Sequence with no accumulation values
I have heard that any sequence that has no accumulation values have at least one improper one at $\infty$ or $-\infty$. But I created a sequence $a_n = n \times (-1)^n$ that seems to have no one. Am I...
View ArticleDerivation of formula for (C,1) summability of integrals
I'm having trouble understanding why the (C,1) formula for integrals is given by $$ \lim_{\lambda\to\infty} \int_{0}^{\lambda}\bigg(1- \frac{x}{\lambda} \bigg)f(x) \,dx $$I understand that we want...
View ArticleIf $\lim\inf_{n\to\infty}(a_n+b_n)=\lim\inf_{n\to\infty}\...
If $\underset{n\to\infty}{\lim\inf}(a_n+b_n)=\underset{n\to\infty}{\lim\inf}\ a_n+\underset{n\to\infty}{\lim\inf}\ b_n$ for any sequence $\{b_n\}$ in $\Bbb R$, does $\{a_n\}$ have to be convergent?My...
View ArticleIf $f$ is a continuous mapping from a metric space $X$ to a metric space $Y$...
In a solution to Baby Rudins excercise 4.4 I found the following statement:If $f$ is a continuous mapping from a metric space $X$ to a metric space $Y$ and $E$ a dense subset of $X$. Then $f(X) =...
View ArticleSingular extremal of a constrained variational problem
Consider the following constrained variational problem: $$\min_{u \in H^{1}(I)} \{\mathcal{F}(u) : u(\pm 1) = 1, \mathcal{G}(u) = 1/3 \},$$ where $I = [-1, 1] \subseteq \mathbb{R}, H^1 (I) := H^{1,...
View ArticleGive example for any 2 sequences which converges but their product diverges...
An example for 2 sequence which converges but their product diverges
View ArticleSharp growth of functions in Gelfand-Shilov spaces
It is well known that a function belongs to $S^\alpha_\alpha$ if and only if there exists $h>0$ such that$$\|f(x)e^{hx^{\frac1\alpha}}\|_\infty+\|\hat...
View ArticleNon-genericity of lower-dimensional functions that can map in the set of...
Supose I have a continuously differentiable function $f:\mathbb{R}^k\rightarrow \mathbb{R}^{n\times m}$, i.e. from the set of $k-$dimensional vectors into the set of $n\times m$ matrices. Suppose that...
View ArticleA Ramanujan style series
Recently, someone asked a question involving the expression$$ \sum_{n=0}^{\infty} (-1)^n (4n+1) \left(\frac{(2n-1)!!}{(2n)!!}\right)^3 $$At first glance, I knew that the expression was the value of a...
View ArticleCalculation of $\int_{1}^{\infty} \frac{\sin^2(\frac{ax}{2})}{x^p}dx$ for $1
I'm having trouble calculating$$\int_{1}^{\infty}\frac{\sin^{2}\left(ax/2\right)}{x^{p}}{\rm d}x\quad\mbox{for}\quad 1 < p < 3$$Could someone please help ?.Everywhere I look I can only find that...
View ArticleInduced norms and inequalities
Background:There was a part of my professors notes that I didn't quite understand. This was the statement:If a norm $|\cdot|$ is induced by a scalar product the following inequality is...
View ArticleUnderstanding the Relationship Between a Function and Its Integral
Body:Hello everyone,I'm currently studying calculus and I've come across a problem that I'm having trouble with. I would appreciate any help or guidance.Here's the problem:Let $$f:[0,3]\to\mathbb{R}$$...
View ArticleFinding limit of a sequence $\sqrt{n}\left(A_{n+1}-A_n\right)$ [duplicate]
QuestionLet $\left\{a_n\right\}, n \geq 1$, be a sequence of real numbers satisfying $\left|a_n\right| \leq 1$ for all $n$. Define$$A_n=\frac{1}{n}\left(a_1+a_2+\cdots+a_n\right),$$for $n \geq 1$. Then...
View ArticleDetermine whether $\int_0^{\infty} \frac{\arctan(x)}{e^x - e}dx$ converges or...
Determine whether $\int_0^{\infty} \frac{\arctan(x)}{e^x - e}dx$ converges or diverges.Attempt: I know that $\arctan x\leq \pi/2$, but how can I proceed from here? I also split the integral...
View ArticleRiemann-Stieljes integration and Total variation
Let $\alpha:[a,b]\to\mathbb{R}$ be a function of bounded variation on $[a,b]$ and $f:[a,b]\to\mathbb{R}$ a bounded function.It is well known that if $f$ is Riemann-Stieljes integrable respect to...
View ArticleIs it true that $\int_a^bf(g(x))\mathrm{d}x
Is my conjecture true?Let $f(x)$ and $g(x)$ be functions that are continuous, real-valued, positive and strictly increasing. Let $a,b,c$ and $d$ be positive real numbers....
View ArticleProve of the completeness of metric space [closed]
$S=\{x=(x_1,...,x_n,...), x_i\in R\}$, $d(x,y)=\sum\limits_{n=1}^{\infty}\frac{1}{2^n}\frac{|x_n-y_n|}{1+|x_n-y_n|}$. is $(S,d)$ complete?
View Articleproduct $\sigma$-algebra on uncountable index set
Let $I$ be an index set and $(\Omega_i, \mathcal{A}_i)$ measurable spaces for every $i \in I$. Then the product $\sigma-$algebra is defined by$$\bigotimes_{i \in I} \mathcal{A}_i := \sigma(E)$$ with$$...
View ArticleRudin's RCA, Theorem $7.16$: The Fundamental Theorem of Calculus.
There is the equality: $$ f(x) - f(a) = \int_a^x f'(t)dt \ \ (a \leq x \leq b). \tag{1}$$There is assumption by Rudin:Suppose $f$ is continuous on $[a,b], f$ is differentiable at almost every point of...
View ArticleSymmetry of functions on the sphere
Let $u$ be a smooth function defined on the sphere $\mathbb{S}^2$, and let $R \in \mathrm{SO}(3)$ be a three-dimensional rotation. Define$$S_R = \{x \in \mathbb{S}^2 : u(x) \neq u(Rx)\}.$$Suppose there...
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