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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4
votes
1answer
28 views

Prove $f(x) = \sum_{n=1}^\infty \frac{\sin nx + \cos nx}{n^3}$ is well-defined and $C^1$.

Prove $f(x) = \sum_{n=1}^\infty \frac{\sin nx + \cos nx}{n^3}$ is well-defined and $C^1$. First of all I need to prove that $f(x)$ is well-defined. I'm not so sure what does it mean. Basically I ...
1
vote
1answer
49 views

For which $\alpha > 0$ we can interchange the integral with the summation?

For which $\alpha > 0$ the equality holds: $$ \int_0^1 \sum_{n=0}^\infty x^\alpha e^{-nx} dx= \sum_{n=0}^\infty \int_0^1 x^\alpha e^{-nx} dx $$ We've learned that an interchange can be ...
1
vote
2answers
32 views

Prove that $f_n(x) = \left(\frac{x}{n}\right)^ne^{-x}$ converges uniformly

Prove that $f_n(x) = \left(\frac{x}{n}\right)^ne^{-x}$ converges uniformly at $[1,\infty)$. So for every $x$, there's $N\in\mathbb{N}$, such that for all $n>N$: $\frac{x}{n} < 1$. ...
0
votes
2answers
11 views

Prove that if $f_n \to f$ uniformly on all closed intervals $[c,d] \subset (a,b)$, then $f_ng\to fg$ uniformly on $[a,b]$

Let $[a, b]$ be a closed bounded interval, $f : [a, b] \to \mathbb{R}$ be bounded, and $g : [a,b] \to \mathbb{R}$ be continuous with $g(a) = g(b) = 0$. Let $f_n$ be a uniformly bounded sequence of ...
1
vote
1answer
40 views

Prove that $\lim_{n \rightarrow \infty} \int_0^1 f_n(x)dx \ne \int_0^1\lim_{n \rightarrow \infty}f_n(x) dx$

If $f_n(x)=nxe^{-nx^2}~\forall~n=1,2,\cdots$ and $x$ real, show that $$\lim_{n \rightarrow \infty} \int_0^1 f_n(x)dx \ne \int_0^1\lim_{n \rightarrow \infty}f_n(x) dx$$ Attempt: By the $Mn$ Test, it ...
2
votes
2answers
36 views

Uniform Convergence of Series $\sum_{n=1}^{\infty}\frac{\sin(x)^n}{n}$

I'm trying to show uniform convergence of a series of complex numbers, but I'm having trouble. The series is as follows: $$\sum_{n=1}^{\infty}\frac{\sin(x)^n}{n} \rm{~~~~~~for}~~~0<x<\pi/2$$ I ...
0
votes
1answer
28 views

If a power series converges uniformly on $\mathbb{R}$ then it must be to $0$?

Let $f(x) = \sum a_n x^n$. Let's assume that $f(x)$ has a radius $R=\infty$ and $f(x)$ converges uniformly. Now, obviously $f(0) = 0$. Meaning, $f(x)$ pointwise converging at $x=0$. Since we assumed ...
1
vote
2answers
46 views

A uniformly convergent sequence of real analytic functions which does not converge to a real analytic function

I am looking for an example of a uniformly convergent sequence of real analytic functions which does not converge to a real analytic function. Also I would appreciate any pointers on how to think of ...
0
votes
0answers
23 views

Weierstrass $M$-test for uniform convergence

I have a problem regarding Weierstrass $M$-test for uniform convergence. Is it necessary to have the hypothesis '$|f_n(x)|\leq M_n$ for all $n\geq 1$'? Or is it sufficient to have it like 'There ...
0
votes
3answers
34 views

Does $f_n(x) = \frac{x^n}{1+x^n}$ converges uniformly on $[0,1]$?

Does $f_n(x) = \frac{x^n}{1+x^n}$ converges uniformly on $[0,1]$? My answer is: No because obviously $f_n(0) = 0$ and $f_n(1)=\frac{1}{2}$, so for every $n\in\mathbb{N}$ it's true that for ...
0
votes
2answers
57 views

Prove that $(f_n)_n$ is uniformly convergent.

