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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2answers
54 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
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1answer
36 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
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2answers
35 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
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0answers
21 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
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1answer
48 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
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1answer
15 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
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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
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1answer
37 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
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1answer
65 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) ...
1
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1answer
67 views

Convergence of $g(x)\cdot f(x)$

Let $g(x)=\frac{1-e^{-x^2}}{x^2}$ for $x \neq 0$,$g(0)=1$ and $f(x)=e^{-(x-n)^2}$. You can assume that g(x) is continuous and bounded with maximum 1 in x=$0$. Show that $\sum_{n=1}^{\infty}g(x)\cdot ...
1
vote
1answer
20 views

Density of $C^{\infty}_{c}(\mathbb{R}^{N})$ in $\mathcal{S}(\mathbb{R}^{N})$

The proof of this fact is quite classical. I tried to find it on my own and, apparently, had "the right idea" since it seems to be the standard way to show it. However, I am stuck with some detail I ...
0
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1answer
27 views

Uniform convergence of a function composition

Let $n \in \mathbb{N}, f_n(x)=e^{-(x-n)^2}$ and $g(x)=\Bigg\{\begin{array}{ll} 1 & x=0 \\ \frac{1-e^{-x^2}}{x^2} & x \neq 0 \\ \end{array} $ You can assume that g is continuous ...
0
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0answers
27 views

Uniform convergence with $\sin^n(x)$

I have to determinate the uniform convergence (set of convergence) and calculate the sum of this series: $$ \sum_{n=1}^\infty \sin^nx \frac{(-1)^{n-1} + n^2}{n} $$ For the convergence, I applied the ...
3
votes
1answer
34 views

Uniform convergence of $\sum_{n=-\infty}^{\infty}\frac{1}{n^2 - z^2}$ on any disc contained in $\mathbb{C}\setminus\mathbb{Z}$

I'm currently revising some complex analysis, and need to show that the series $$\sum_{n=-\infty}^{\infty}\frac{1}{n^2 - z^2}$$ defines a holomorphic function on $\mathbb{C}\setminus\mathbb{Z}$. The ...
1
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0answers
27 views

Limit of improper integrals of uniformly convergent function

I've got a problem. Let $g(t)\ge0$ and it has improper integral on interval $[A, B)$. Furthermore, sequence of integrable functions $f_{n}(t)$ is uniformly convergent do $f(t)$ on every subinterval ...
0
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1answer
40 views

Establish if $g_n (\alpha)=\int_a^b \ \alpha(x) \ \sin (nx) \ \cos(nx) $ converges uniformly

$$X=\{ \alpha:[a,b] \rightarrow \mathbb{R} \}$$ $\alpha''$ exists and it is continuous $$\exists \ K>0 \ : \forall \ x \in [a,b], \forall \alpha \in X: \\ \ \\ \rvert \alpha(x) \rvert, \rvert ...
5
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2answers
94 views

Show that $\sum\limits_{n=1}^{\infty}\frac{x}{1+n^2x^2}$ is not uniformly convergent in $[0,1]$

Show that $\sum\limits_{n=1}^{\infty}\frac{x}{1+n^2x^2}$ is not uniformly convergent in $[0,1]$. I was thinking in the direction of taking the maximum value of each term $\frac{x}{1+n^2x^2}$, ...
1
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2answers
34 views

Term-by-Term Differentiation and UNIFORM CONVERGENCE: True Relation

For a series with $\sum u_n'(x)$ not uniformly convergent, and If $f '(x) = \lim_{n\to\infty} f_n'(x) $ where $f(x)=\lim_{n\to\infty} f_n(x) $ and $ f_n(x) $ $=u_1+u_2+ . . . +u_n$ Then the ...
0
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2answers
22 views

Pointwise convergence of Lipschitz functions from a compact space implies uniform convergence

Let $(f_n)$ be a sequence of $1$-Lipschitz functions from $(X, d_X)$ to $(Y,d_Y)$ where the first one is compact and the latter is complete (I am not sure if this matters). Let $f_n \to f$ pointwise. ...
0
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1answer
23 views

Does this function series converge uniformly? [closed]

Prove/Disprove: If the functions $f_n (x)$ are monotonic for every $x \in [a,b]$ and $\sum_{n=1}^\infty f_n (a)$ and $\sum_{n=1}^\infty f_n (b)$ absolutely convergent, then $\sum_{n=1}^\infty f_n (x)$ ...
1
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1answer
29 views

$f_n$ converges uniformly on $\partial \Omega$ then $f_n$ converges uniformly on $\bar{\Omega}$

The problem states that $f_n$ is a sequence of functions which are continuous on the closure of $\Omega$ and holomorphic on $\Omega$ where $\Omega$ is a bounded region and were asked to show that if ...
0
votes
1answer
31 views

