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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...
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Are there any $f(x)$ whose $\xi_n$ of the Lagrange's remainder does not...
Consider the Maclaurin's series of an analytic function $f$ with Lagrange's remainder.Define $\xi_n$ as$$f(1)=\sum_{k=0}^{n-1}\frac{f^{(k)}(0)}{k!}+\frac{f^{(n)}(\xi_n)}{n!}$$where $0<\xi_n<1$,...
View ArticleHelp me show that $\max_{k \leq n}|X_k|$ is an adapted process
Setup$(\Omega, F, P)$; probability spaceLet $\{F_n\}_{n=0,1,2,\ldots}$ be a filtration, and $X=\{X_n\}_{n=0,1,2,\ldots}$ be an adapted process.In addition, we define$$X^\ast_n = \max_{k\leq...
View ArticleHow many removable discontinuities can a real function have?
Let $f: \mathbb{R} \to \mathbb{R}$ be a function. We say that $x_0 \in \mathbb{R}$ is a point of removable discontinuity for the function $f$ if $f$ is discontinuous at $x_0$, but $\lim_{x \to x_0}...
View ArticleWeak convergence in $L^p$ preserves $L^\infty$-norm
Let $\Omega \subseteq \mathbb{R}^N$ be open. I thought about a generalization of another question, leading me to this:Assume $f_n \rightharpoonup f$ in $L^p(\Omega)$ for some $p \in [1, \infty)$ (i.e....
View ArticleLebl, Jiří. Basic Analysis I: Introduction to Real Analysis, Volume 1 -...
This is not a math problem/question.I was hoping if any one of y'all know where I could find the solution manual for the book: Basic Analysis I: Introduction to Real Analysis. I am self-studying Real...
View ArticleHow do I prove a set is a σ-algebra
My question is exactly how I said it in the title, how do I prove a set is a σ-algebra, also now that I am writing this how do I prove that σ(G) is a Borel σ-algebra? I would really appreciate any...
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Example from Real analysis and probability
There is the definition of sequence of sets being $\downarrow \varnothing$.$A_n:=[n,\infty)$. $A_n = [n,\infty), A_{n+1} = [n+1,\infty)$, so $A_{n+1} \supset A_n$. But $\cap_n A_n = A_n$. On the other...
View ArticleWhy do we require assignment of delta for a given epsilon to be independent...
After seeing some proofs of epsilon delta x^2 , e^x, x^3, I notice that the common proof pattern is we take $|f(x)-f(c)|$ and find some expression $h(c,\delta): \mathbb{R} \times \mathbb{R}_{\geq 0}...
View ArticleWhat's the limit of the series $\log_2(1-x)+x+x^2+x^4+x^8+\cdots$.
Find $$\lim_{x\to1^-}\log_2(1-x)+x+x^2+x^4+x^8+\cdots$$I have found $1-\dfrac{1}{\ln2}$ as a lower bound, but not further than that
View ArticlePart of proof from Real analysis and probability
There is the citation which is self-sufficient."An interval J := (c, d] isa union of countably many disjoint intervals $J_i := (c_i, d_i]$. For each finiten, J is a union of the $J_i$ for $i = 1,...,...
View Articlespectrum of weighted sum of projection operators
Let $P_i$ be a collection of projection operators on some Hilbert space $H$, so that $P_iP_j=0$, and so that $\sum_i P_i=Id$, where $Id$ is the identity operator. Assume that there exists a bounded...
View ArticleMaximal lower bound of $|a_n|$, the coefficient of $x^n$ in a $2n$-degree...
Inspired by this quesion. I'm curious about the true maximal lower bound of $|a_n|$, where $n\geq 1$ and $a_n$ is the coefficient of $x^n$ in the following symmetric-coefficient polynomial$$ P(x) =...
View ArticleIf $a_n\ge0$ and $\sum a_n$ converges then $\sum\sqrt{a_na_{n-1}}$ converges,...
Suppose the series $\sum_{n=1}^{\infty}{a_n}$ is convergent ($a_n \geq0$), Is it true that $\sum_{n=1}^{\infty}\sqrt{a_na_{n-1}}$ is convergent ?Is the converse true?My attempt:The first part I was...
View ArticleInfinite power tower compressing number line to finite interval?
I’ve been exploring an intriguing infinite power tower and noticed something surprising: it appears to compress the entire infinite positive number line into a finite interval.Specifically, the...
View ArticleStone-Weierstrass and constant functions
In Real Analysis: Modern Techniques and Their Applications by Gerald Folland, (2nd edition, page 141, Corollary 4.50), we find the following statement of the Stone-Weierstrass theorem:Let $X$ be a...
View ArticleUniformly converging sequence of functions and analyticity
Let $f_n:\mathbb R \rightarrow \mathbb R$ be a sequence of functions such thateach $f_n$ can be extended to an entire function $g_n: \mathbb C \rightarrow \mathbb C$.Let$$\Omega = \{ x + y i \in...
View ArticleIf A is a Lebesgue Measurable set, find measure of reflection
Suppose we have a set $A \subset \Bbb{R^2}$ that is Lebesgue measurable.Now define $B$ = {$(x,-y) : (x,y) \in A$). We want to find the Lebesgue Measure of BIt’s seems to me that $B$ has the same...
View ArticleProving the Unit Sphere without the North Pole is Homeomorphic to the Plane
I'm having trouble proving that the real plane and unit sphere with the north pole removed are homeomorphic.Even considering the function that maps from the sphere to the plane, I can't seem to...
View ArticleProve a continuous bijection between the space of probability distributions...
Let $\Delta([\underline p, \overline p])$ denote the set of all cumulative distribution functions (CDF) supported on the compact set $[\underline p, \overline p]$. $G_i\in \Delta([\underline p_i,...
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