Clik here to view.
Does the limit $\lim_{x\to1^{-}}(1-x)\sum_{n=1}^{\infty}2^{n} x^{2^n}$ exist?
Let the lacunary power series $$f(x):=(1-x)\sum_{n=1}^{\infty}2^{n} x^{2^n},0<x<1.$$ We use Matlab to draw the function $f(x)$ graph, which shows that the function $f(x)$ has limits when...
View ArticleComputing eigenvalues for specific initial conditions
Take the following equation $-x'' = \gamma x$ which has the known solution $$x = A\cos(\sqrt{\gamma}t ) + B \sin(\sqrt{\gamma}t).$$Now assume that $x(0) = x(\alpha)$ and $x'(0) = -x'(\alpha)$. I am...
View ArticleConvex combinations sequence question
Let $a_0,a_1,\beta$ be given with $0<\beta<1$ Let the sequence be defined by $a_{n+2} = \beta a_{n+1} + (1-\beta)a_n$ for $n\geq0$Show that $\{a_n\}$ converges and find its limit..How to find...
View ArticleUnderstanding least upper bound under Dedekind cuts
I'm studying real numbers under Dedekind's cuts construction ($x=\{x\in \mathbb{Q}:x<q \text{ for some } q\in\mathbb{Q\}}$ would be a real number).The theorem I'm trying to understand is this: for...
View Article$\mu$ is a finite signed measure on $(\mathbb{R},\mathscr{B}(\mathbb{R}))$....
I want to prove the following:Claim$\quad$Let $\mu$ be a finite signed measure on $(\mathbb{R},\mathscr{B}(\mathbb{R}))$. Show that $V_{F_{\mu}}(-\infty,x]=|\mu|((-\infty,x])$ for all...
View ArticleLimit of integrals over compact $J$-measurale sets
Question. Let $U \subset \mathbb{R}^m$ be an open $J$-measurable set and $(K_i)$ a sequence of compact $J$-measurable sets such that $K_i \subset \operatorname{int} K_{i+1}$ for every $i \in...
View ArticleQuestion About Absolute Continuity of A Finite Signed Measure (Proof of...
I am self-studying measure theory using Measure Theory by Donald Cohn. I got stuck on a step in his proof of Proposition 4.4.5. Here is the statement of the proposition:Proposition 4.4.5$\quad$Let...
View ArticleJensen's inequality for integrals
What nice ways do you know in order to prove Jensen's inequality for integrals? I'm looking for some various approaching ways.Supposing that $\varphi$ is a convex function on the real line and $f$ is...
View ArticleOpen Cover of $0 < |x| \leq 1$ in $\mathbb{R}$?
I have to give an example of the following:Let $A = \{x \in \mathbb{R^2} \ |\ 0 < |x| \leq 1\}$. Give an open cover of $A$ that has no finite subcover of $A$. Furthermore, I have to prove this is...
View ArticleProve that if $x \in R,$ then there exists $n \in Z$ satisfying $x \leq n < x+1$
So this question in my book looks like it's essentially asking me to prove the ceiling function exists. This question is slightly different to other things I found in related question because we're...
View ArticleProving that a function defined as an expectation is Lipschitz
I would like to prove thisLet $G(y)\doteq\mathbb{E}[X|\gamma X+\gamma^{1/2}Z=y]$ where $Z\sim\mathcal{N }(0,1)$ and $X\sim\pi(x)$ where $\pi(x)$ has bounded support. $Z$ and $X$ are indepenedent. I...
View ArticleFrom Taylor Series to Rational Function
I am considering the reverse process of the Taylor expansion of a rational function.Question: suppose that I have$$f(x) = a_{0}+a_{1}x+a_{2}x^{2}+\cdots,$$and I know that $f(x)$ is a rational function...
View ArticleHow to evaluate the integral $\int_0^{\infty} \frac{\sin(k \ln(t))}{\sqrt{t}}...
I am trying to evaluate the integral$$\int_{0}^{\infty}\frac{\sin\left(k\ln\left(t\right)\right)}{\sqrt{t}}\,\left\{\frac{1}{t}\right\} \, dt,$$ where $\left\{\cdots\right\}$ denotes fractional part...
View ArticleIf $\lim\limits_{x \to \pm\infty}f(x)=0$, does it imply that $\lim\limits_{x...
Suppose $f:\mathbb{R} \rightarrow \mathbb{R}$ is everywhere differentiable and$\lim_{x \to \infty}f(x)=\lim_{x \to -\infty}f(x)=0$,there exists $c \in \mathbb{R}$ such that $f(c) \gt 0$.Can we say...
View ArticleIs this functional in equality $\left(\int_0^1 |f'(x)| dx\right)^2 \geq C...
Inspired by Sobolev inequalities, I would like to find a universal constant $C>0$ such that$$ \left(\int_0^1 |f'(x)| dx\right)^2 \geq C \left(\int_0^1 f(x)dx -1\right) $$for all functions $f \in...
View Article$0.101001000100001000001$... is irrational. [duplicate]
How do I show that $0.101001000100001000001...$ is irrational? How do I generalize this to decimal expansions of a similar type, i.e. what is a criterion for decimal expansions with $0$'s and $1$'s...
View ArticleCan we replace finite with countably infinite sequences in the definition of...
Absolute Continuous functions require a finite sequence of subintervals of the domain. Can we replace this finite sequence with a countably infinite one in this definition?Please note that an...
View ArticleNumerical integration of $\int_0^{\infty} t^{a-1} \cos(k \ln(t))...
For $0<a<1$ and $k>0$, I am studying the following integral, $$h(a,k)=\int_0^{\infty} t^{a-1} \cos(k \ln(t)) \left\{\frac{1}{t}\right\} \, dt .$$How to numerically integrate this integral in...
View ArticleSmooth version of Tietze extension theorem
Thanks to the properties of mollifications, it is very easy to prove Urysohn's lemma in the euclidean space with the big plus that the constructed function is smooth.I was wondering if something of the...
View ArticleSum of two function sequences
Let (fn) and (gn) be two function sequences which converge uniformly to the functions f and g respectively. Let a and b two real numbers.Then the function sequence (a.fn + b.gn) converge uniformly to...
View Article