Convergence of a sequence of roots of continous functions

Let $(f^n,n\in\mathbb{N})$ be a sequence of complex continous functions so that $f^n(u)\longrightarrow f(u)$ uniformly to a complex continous function f if $n \longrightarrow \infty$. I addition I ...
1
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0answers
37 views

uniform convergence of an alternating series

I have this series of functions $\sum_{n=1}^\infty (-1)^n\log(1+\frac{x}n)$ with $x\geq0$. It's easy to see the pointwise convergence of the series and I also prove that it converges uniform on ...
0
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1answer
17 views

continuity of function series

So here it is: $$\sum_{n=1}^\infty \frac{\sin(\frac{1}{nx^2})}{1+(x-1)\ln^4(xn)}$$ $$x \in (1,\infty)$$ My task is to prove its continuity if possible. My lead was to try proving it through ...
2
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0answers
43 views

Condition for term-by-term differentiation of a non-convergent series

In a problem I have seen, a series $\sum_n u_n(x)$ with $$ f_n(x) = \frac {\log(1+n^4x^2)}{2n^2}$$ here $\sum u_n'(x)$ is not uniformly convergent, BUT If $f '(x) = \lim_{n\to\infty} f_n'(x) $ ...
1
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0answers
36 views

Uniform convergence fourier series |sin(x)|

As an exercise we have to calculate the fourier series of |sin(x)| (was no problem) and after that we are meant to show that this series converges uniformly towards |sin(x)|. After thinking about it ...
0
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2answers
26 views

Let $f_n = (\frac{1}{n})\chi_{[0,n]}$ and $f = 0$. Show that $(f_n)$ converges uniformly to $f$.

Let $f_n = (\frac{1}{n})\chi_{[0,n]}$ and $f = 0$. Show that $(f_n)$ converges uniformly to $f$. I have never done an example of convergence of sequences that have characteristic (indicator) ...
0
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1answer
34 views

Uniformly bounded sequence of analytic functions in the unit disk

Suppose $\{f_n\}$ is sequence of analytic functions that is uniformly bounded in the open unit disk and for every positive integer $k$, $f_n(\frac{1}{1+k})\to 0$ pointwise. Then, $\{f_n\}$ converges ...
3
votes
1answer
80 views

Is series $\sum_{n=0}^{\infty} \Big(\frac{x}{1-x}\Big)^n \frac1{(1-x)^2}$ uniformly convergent for $x <\frac1{2}$?

I believe by the ratio test that it converges for $x < \frac1{2}$ but I can't seem to apply a Weierstrauss M test or other test to show uniform convergence. Maybe it is not.
0
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1answer
28 views

Prove that the integral of a sequence converges uniformly

I've been stuck on this problem for a bit now: Let $g$ and $f_0$ be continuous functions on $[0,1]$. Define the sequence on $[0,1]$ by $$f_n(x) = \int_0^t g(t)f_{n-1}dt.$$ I have to prove that the ...
1
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1answer
30 views

Sequence of funtions $f_n = n(f(x+ \frac{1}{n})-f(x))$ for the continous differentiable function $f$ on $\mathbb R$

Let $f$ be a continous differentiable function on $\mathbb R$. Let $f_n$ be a Sequence of funtions $f_n = n(f(x+ \frac{1}{n})-f(x))$. Then (a) $f_n$ converges uniformly on $\mathbb R$ (b) $f_n$ ...
1
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1answer
23 views

Prove that the sequence $[f_n]_{n \in N}$, $f_n(x)=\frac{x^{2n}}{1+x^{2n}}$ converges uniformly on $x \in [1+\delta,\infty[$

I've been asked to prove that the sequence $[f_n]_{n \in N}$ with $f_{n}(x)=\frac{x^{2n}}{1+x^{2n}}$ converges uniformly on $x \in [1+\delta,\infty[$ where $\delta > 0 $. So far I've found that ...
0
votes
0answers
12 views

Uniform convergence of a piecewise function

Let $(p_n)\subseteq\mathbb{R}$ and $(q_n)\subseteq(0,1)$ be sequences such that $\lim_{n\to\infty}{q_n}=0$. Define the functional sequence $(a_n(x))\subseteq C[0,1]$ for $x\in[0,1]$ as: $$a_n(x) = ...
0
votes
2answers
53 views

Continuity of sum 1/(n^2-x^2)

Given the function $$f(x) = \sum_{n=5}^\infty \frac{1}{n^2-x^2} \text{ for } x \in (-5,5)$$ prove that it's continuous and differentiable. So i tried doing something like $$ \sum_{n=5}^\infty ...
1
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1answer
28 views

Convergence of $f_n=(1-x)^n$ on different intervals.

