence of a series

Let $(a_n(z))$ is a sequence of holomorphic functions defined on $\mathbb{C}\setminus A$, where $A$ is a set of simple poles. I am thinking about proving that $\sum_{n=1}^{\infty}\left | a_n(z) \...
0
votes
1answer
59 views

Uniform convergence of $\frac{\cos(nx)}{e^{nx}}$

We have a sequence $(f_n)$ on $[0,\infty)$, defined by $f_n(x)=\frac{\cos(nx)}{e^{(nx)}}$. The limit function $(f_n)$ of this sequence is $0$ for $x>0$ and $1$ for $x=0$. First part of the question ...
0
votes
1answer
17 views

Uniform convergence of supremum

If a sequence $\{f_n\}$ converges uniformly to a limit $f$ on the domain $D$, then the sequence $\{M_n\}$, with $M_n = \sup_{x} |f_n(x)-f(x)| $, converges to zero. So what I thought was since $\{...
1
vote
1answer
31 views

Finding the values of $z$ such that $\sum_{n=0}^{\infty}(1+\sin{n})^nz^n$ converges

I'm trying to apply the nth root test to $$\sum_{n=0}^{\infty}(1+\sin{n})^nz^n$$ Hence I use that $\hat{R}=\left (\limsup |a_n|^{\frac{1}{n}}\right )^{-1}$ and get $$\hat{R}=\left (\limsup (1+\sin{n}) ...
0
votes
1answer
40 views

Prove uniform convergence for $f_n(x) = \sqrt[n]{x^2+xn + 1}$ on $(0,1)$ and $(1, +\infty)$ [closed]

By what methods and means I can prove that function's UC on these intervals? I know the answer, it uniformly converged on first interval and not uniformly on second to $f(x) = 1$, but I need to prove ...
1
vote
1answer
67 views

Prove uniform convergence of series

I'm given to functions, $f_{n}(x)=e^{-(x-n)^2}$ and $g(x)= \begin{cases} \frac{1-e^{-x^2}}{x^2} & x \neq 0 \\ 1 & x=0 \end{cases}$. I have to prove that $$\sum_{n=0}^{\infty} g(x) \cdot f_n(x) ...