How do I prove that the set of whole numbers is not bounded neither above nor...
How do I prove that the set of integers is bounded neither above nor below? I know that the set of natural numbers is not bounded above. I want to use this knowledge, but I don’t know how.
View ArticleAn outer measure is built from a collection of sets, then are these sets...
We knowLet $\epsilon\subset\mathscr{P}(X)$ and $\rho:\epsilon\rightarrow[0,\infty]$ be such that $\emptyset\in\epsilon, X\in\epsilon,\rho(\emptyset)=0$, then the set function defined on...
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Conceptual question about applying the uncertainty principle to a solution of...
This question is related to the following previous questions question 1, question 2. I plan to share it with a teacher so please don’t vote for delete it until he reads it.In brief, this is a...
View ArticleSolving the integral $\int_{-\infty}^{\infty} \frac{\ln(1 + (x + x^2)^2) +...
I saw the integral $$\int_{-\infty}^{\infty} \frac{\ln(1 + (x + x^2)^2) + \ln(1 + x^2)}{x^2} \, dx$$ on a hard integral of the day page. The solution I found on the page used complex analysis, but I am...
View ArticleSeries $\sum _{k=0}^{n-1}\frac{1}{2^k }\binom{2 k}{k}$
I'm looking for series alike in the literature containing in their summands numbers of the type $\displaystyle Q_n=\sum _{k=0}^{n-1}\frac{1}{2^k }\binom{2 k}{k}$. The following three series have...
View ArticleProof that non-monotonic sequence is unbounded
I am looking at a non-monotonic recursive sequence given by$$a_0 =2,$$$$a_{n+1} = \frac{1}{2}\left( a_n - \frac{1}{a_n}\right).$$Is the sequence bounded or unbounded?I suspect that the sequence is...
View Articleuniqueness backwards in time of ODE: no solution starting at $x_0$ can reach...
Use uniqueness backwards in time to show that if $\dot{x} = f(x)$ and f is locally Lipschitz,if $x_0 \ne y_0$ and $f(y_0) = 0$, no solution starting at $x_0$ can reach the fixed point at $y_0$ ina...
View ArticleIf $f'$ tends to a positive limit as $x$ approaches infinity, then $f$...
Some time ago, I asked this here. A restricted form of the second question could be this:If $f$ is a function with continuous first derivative in $\mathbb{R}$ and such that $$\lim_{x\to \infty} f'(x)...
View ArticleConvergence of series $\sum_{n=1}^\infty (-1)^n \sin(\log(n))\cdot...
Find the range of values $p$ such that the series $$\sum_{n=1}^\infty (-1)^n \sin(\log(n))\cdot \frac{1}{n^p}$$ converges.Context: this question was on an undergraduate real analysis exam at a...
View ArticleProving that $\lim\limits_{x\to\infty}f'(x) = 0$ when...
I've been trying to solve the following problem:Suppose that $f$ and $f'$ are continuous functions on $\mathbb{R}$, and that $\displaystyle\lim_{x\to\infty}f(x)$ and...
View ArticleLimit of a function. Integration problem
I have this problem:Let $c\in \mathbb{R}$. If $\int_c^\infty f(x)dx$ converges, then $$\lim_{x\to \infty} f(x)$$ exist and is $0$. Moreover, if $f$ is monotonic, $\lim_{x\to \infty} xf(x)$ exist and is...
View Article$E$ measurable implies $\forall \varepsilon>0\exists A\in\mathcal S_\delta$...
From Royden's Real Analysis 4th edition, section 17.5, problem 36:Let $\mu$ be a finite premeasure on an algebra $\mathcal S$, and $\mu^*$ the induced outer measure. Show that a subset $E$ of $X$ is...
View ArticleEstimation of Riemann integral over subinterval
ProblemSuppose a real-valued function $f$ be Riemann integrable over an interval of length $L$ and its integral is $I$. For what positive number $r$ can we guarantee that there must exists a...
View ArticleThe Cauchy Product of $(\operatorname{Li}_2(x))^3$
Here is a bunch of interesting and useful Cauchy products in the work with the harmonic series,$$i) \ \log^2(1-x)\operatorname{Li}_2(x)$$$$=\sum _{n=1}^{\infty } x^n...
View ArticleCalculate...
How to integrate this using cylindrical coordinates: $$\int_{-1}^1\int_{-\sqrt{1-y^2}}^1\int_{-1}^1(x^2+y^2)^{3/2}\,dz\,dx\,dy$$Now what I can understand is that we can directly integrate the $z$ part....
View Articlethe solution of Fredholm´s integral equation
Be $\lambda \in \mathbb{R}$ such that $\left | \lambda \right |> \left \| \kappa \right \|_{\infty }(b-a)$.Prove that the solution $f^*$ of the integral equation of Fredholm $$\lambda f...
View ArticleGradient of the function and the contour line
I do not understand, reading the chapter in the book about Lagrange multipliers, why the gradient of the function $f$ is perpendicular to the contour line? There is no sufficient explanation there,...
View ArticleIf for any $D>0$ there exists some constant $C_D$ such that $P(|X_n|\leq...
Statement: Let $\{X_n\}$ be a sequence of random variables. If for any $D>0$ there exists some constant $C_D$ such that $P(|X_n|\leq \frac{C_D}{n^{1/7}})\geq 1-C_Dn^{-D}$, then $X_n\rightarrow_p...
View ArticleLooking for closed-forms of $\int_0^{\pi/4}\ln^2(\sin x)\,dx$ and...
A few days ago, I posted the following problemsProve that\begin{equation}\int_0^{\pi/2}\ln^2(\cos x)\,dx=\frac{\pi}{2}\ln^2 2+\frac{\pi^3}{24}\\[20pt]-\int_0^{\pi/2}\ln^3(\cos x)\,dx=\frac{\pi}{2}\ln^3...
View ArticleWhen is $f_c(x,y) = xy +\lfloor cx\rfloor \lfloor cy\rfloor $ associative and...
Let $c > 0$ be a positive real number. Define $f_c : \mathbb{Z} \times \mathbb{Z} \to \mathbb{Z}$ by$$f_c(x,y) = xy +\lfloor cx\rfloor \lfloor cy\rfloor $$Now clearly we have $f_c(x,y) = f_c(y,x)$...
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