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
25 views

Arc length of a sequence of semicircles.

Let $\gamma_1$ be the semicircle above the x axis joining $-1$ and $1$. Now divide this interval into $[-1,0]$ and $[0,1]$, and trace a semicircle joining $-1,0$ above the $x$-axis and another one ...
2
votes
1answer
25 views

Weak convergence of probability measures and uniform convergence of functions

I am stuck on Problem 4.12 of Karatzas and Shreve's book Stochastic Calculus and Brownian Motion: Suppose that $\{ \mathbb{P}_n \}$ is a sequence of probability measures on $(C[0, \infty), ...
0
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2answers
21 views

What values of $b$ such that $f_n(x)=b\cos\left(\frac{x}{n}\right)$ converges uniformly?

For what values of $b$ does the sequence of functions: for each $n\in\mathbb{N}$, let $$f_n(x)=b\cos\left(\frac{x}{n}\right), \text{ } x\in[0,1]$$ converge uniformly in the space $C[0,1]$ equipped ...
0
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2answers
50 views

A sequence of functions converges in $C[0,1]$ iff it is Cauchy? Is it pointwise or uniform convergence?

In my notes, there is this theorem: A sequence in $R^n$ converges (to a limit in $R^n$) iff it is Cauchy. I understand that this theorem applies to all complete metric spaces, not just to $R^n$. ...
0
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1answer
17 views

How to prove that the steeple function is not uniformly convergent?

In class we encountered this function $$f_n(x)=\begin{cases} n^2x, & 0 \leq x \leq 1/n\\ 2n - n^2x, & 1/n \leq x \leq 2/n\\ 0, & 2/n \leq x \leq1 \end{cases}$$ The prof said ...
0
votes
2answers
25 views

Completeness of a certain normed space

let $(X, \| \|$) be the normed linear space of bounded uniformly continuous real valued functions defined on $R$. I need to prove that X is complete (under sup norm). My attempt: Let {$f_n$} be a ...
0
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2answers
26 views

Show that the series is not absolutely convergent but is uniformly convergent in the whole complex plane

Show that the series is not absolutely convergent but is uniformly convergent in the whole complex plane. ...
1
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2answers
38 views

On proving uniform convergence on an interval.

I have a doubt on uniform convergence. We have said in class that a series of functions $f_n(x)$ converges uniformly on an interval $I$ iff $$ \lim_{n \rightarrow \infty} \sup_{I} | \sum_{k = n ...
1
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0answers
15 views

Weak uniform convergence

Let $(X,\|\cdot\|)$ a reflexive and separable Banach space, and note by $X^{*}$ its topological dual and $\omega$ its weak topology. Also, put $C_{\omega}(I,X)$ the space of the continuous mappings ...
0
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0answers
10 views

Sufficient conditions differentiability in quadratic mean

I'm trying to show lemma 7.6 in van der Vaart "Asymptotic Theory" on the sufficient conditions for differentiability in quadratic mean of a probability density function but I have some doubts when it ...
2
votes
1answer
22 views

Sequence of functions, pointwise but not almost uniform convergence to $0$.

I have the function sequence: $$f_{n}(x)=\chi_{(n,\infty)}(x)=\begin{cases}1, & \text{ if } x>n\\0, & \text{ otherwise}\end{cases}$$ Why does this sequence converges almost everywhere on ...
0
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2answers
45 views

Uniform convergence, wrong answer?

I have the functions $$ f_n(x) = x + x^n(1 - x)^n $$ that $\to x$ as $n \to \infty $ (pointwise convergence). Now I have to look whether the sequence converges uniformly, so I used the theorem and ...
1
vote
0answers
14 views

Limit of convergent sequence of contraction maps

Let $f_n$ a sequence of contractions on a metric space $(Y,d)$, with a Lipschitz constant $0<\lambda<1$. Suppose that for all $y\in Y$ the sequence $f_n(y)$ converges to $f(y)$. Then $F$ is also ...
-1
votes
0answers
23 views

Uniform Convergence for Digamma function sum representation

I am dealing with the following summation: $$ \frac{1}{\psi_{(1)}(1)} \sum_{n=0}^{\infty} \frac{1}{(1+n-x)(1+n-y)} = \frac{-1}{\psi_{(1)}(1)}\frac{1}{x-y} \left( \psi_{(0)}(1-x) - \psi_{(0)}(1-y) ...
0
votes
1answer
21 views