Let $g$: $[0,1]\to\mathbb{R}$ be continuous and $g({1})=0$. Define $f_n(x)= x^{n}{g(x)}$. Prove that $(f_n)_n$ is uniformly convergent.
1
vote
1answer
30 views

prove $({f_n})_n$ is uniformly convergent on ${[0,1]}$

The real function ${g}$ is continuous on $[0,1]$ .we define ${f_n}$ on ${[0,1]}$: $$f_n(x)=\frac{{{g(x)\sin^{n} (x)}}}{{{1+nx}}}$$ prove $({f_n})_n$ is uniformly convergent on ${[0,1]}$ .
-2
votes
1answer
22 views

Uniformly convergent & point convergent

Let ${({f_n})_n}$ is a sequence that ${f_n}(x)=tan^{-1}(nx), x\in [0,\infty)$. Prove for every $[a,b]$ that $a>0$ is Uniformly convergent and on $[0,b]$ just point wise convergent.
1
vote
0answers
51 views

Problem proving monotonicity while showing uniform convergence of $\sum_{k} \frac{1}{k} \sin \left ( \frac{\pi k^2}{x + k} \right )$

Remark This is a homework type of question and since there is no homework tag anymore I ask you to tell me how would you solve this (hints) and let me solve it for myself so I can practice. I can ...
1
vote
1answer
13 views

Supremum of cadlag functions

Let $f_n,f$, $n\in\mathbb{N}$, be (real-valued) cadlag functions on $[0,1]$ such that $$\sup_{0\le t\le 1}|f_n(t)-f(t)|\to 0\text{ as }n\to\infty.$$ Does someone have an idea how to prove that ...
1
vote
1answer
59 views

Is this equality true? Why? Why not?

Let $$ \lim_{a\to 0} \frac{1}{2} \left( \left( \sum_{n=-\infty}^\infty \frac{1}{(n+a)^2} - \frac{1}{a^2} \right) \right) = \sum_{n=1}^\infty \frac{1}{n^2}$$ I already know that ...
3
votes
0answers
50 views

Real analytic function with radius of convergence 1 at non-negative integers

So, as the title states, the problem I was confronted with was to find a real-valued everywhere analytic function $$f:\mathbb{R}\to \mathbb{R}$$ s.t. at every non-negative integer, k ...
0
votes
0answers
13 views

K-Uniformity of Infinite Sequence

A book on random number generators refers to the subject of infinite-uniform infinite sequences as being "random." I was wondering if anyone could shed light on the definition of K-Uniform Infinite ...
-1
votes
0answers
29 views

Does uniform convergence on $D\subseteq \mathbb{C}$ imply uniform convergence on all subsets of $D$?

Let $f_n:D\rightarrow \mathbb{C}~\forall n\in \mathbb{N}$. If $(f_n)_{n\in\mathbb{N}}$ converges uniformly on $D\subseteq \mathbb{C}$ against $f:D\rightarrow \mathbb{C}$, does $(f_n)_{n\in\mathbb{N}}$ ...
2
votes
2answers
31 views

A sequence of Continuous Functions Converges Uniformly over $\mathbb{R}$ if it Converges Uniformly over $\mathbb{Q}$

I'm trying to show that if ${f_n}$ is a sequence of real functions that is continuous over all of $\mathbb{R}$ and that converges uniformly to $f$ over $\mathbb{Q}$, then it converges uniformly to $f$ ...
2
votes
2answers
32 views

Analyticity: Uniform Limit

Problem Consider a uniformly bounded sequence over the real line: $$f_n:\mathbb{R}\to\mathbb{C}:\quad|f_n(x)|\leq L$$ Suppose they have analytic continuations with common domain: ...
-1
votes
1answer
12 views

does function converge uniformly

Does the sequence $(f_n)$ of functions $f_n:(0,\infty)\to\mathbb R$ $$f_n(x) = \begin{cases} 0 & x \in (0,\frac1{n+1}) \\ \sin^2 \frac\pi x& x \in [\frac1{n+1},\frac1n]\\ 0 ...
4
votes
1answer
71 views

$f_n(x_n) \rightarrow f(x) $ by uniform convergence

I am very nearly done with this problem, but I have a concern that someone must help me alleviate. Suppose $f_n\rightarrow f$ uniformly, $f_n$ are continuous, and $x_n\rightarrow x$. Prove that ...
0
votes
1answer
20 views

