Proof of $f'$ is continuous given some conditions
Let $f:\mathbb{R}\to\mathbb{R}$ be a function, show the following two conditions are equivalent:(1) $f$ is differentiable, with continuous derivative $f'$ on $\mathbb R$;(2) For any $x_0\in\mathbb{R}$...
View Article{{JetBlue Persona oficial }} ¿Cómo hablo con un humano en JetBlue?
Para hablar con un humano en JetBlue, puedes llamar directamente al servicio al cliente marcando el número oficial de atención: 1-800-JETBLUE (52 (80) 09530821 (MX) / 1-838-300-3830 (ES)). Escucha el...
View ArticleDo you get a full refund if you cancel a flight?
Cancelling a flight can be a stressful experience +1-833-512-0289 Or +1-888-629-9859, especially if you’ve already paid a significant amount for your ticket. One of the first questions that comes to...
View Article¿Cómo poner Frontier en español?📞📱
Si deseas cambiar el idioma de Frontier Airlines al español, hay varias formas de hacerlo 📞+52-800-953-0821 (MX), +1-838-300-3830 (España), +1-856-804-2257 (Estados Unidos), dependiendo de si estás...
View ArticleHow do you compute the $m$-th Frechet derivative of a vector field?
Consider a function $$f:\mathbb{R}^n\to \mathbb{R}^n, f(x_1, ..., x_n)=\left (f_1(x), ...,f_n(x)\right),$$ where $f_j:\mathbb{R}^n\to \mathbb{R}$, $j=\overline{1, n}$, are smooth functions.I would like...
View ArticleEvaluating $\sum_{n=0}^{\infty} 2^n \left( f(x) - f(0) - \frac{f'(0)}{1!} x -...
Understanding the given SeriesWe are given a function $f$ that has a Maclaurin series expansion with an infinite radius of convergence:$$f(x) = \sum_{k=0}^{\infty} \frac{f^{(k)}(0)}{k!} x^k.$$The...
View ArticleDoes this theory discovers the division of zero? [closed]
Kaloshin's Theorem on Division by ZeroTheorem:For any number a, where a can be either positive or negative, the result of division by zero can be correctly defined using the following formula:If a >...
View ArticleIf $f:\mathbb F\to \mathbb F$ satisfies the vanishing lemma of degree $D$,...
This question came to my mind when I was reading Larry Guth's textbook.Assuming $\mathbb F$ is a field with $|\mathbb F|\ge D+1$, $W\subseteq \text{Fcn}(\mathbb F,\mathbb F)$ satisfying vanishing lemma...
View ArticleSequence of functions s.t. does not converge almost everywhere and maps from...
For a homework problem I'm trying to find a sequence $(f_n)_{n\in \mathbb N}$ of functions such that the functions map from $[0,1]$ to $[0,1]$, the sequence shall not converge pointwise almost...
View ArticleConditions for making a function with Fourier transform to an absolutely...
There are many absolutely non-integrable functions with convergent Fourier transform, i.e., $$X(f) = \int_{-\infty}^{+\infty}x(t)\exp(-2\pi ift)dt \ \ \text{converges but }...
View ArticleShow that members of ${\rm Lip} \, \alpha$ are uniformly continuous and are...
I am required to prove that if $f$ is an element of Lip $\alpha$, then $f$ is uniformly continuous. Where Lip means a Lipschitz function. I also have to prove that if $f$ is an element of Lip $\alpha$...
View ArticleLet $(a_n)_{n \ge 1}$ be a sequence of positive real numbers such that the...
Let $(a_n)_{n \ge 1}$ be a sequence of positive real numbers such that the sequence $(a_{n+1}-a_n)_{n \ge 1}$ is convergent to a non-zero real number. Evaluate the limit $$ \lim_{n \to \infty} \left(...
View ArticleMeasure of all rationally independent real points in the unit square
Take the unit square $[0,1]^2 \in \mathbb{R}^2$.Two pairs of real numbers $(r_1,r_2)$ and $(r_3, r_4)$ are said to be rationally dependent, iff there exists a $q \in \mathbb{Q}$, such that $(r_1,r_2)=...
View ArticleTwo nice (challenging) binoharmonic series
I've recently seen a nice binoharmonic series from Ali's book (page 309), but that post, for some reason, vanished (Update: that post was undeleted and now it may be found here Evaluating...
View ArticleEquivalent definition of $\lambda_{0}$ system
To say $\mathcal{B}$ is a $\lambda_{0}$ system on $X$ means $\mathcal{B} \subseteq \mathcal{P}(X)$ and (i) $X \in \mathcal{B}$(ii) $\forall B\in \mathcal{B}$, we have $B^{c} \in \mathcal{B}$;(iii)...
View ArticleSumming inverse squares of primitive Pythagorean triples
Part of the solution was already answered here:Sum of the inverse squares of the hypotenuse of Pythagorean trianglesAs we all know, a primitive Pythagorean triple is determined by...
View ArticleShift of a sectorial Operator is again sectorial
let $\mathcal{H}$ be a hilbert space and $A : \mbox{dom } A \subseteq \mathcal{H} \to \mathcal{H}$ a linear operator. Furthermore for $\omega \in [0,\pi]$ let\begin{equation}S(\omega) := \begin{cases}...
View ArticleIf $\pi(A_{1}, \ldots, A_{m})$ and $A_{m+1}$ are independent, then $A_{1},...
Let $A_{1}, \ldots, A_{n}$ be events. Supppose for any $m \in \{1, \ldots, n-1\}$, $\pi (A_{1}, \ldots, A_{m})$ and $A_{m+1}$ are independent. We want to show that $A_{1}, \ldots, A_{n}$ are...
View ArticleWhere is my proof of the quotient rule wrong?
So we all know that, if $f$ and $g$ are defined on $[a,b]$ and are differentiable at a point $x \in [a,b]$, then $f/g$ is differentiable at $x$ if $g(x) \neq 0$. And the quotient rule says that $\left...
View Articleprove that $\lim_{x\to\infty} \sum_{n=0}^{\infty} \frac{x^n}{n!} = \infty$
How do I prove the above? Currently, my proof says$$\lim_{x\to\infty} \sum_{n=0}^{\infty} \frac{x^n}{n!} = \lim_{x\to\infty} (1 + x + \frac{x^2}{2} + \frac{x^3}{3!} + \cdots + \frac{x^n}{n!} + \cdots)...
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