Find intervals where a series converges and uniformly converges

I'm not sure if I'm doing better. Here is the stuff. Consider the sequence of functions $$f_n(x) = nx \left(\frac{x}{n}\right)^n\text{sinc}^n\left(\frac{x}{n}\right)$$ and the series $$s(x) = ...
1
vote
1answer
38 views

Number of roots of a sequence of a uniformly convergent holomorphic functions implies an upper bound for the number of roots of their limit

Let $G$ be an open, simply connected region in $\mathbb{C}$. We define a sequence of holomorphic functions $(f_n)_{n \in \mathbb{N}}, f_n: G \to \mathbb{C}$ as almost uniformly convergent iff $(f_n)$ ...
0
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0answers
28 views

Sequence of Functions - Pointwise and Uniform Convergence

I'm learning about sequences of functions and need some help with this problem: Show that the sequence of function $f_n(x)$ where $$f_n(x) = \begin{cases} \frac{x}{n}, & \text{if $n$ is ...
0
votes
1answer
32 views

Pointwise and Uniform converge of $f_n(x) = \frac{nx^2}{1 + nx}, x \in [0, 1]$

I'm learning about sequences of functions and need some help with this problem: Investigate pointwise and uniform convergence of the sequence of functions $$f_n(x) = \frac{nx^2}{1 + nx}, x \in ...
1
vote
2answers
41 views

Convergence of Series of Functions $\sum_{n = 1}^{\infty}(n + 1)e^{1 - nx}$

I'm learning about series of functions and need some help with this problem: Given the series of function $\sum_{n = 1}^{\infty}(n + 1)e^{1 - nx}$ show that (i) it converges pointwise but ...
0
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0answers
27 views

Uniform approximation by polynomials in $\frac{1}{z-z_1}$

In Princeton's Complex Analysis by Stein, E.M. and Shakarchi, R., in order to prove Runge's approximation theorem, the author established the following lemma (Lemma 5.10, page 63): If $K$ is ...
0
votes
1answer
28 views

From pointwise convergence to uniform convergence

Consider a real-valued random variable $X$ defined on the probability space $(\Omega, \mathcal{F}, \mathbb{P})$. Let $a\in \mathbb{R}$. Let $\{f_n(X, a)\}_n$ be a sequence of real-valued random ...
0
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0answers
36 views

Prove that $\lim_{x\to\infty} f(x) = \alpha$

Let $f_n\to f$, uniformly on $[0,\infty)$ and let's assume that $\lim_{x\to\infty} f_n(x) = \alpha_n$ and $\alpha_n\to \alpha$. Show that: $$\lim_{x\to\infty} f(x) = \alpha$$ Now, the proof goes ...
3
votes
1answer
55 views

Exchange of limits in integration

Let $f_n(x):=\frac{x^n}{n+x^n}$ for $x \in [0,+\infty)$. Let $f$ indicate its pointwise limit, i.e. $$f:\begin{cases}0 &\quad~ 0\le x< 1 \\ 1 & \quad \quad x\ge 1\end{cases}$$ I observed ...
1
vote
1answer
38 views

Do these series converge uniformly?

$$\sum_{n \geq 0}e^{-nx}\cos(nx),\quad x \in \mathbb R$$ $$\sum_n (-1)^n \frac {x^2+n}{n^2},\quad x\in \mathbb R$$ I tried using Weierstrass M-test but that does not work here. I think these series ...
2
votes
2answers
34 views

Show that the series converges to a function $S(x)\in C^1$.

Let $$\sum_{n=1}^\infty \frac{(-1)^n}{\sqrt n} \arctan(\frac{x}{\sqrt n})$$. Show that the function converges uniformly for every $x\in\mathbb{R}$ to a function, $S(x)$ which is in $C^1$. So I ...
0
votes
2answers
38 views

Prove that $\sum_{n=1}^\infty \frac{e^{inx}}{n}$ is uniformly convergente by variable change

Prove that $$\sum_{n=1}^\infty \frac{e^{inx}}{n}$$ is uniformly convergente in $[\delta,2\pi-\delta]$ with $\delta \in(0,\pi)$ It could be proved by the  Dirichelt's criterion; but if we want to ...
0
votes
0answers
29 views

$e^{e-1}-e$ and a series involving Stirling numbers of second kind

First, using the generating function for the Stirling numbers of second kind ${m\brace n}$ $$\frac{(e^x-1)^n}{n!}=\sum_{m=n}^{\infty}{m\brace n}\frac{x^m}{m!},$$ multiplying by $e^x$, and integrating ...
1
vote
3answers
42 views

Does $f_n(x)=\frac{x}{n}\log(\frac{x}{n})$ converge uniformly in $(0,1)$?