Basic properties of uniform limits in Banach spaces

Where can I find infos (books, keywords, online materials, etc.) about when the uniform limit of a sequence of continuously differentiable functions $f_n:U\subseteq E\rightarrow F$ between arbitrary ...
1
vote
1answer
49 views

Uniform Convergence to the Exponential Function over a Compact Interval

I'm trying to show that the sequence of functions $f_n(x)=(1+(x/n))^n$ converges uniformly to $f(x)=e^x$ over any compact interval of the real line. We're assuming that it converges pointwise. Here is ...
0
votes
1answer
31 views

Uniform continous functions? [duplicate]

Please if someone could tell me how to show that $1\over x$ is not uniformly continuous on $(0,1)$. I hope I've been clear enough, thanks.
0
votes
2answers
30 views

Uniform convergence $f_n(a) = a^{4n} + \frac1{n^2}$

I have $f_n(a) = a^{4n} + \frac1{n^2}$ which I know converges to $f(a)=0$ uniform on theinterval $[0,1)$ This works? $\lim \limits_{n \to \infty} a^{4n} + \frac1{n^2} = \lim \limits_{n\to\infty} ...
1
vote
2answers
25 views

Example for a sequence of continuous functions which converge to a continuous function pointwise but not uniformly on a compact set.

My actual aim is to verify that each of the conditions in Dini's theorem is essential or not.Theorem says that A sequence {$f_{n}$} of continuous functions defined on a compact set $E$ converges to ...
0
votes
1answer
26 views

Uniform convergency of a given sequence of function

Consider the sequence of function $f_{n}(x)=n^{2}x\bigl(1-x^{2}\bigr)^{n}$. Is this sequence of function convergent uniformly in $[0,1]$ ? We find that $f$ converges point wise to the function ...
1
vote
1answer
18 views

Tell if a series converges uniformly

Let $f(x) = \sum_{n=1}^\infty \frac{-2x}{(x^2+n^2)^2}$. Check if $f_n(x)$ converges to a continuous function. So I've seen a solution that uses the fact that if $f(x)$ converges uniformly and ...
0
votes
0answers
46 views

Is this sequence uniformly convergent?

Consider the sequence of functions $f_t\colon [0,1]^2\rightarrow\mathbb{R}$ defined by $f_t(x_1,x_2)=x_1x_2(1+\alpha(1-x_1^{1/t})(1-x_2^{1/t}))^t.$ Here $-1\leq\alpha\leq 1$ is a constant. Then ...
0
votes
1answer
30 views

Uniform convergence of series of arccosines

The series $$\sum_{n=1}^\infty\arccos{{n^4x^4}\over{1+n^4x^4}}$$ supposedly converges uniformly on any interval $I$ for which $0\notin\overline{I}$ while ...
1
vote
1answer
29 views

Uniform convergence of $A^n/n!$

In a proof regarding finite space Markov Jump Processes in which the function $P(t)=e^{tG}$ is a solution to both the backward and forward Chapman-Kolmogrov equations, one of the steps assumes that ...
3
votes
1answer
67 views

Proving $\Gamma(x)$ is holomorphic

My professor defined Gamma function in the following way:$$\Gamma(z)= \lim \limits_{n \rightarrow \infty} \frac{n!n^z}{z(z+1)....(z+n)}$$ Now we first observe that $f_n(z)= ...
1
vote
2answers
88 views

Uniformly convergent series on $\vert x\vert \leq k$ where $k\in (0,1)$

I want show that each of the series $\sum^{\infty}_{n=1}\frac{nx^n}{1-x^n}$ and $\sum^{\infty}_{n=1}\frac{x^n}{(1-x^n)^2}$ converges uniformly on $\vert x\vert \leq k$ where $k\in (0,1)$; and prove ...
1
vote
1answer
28 views

Pointwise and Uniform convergence with one-sided limit

Consider a function $g(x,y)$, $x\in X$ and $y\geq 0$ where $X$ is a compact subset of $\mathbb{R}$. Assume that $g(x,y)$ converges pointwise to zero as $y\downarrow 0$, for all $x\in X$. Is the ...
0
votes
0answers
33 views

