Limit superior for $(1 - \alpha)$ quantile asymptotic upper bound on...
Question.I am having difficulty working through the limit superior part of the following derivation of an asymptotic upper bound on the $(1 - \alpha)$ quantile on the probability of the most frequent...
View ArticleCalculus problem involving differentiable function [closed]
Question: Let $f: \mathbb{R} \to \mathbb{R}$ be a twice differentiable function, and let $a$ and $b$ be real numbers s.t. $a<b$ and$$\ln...
View ArticleA reverse arithmetic geometric inequality
This question arose from my question here which I cannot answer.It is well known that$$x(x-1)(x-2)\cdots(x-k+1)\leq \left(\frac{2x-k+1}{2}\right)^k.$$For given $x,k$ and $x-k+1\geq 0,$ what is the...
View ArticleProof: $\lim \limits_{n\to\infty}(1+\frac{z}{n})^n = \exp(z)$
I define $\exp: \mathbb C \to \mathbb C$ as $z \mapsto \sum \limits_ {k=0}^{\infty}\frac{z^k}{k!}$. I would like to show that $\lim \limits_{n\to\infty}(1+\frac{z}{n})^n = \exp(z)$. I have a proof for...
View ArticleUnderstanding the definition of continuity in real analysis
I'm reading the definition of continuity at a point from Introduction to Real Analysis by Bartle-Sherbert text."Real Analysis" definition of continuity:5.1.1 Definition Let $A \subseteq \mathbb{R}$,...
View ArticleEvaluate $\int_{0}^{\infty}\frac{\ln^2(x)\cdot\sin(x)}{x^2+1}dx$
Evaluate $$A=\int_{0}^{\infty}\frac{\ln^2(x)\cdot\sin(x)}{x^2+1}dx$$I rewrote it as $$A=\int_{0}^{\infty}\Im\left(\frac{\ln^2(x)\cdot...
View ArticleHow to extract an specific sequence on a Convex set
I am trying to prove that a convex map $f: S \rightarrow \Bbb{R}$ where $S \subset \Bbb{R}^n$ is an open and convex set and $n \in \Bbb{N}$ arbitrary is continuous by generalizing this proof . This...
View ArticleProve the given inequality using only classical methods.
Prove, using only classical methods, that $e^x-x^2\sqrt{x+1}-\frac{\ln x}{x}>0$$(x>0)$The CLASSICAL means that:You can take any derivative.You can use the properties of the inequalities. For...
View ArticleProve that a monotone and surjective function is continuous
Let $I$ be a interval and $f:I \rightarrow \mathbb{R}$ monotone and surjective prove that $f$ is continuous.I tried using the definition of $\epsilon$-$\delta$ and supposing that $f$ is not continuous...
View ArticleEvaluate $ \int_{0}^{1} \log\left(\frac{x^2-2x-4}{x^2+2x-4}\right)...
Evaluate :$$ \int_{0}^{1} \log\left(\dfrac{x^2-2x-4}{x^2+2x-4}\right) \dfrac{\mathrm{d}x}{\sqrt{1-x^2}} $$ Introduction : I have a friend on another math platform who proposed a summation question and...
View ArticleAlternative Representation of the Lipschitz Condition
In some papers, the following condition is often used as an assumption for a function$b: \mathbb{R}^n \to \mathbb{R}^n$:$$\langle b(x)-b(y), x-y\rangle \le K |x-y|^2, \forall x, y \in \mathbb{R}^n.$$I...
View ArticleUnion of arbitrary neighborhoods of rational numbers
Let $Q$ be the set of rational numbers, which we enumerate to write as $\{ r_n \}_{n=1}^\infty$.For each $r_n$, let us allocate some $\epsilon_n >0$ and consider the interval $[r_n -\epsilon_n,...
View ArticleWriting a sum as a contour integral
It is claimed that, if we have two complex functions $f$ and $g$, that if we sum $g$ over the zeros $\rho=\beta+i\gamma$ of $f$ where $0<\gamma\le T$, then we can write $$\sum_{0<\gamma\le...
View ArticleProve for every polynomial $p(x)$ of degree less or equal than $n$ satisfying...
For every polynomial $p(x)$ of degree less or equal than $n$ satisfying$$\lim_{x \to x_0} \frac{p(x)}{(x - x_0)^n} = 0, $$we have $p \equiv 0$.This problem is from my notes booklet on Taylor and it is...
View ArticleIf $f$ satisfies a Lipschitz condition then f is of bounded variation in...
If $f$ satisfies Lipschitz condition then $f$ is bounded variation in $\mathbb{R}$? A function $f:\mathbb{R} \rightarrow \mathbb{R}$ satisfies a Lipschitz condition at if $\exists M>0$ such that...
View ArticleA power series that converges for $|z| \leq 1$ and diverges otherwise.
I need to find a power series $\sum a_n z^n$ that converges for $|z| \leq 1$ and diverges otherwise.I think I have one I just want to be sure.So, the series:$$\sum \frac{z^n}{n^2}$$has radius of...
View ArticleShow that for all $x \ge 1$, $\log(x) \le \sqrt{x} - \frac{1}{\sqrt{x}}$...
This is problem #9 from section 3.5 of Buck's Advanced Calculus. The section is about Taylor's Theorem so my instinct was to try to find some appropriate function with a Taylor polynomial that could...
View ArticleIrrational number and asymptotic equality
If $\alpha$ is an irrational number and there exists integer sequences $A_n, B_n, C_n, D_n$ such that as $n\to\infty$$$A_n+B_n\alpha\sim (0.9)^n$$ and $$C_n+D_n\alpha\sim (0.2)^n$$Prove that these two...
View ArticleCheck whether the series $\sum_{n=1}^{\infty} \frac{(2^n n!)^2}{(2n+1)!}$...
Check whether the given series $$\sum_{n=1}^{\infty} \frac{(2^n n!)^2}{(2n+1)!}$$ converges or diverges.I did this question by using Stirling's approximation. By using the approximation, I found out...
View ArticleA proof for weak convergence of a sequence of distributions to a continuous...
There is this post giving a counterexample that has a sequence of measures that converges to Lesbegue measure on (0,1) in distribution but not in total variation as below:Let $\mu_n (E)=\int_E(1-\cos...
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