$f_n(x)=\frac{x}{n}\log\frac{x}{n}$ I tried to expand the bounds to $[0,1]$ and prove that the hypotheses of Dini's uniform convergence criterion are satisfied, but I'm not even sure I can expand the ...
2
votes
0answers
46 views

The limit of a uniform convergent sequence of isometries is an isometry (problem 6-3 of Lee's “Riemannian manifolds”)

I'm trying to prove the following theorem: let $f_n : M \to N $ a sequence of isometries of Riemannian manifolds that converges uniformly to a function $f:M \to N$: prove that $f$ is an isometry too. ...
0
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0answers
38 views

Show that $\sup_{x\in [0,1]} f_n(x) \to \sup_{x\in [0,1]} f(x)$

Let $f_n(x):[0,1]\to\mathbb{R}$, a sequence of functions which converges uniformly to $f(x)$, a bounded function. Show that $\sup_{x\in [0,1]} f_n(x) \to \sup_{x\in [0,1]} f(x)$ $|f(x)|< M$ ...
0
votes
2answers
32 views

How to preform a uniform convergence test on a sequence of functions with intervals that depend on $n$?

Let $$ f_{n}(x) = \begin{cases} \dfrac{1}{x} & \dfrac{1}{n}\leq x \leq 2 \\[4pt] 0 & 0 < x < \dfrac{1}{n} \end{cases} $$ $f_{n}(x)$ converges point-wise to $f(x)=\frac{1}{x}$, ...
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0answers
37 views

What is the name of the test that is used to test whether a sequence $f_{n}$ is uniformly convergent?

As I understand there are two tests where you test whether a function is uniformly convergent. 1) Where you use $\epsilon>0$ and find $N$ such that $|f_{n}-f|<\epsilon$, $\forall n>N$, what ...
3
votes
4answers
92 views

Is it true that if $f_{n}\rightarrow f$ uniformly converges then $f^{\prime}_{n}\rightarrow f^{\prime}$?

Let $f_{n}$ be some sequence of functions. If $f_{n}$ uniformly converges to $f$ then is it true that $f^{\prime}_{n}\rightarrow f^{\prime}$? Is there an example that proves/disproves this?
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0answers
17 views

Uniform convergeness and polynomials

If $f(x)$ is continuous function on $[a,b]\Rightarrow\exists <P_{n}(x)>$ sequence of polynomials such that $P_{n}(x)$ uniformly converges to $f(x)$ on $[a,b]$. Is this statement true or false? ...
2
votes
3answers
44 views

Is $f_{n}(x)=x^{n}$ uniformly convergent on the interval $[0,1]$?

Let $f_{n}(x)=x^{n}$. I know that $f_{n}(x)$ converges to $0$ on the interval $[0,1)$ and converges to $1$ on $x=1$. But is $f_{n}(x)=x^{n}$ uniformly convergent on the interval $[0,1]$?
1
vote
2answers
38 views

Using Arzelà–Ascoli Theorem for the whole sequence

Let $K \subset \mathbb{R}^n$ be compact and $(f_n)$ a sequence such that $f_n \colon K \to \mathbb{R}^M$ is continuous for all $n ∈ \mathbb{N}$. Suppose that $(f_n)$ is pointwise convergent and ...
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votes
1answer
73 views

Prove that $f_{n}(x) = \frac{1}{x} \chi _{[\frac 1 n, 2]}$ is not uniformly convergent on $[0,2]$?

Let the sequence of functions $\{f_{n}\}_{n=1}^{\infty}$ be defined by $f_{n}:[0,2] \to \mathbb{R}$ where $f_{n}(x) = \left\{ \begin{eqnarray} 0 &,& 0 \le x < \frac 1 n \\ \frac 1 x ...
0
votes
1answer
31 views

Find if $\sum_{n=1}^{\infty} (nx-n+1)x^n$ converges uniformly for $|x| < 1$.