Is absolute continuity enough for uniform convergence of Chebyshev interpolation

Wikipedia says For every absolutely continuous function on [−1, 1] the sequence of interpolating polynomials constructed on Chebyshev nodes converges to f(x) uniformly. $^{[\text{citation ...
1
vote
2answers
79 views

Uniform convergence of the sequence $f_n(x)=f(x+1/n)$ for uniformly continuous $f$

Let $f$ be a uniformly continuous real-valued function on $(-\infty, +\infty)$, and for each $n\in I$ let $f_n(x)=f\left(x+\frac{1}{n}\right)$. Prove that $\{f_n\}_{1}^{\infty}$ converges ...
1
vote
2answers
39 views

$(f_n(x))$ converges to a discontinuous function Counterexample

If the sequence of functions $(f_n(x))$ converges to a discontinuous function $f(x)$ on a set S, then the sequence does not converge uniformly on S. If a function $f(x)$ is defined on [−1, 1] and ...
1
vote
1answer
53 views

Proving the existence of a sequence of polynomials convergent to a continuous function $f$.

I need to show that if $f$ is continuous function ($f:\mathbb{R}\rightarrow \mathbb{R}$), then there exists a sequence of polynomials which converges to $f$ on any compact subset of $\mathbb{R}$. I ...
1
vote
2answers
41 views

Comparison of the consequences of uniform convergence between the real and complex variable cases,

In the real variable case, I think that uniform convergence preserves continuity and integrability, i.e., for an integral of a sequence of continuous (or integrable) functions, which converge ...
5
votes
2answers
65 views

Convergence of series of $1/n^x$ - pointwise and uniformly,

Consider the series $$\zeta(x) = \sum_{n\ge 1}\frac {1}{n^x}.$$ For which $x \in[0,\infty)$ does it converge pointwise? On which intervals of $[0,\infty)$ does it converge uniformly? My work: I ...
0
votes
1answer
19 views

Infinite products of scaled indicator variables: almost sure convergence vs. uniform convergence of the sample mean

Let $\frac{X_i}{2}\sim Ber(0.5) \implies E[X_i]=1$, and let $Y_n=\prod\limits_{i=1}^n X_i$. Since the $X_i$ are iid, $E[Y_n]=1,\;\forall n<\infty$. However, something weird appears to be happening ...
0
votes
1answer
23 views

uniform convergence of diff'ble functions with derivatives converging in $L^1$

Suppose we have a sequence of differentiable functions $f_n$ from a closed interval $I\subseteq\mathbb{R}$ to $\mathbb{R}$ with the following properties: $f_n$ converges uniformly to some function ...
7
votes
3answers
93 views

What does it mean for a topology on a mapping space to correspond to a type of convergence?

I've been asked to prove that three different topologies on $Y^X = \{f : X \to Y | f \text{ is continuous} \}$ correspond to three different types of convergence, but I don't understand exactly what ...
0
votes
1answer
40 views

Iterating average

If $f$ is a continuous function $[0,1]\to \mathbb R$, we define a linear application $T$ as follows $$T(f)(x)=\begin{cases} f(0) & \mathrm{if }~ x=0 \\[0.2cm] \displaystyle ...
1
vote
1answer
32 views

Uniform convergence of series and continuity of $f$

This is from Ross's Elementary Analysis Textbook: The series $(2^{-n})(x^n)$ from $n=1$ to $n= \infty$ represents a continuous function on $(-2,2)$, but the convergence isn't uniform. He points out ...
1
vote
1answer
40 views

Weierstrass M-test help

I am supposed to use M-test on this one $$\sum \frac {n\ln (1+nx)}{x^n}$$ on $$1<x< \infty$$ But I face problems finding an appropriate $M_n$, thanks for help
4
votes
1answer
63 views

Determine whether $\sum\limits_{n=1}^\infty \frac{1}{n^x}$ converges uniformly on $(1,\infty)$

Detemine whether $\sum\limits_{n=1}^\infty \frac{1}{n^x}$ converges uniformly on $(1,\infty)$. My attempt: Upon attempting to use the Weierstrauss M-test I get ...
1
vote
2answers
43 views

limit and uniform convergence of sequence of function

I have a sequence of function $g_n$: $$ g_n(x) = xe^{-nx}, \qquad \text{for } x \in [0,\infty ) $$ I need to find the limit, and determine the uniform convergnece of the sequence. I think the ...