Let $f_n=(1-x)^n$. Show the convergence is uniform on $[a,2-a]$ where $a \in (0,1)$ I found the pointwise limit function $f$ to be $1$ is $x=0$ and $0$ on $x\leq 1$. If $x>1$ the limit ...
1
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1answer
40 views

Uniform convergence of integral and derivatives of $f_n(x)=\frac{\sin nx}{n}$

Let $f_n$ on $[-\pi,\pi]$ be: $$f_n(x)=\frac{\sin nx}{n}$$ Let $f$ be the pointwise limit of $f_n$ Denote $F(x)=\int_{-\pi}^x f(y)dy$ and $F_n(x)=\int_{-\pi}^x f_n(y)dy$. On $[-\pi,\pi]$: ...
0
votes
1answer
25 views

Does the sequence of characteristic functions converge uniformly?

$f_n(x) = 1$ if $x ∈ [0, \frac{1}{n}$] and $f_n(x) = 0$ otherwise. I know that it converges pointwise. My guess is that it doesn't converge uniformly since lim $sup (f_n(x)) = 1$ as n goes to ...
1
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0answers
40 views

Convergence in measure implies uniform convergence on a set of finite measure

Question is : Does there exists a sequence $(f_k)$ of lebesgue measurable functions such that $f_k$ converges to $0$ in measure in $\mathbb{R}$ but no subsequence converges uniformly on any subset ...
2
votes
2answers
59 views

Is $x^n$ uniformly convergent on $[0,1)$?

I know that it's not uniformly convergent on $[0,1]$ but on $[0,a]$ with $a < 1$. Does that mean it converges uniformly on $[0,1)$?
0
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0answers
22 views

Bernstein polynomial in Banach

For a continuous function $f : [0,1] \to R$, there exists a sequence of polynomial functions: $P_n(x)=\sum_{k=0}^n C^k_n x^k(1-x)^{n-k} f(\frac{k}{n})$ (Bernstein's polynomes) which converges ...
0
votes
2answers
20 views

Proving a function has second derivative / Uniform convergence of a series

I am studying sequences and series of functions and in the course notes there is this excercise: Prove the function $$f(x)=\sum_{n=0}^{\infty}\frac{(-1)^n}{(2n+1)!}x^{2n+1}$$ has second derivative. ...
2
votes
1answer
20 views

Showing uniform convergence to origin in 3rd quadrant for $x(t)=\frac{1}{\frac{1}{x_0}-t}$ as $t\ \rightarrow \infty$

I want to show that for the system $\dot{x}=x^2, \dot{y}=y^2$,any solutions starting in the 3rd quadrant not including 0, converge uniformly to the origin. For an initial point $(x_0,y_0)$, (note both ...
1
vote
1answer
35 views

Weierstrass uniform convergence - Stuck to the point.

$$\sum_{n=1}^\infty \frac{\sin (nx)}{n!}$$ Interval: $x \in(- \infty, + \infty)$ I've been trying to do this all day, but I just cant get to the end of it. It's not that I do not understand the ...
-1
votes
0answers
15 views

Sequence of monotonoe fuctions

I have this problem but I'm not sure if it is true. I cannot find a counterexample so I suppose that it is. Could you give me a hint? The problem is: Let $\{ f_n \}$ a sequence of monotonoe functions ...
1
vote
1answer
32 views

Describe the intervals where $f_n(x)$ converges uniformly

Problem: $f_n(x) = \frac{6}{(1+x^{2n})}$ and $x\in\mathbb{R}$. Find all real numbers $x$ where $f_n(x)$ converges and describe the limit function. I found the limit function to be ...
0
votes
3answers
44 views

Show pointwise convergence and (potentially) uniform convergence $\sum_{k=1}^\infty\frac{x^k}{k}$

I am looking to show pointwise convergence and (potentially) uniform convergence of the following: $$\sum_{k=1}^\infty\frac{x^k}{k}$$ I know (from my book) this converges for my given values of $x ...
0
votes
1answer
48 views

The theory of riemann zeta function titchmarsh page 15 question in the proof of the functional equation

I am currently reading Titchmarsh's book about the Riemann Zeta function and came across a problem in a proof of the functional equation that I cannot solve. To be precise, I am referring to this ...
2
votes
1answer
30 views

Uniform convergence on compact subset

Let there be two functional squences $$a_n(x)=\sqrt[n]{x} \quad \textrm{ for $x\in(0,\infty)$}$$ $$b_n(x)=\sum_{k=0}^{n}x^k(1-x)^k=\frac{1-x^{n+1}(1-x)^{n+1}}{x^2-x+1} \quad \textrm{ for $x\in ...
0
votes
1answer
20 views

Prove this series converges to a continuous function

My problem: Prove that the series $\sum\limits_{n=0}^\infty e^{n(\sin(nx)-2)}$ converges for all $x\in\mathbb{R}$ to a continuous function. By the root test it converges, but as far as the continuous ...
0
votes
1answer
29 views

Uniform and pointwise convergence

So we had the pointwise and uniform convergence, and I do get that a sequence of function can converge to a function, just like ordinary sequences do. But what I don't quite get is this pointwise and ...