Find if $$\sum_{n=1}^{\infty} (nx-n+1)x^n$$ converges uniformly for $|x| < 1$. I can tell that the partial sum is $s_n(x) =nx^{n+1}$, thus the sequence converges to 0. What can I do from ...
0
votes
1answer
18 views

Differentiable function impleis uniform convergence

If $f_n$ is differentiable, does that imply $f_n \to f$ uniformly if $f$ is the limit of $f_n$? If so, is it a good idea to use the formal definition of differentiability, then show from this that ...
4
votes
3answers
56 views

Uniform convergence of the series on unbounded domain

1 $$\sum_{n=1}^\infty (-1)^n \frac{x^2 + n}{n^2} $$ Is the series converges uniformly $\mathbb R$ I have tired by this result if $\{f_n(x)\}$ is a sequence of a function defined on ...
1
vote
1answer
28 views

Uniform convergence in Mercer Theorem for bounded kernels

Let $\mu$ be a finite, strictly positive measure on $\mathbb{R}$, and let $k$ be a measurable positive-definite kernel. Assume $k$ is bounded, and let $T:L^2(\mu)\rightarrow L^2(\mu)$ be defined by $$ ...
2
votes
0answers
17 views

Stochastic Convergence

I need help figuring out if a series of "apparently random" digits are the result of the same (possibly non-polynomial) function, ergo, not-random but deterministic. The highest level math I know is ...
1
vote
2answers
41 views

Does $f_n :$ $ ]0;1[ \to \Bbb R, f_n = \frac{nx}{nx+1}$ converge uniformly?

Let $f_n :$ $ ]0;1[ \to \Bbb R, f_n = \frac{nx}{nx+1}$ be a sequence of functions. Is the convergence pointwise or uniform? It is easy to show, for $x$ fixed, that the numeric series defined by ...
0
votes
2answers
44 views

Show that the radius of convergence of $e^x$ is infinite

I am a bit confused as to whether I am doing this question correctly. Firstly, we have defined the radius of convergence of a power series centered at a $$\sum_{n=0}^{\infty} a_n(x-a)^n$$ to be the ...
0
votes
1answer
37 views

Determine whether a function series is uniformly convergent

Determine whether $\sum_{j=0}^{\infty} \frac{\sin(jx)}{(2+x^2)^j}$ is uniformly convergent for $x\in\mathbb{R}$ So I started by saying as $|\sin(jx)|\le1$ so $\sum_{j=0}^{\infty} ...
0
votes
1answer
24 views

Prove a functional series is pointwise and/or uniformly convergent

Determine whether the following sequence is pointwise and/or uniformly convergent $(f_n)_{n\in\mathbb{N}}$ where $x\in\mathbb{R}$ and $$f_n(x)= \left\{ \begin{array}{ll} n & ...
1
vote
1answer
35 views

Convergence of the series (N.B.H.M)

Which of the following are true $\sum_{n=1}^{\infty}\frac{(-1)^n + \frac{1}{2}}{n} $ is convergent. I have tried by Lebnitz rule., but $|a_n|$ is not a decreasing sequence, So ...
0
votes
1answer
34 views

Show a sequence of functions is pointwise but not uniformly convergnt

Let $f:\mathbb{R}\rightarrow\mathbb{R}$ be a bounded function with the properties $f(0)=0$ $\lim_{x\to\infty} f(x)=1$. For each $n\in\mathbb{N}$ define $f_n(x)=f(x+e^n)$ with $(x\in\mathbb{R})$ ...
0
votes
1answer
43 views

Determine whether the functional series $\sum\frac1{(x+j)^2}$ and $(\sin\frac{x}{j})^j$ are pointwise and/or uniform convergent

Determine whether the following functional series is pointwise and/or uniformly convergent 1) $\sum_{j=1}^{\infty} \frac{1}{(x+j)^2}$ with $(x>0)$ 2)$\sum_{j=1}^{\infty} ...
1
vote
1answer
56 views

Is $f_n$ uniformly convergent?

Prove that $f_n(x)=(\sqrt{x^2+\frac{1}{n}})_{n\in\mathbb{N}}$ $(x\in\mathbb{R})$ is pointwise convergent, and then check to see if its uniformly convergent So I can prove it is pointwise: ...