Are Darboux and Riemann's integral equivalent when using standard partitions?
While studying Analysis, I used many different textbooks: "Understanding Analysis" (Stephen Abbott), "Elements of Real Analysis" (Denlinger), "Calculus" (Michael Spivak).Initially when dealing with...
View ArticleMeasurability of the Projection
I was looking for a projection measurability theorem with the least restrictions and as self-contained as possible, without the need to use capacity theory, for example.I thought I found what I was...
View ArticleLet $u_1=\frac1N$, where $N\in\mathbb{Z^+}$, and $u_n=u_{n-1}+(u_{n-1})^2$....
I was playing with recursively defined sequences, and stumbled upon a curious apparent result.Let $u_1=\frac1N$, where $N\in\mathbb{Z^+}$, and $u_n=u_{n-1}+(u_{n-1})^2$.Let $f(N)=$ number of terms in...
View ArticleMCQ: Limit inferior/superior of a sequence question
Q: Let $\left\{x_n\right\}_{n \geq 0}$ be a sequence in $\mathbb{R}$ such that $x_n \geq(-2)^n$ for every $n \in \mathbb{N}$. Which one is true?(A) $\liminf \limits_{n \rightarrow \infty}...
View ArticleSimple chain rule for a continuous and bounded variation function [closed]
Is the property below correct?$$\forall t \geq 0, \: F(t)^2 = F(0)^2 + 2 \int_0^t F(s) d F(s), $$where $F:\mathbb{R} \to \mathbb{R} $ is continuous and has bounded variations? Is it shown somewhere?
View ArticleDoes $\limsup_{n\to\infty}\sqrt[n]{a_n}=1$ imply...
I am working on an exercise from calculus:Let $\{a_n\}_{n\ge 1}$ be a positive sequence such that $\limsup_{n\to\infty}\sqrt[n]{a_n}=1$.Show that $\limsup_{n\to\infty}\sqrt[n]{a_1+\cdots+a_n}=1$.Here...
View ArticleOn the solution of these equations
Do not forget to see the Good News at the end of the problem.This problem is linked to the previous one, up to a changes of coordinates. However, that question is actually about only the first three...
View ArticleUpper bound on absolute value of Confluent Hypergeometric Function with real...
A hypergeometic E-function is$${}_pF_q\left(\left.\begin{array}{ll}a_{1}, \ldots, & a_{p} \\b_{1}, \ldots, & b_{q}\end{array} \right\rvert\, \lambda z^{q-p}\right)=\sum_{n=0}^{\infty}...
View ArticleIs the set of rationals an interval? [closed]
$I(\neq\phi)\subset\mathbb{R}$ is said to be an interval iff $\forall a<b \in I, \exists x\in I: a<x<b$ This was the definition taught to us in college. And by this definition the set of...
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Prove function with harmonic number continuity
I am looking for a function $f(n,k)$ in which for $0\leq k\leq1$ the function image contains all the values of the interval $[1,nH_n)$, where $H_n = \sum_{k=1}^n \frac{1}{k}$ is the $n^{th}$ harmonic...
View ArticleDo you need to always specify a measure when integrating?
I've seen people name the "Riemann measure". I did some research, and they were likely referring to the Peano-Jordan measure, but I'm not sure about that. So, in order to formally define the Riemann...
View ArticleProve or disprove: $ \lim _{n \to \infty} \lim_{k \to \infty} a_{k, n} =...
Let $\{a_{k,n}\}_{k\geq 1, n \geq 1}$ be a sequence of non-negative real numbers where $a_{k,n} \leq M $ for all $k \geq 1$ and $n \geq 1$. Assume that $a_{k,n}$ is increasing in $k$ and decreasing in...
View ArticleDivergence of...
I was looking for a function $f$ which verifies (I do not know if it exists) :$f:\mathbb{R}\rightarrow\mathbb{R}$$\forall x\in\mathbb{R},\;f(x)\geq...
View ArticleShow that $\bigcap_{n\in\mathbb{N}}=(0,1) $ if {$ {A_n}$}$_{n\in\mathbb{N}}...
I'm using the property of $A=X\cap Y$ for n sets i.e.$A=\bigcap_{ n\in\mathbb{N}}A_n \iff$(($\forall n\in\mathbb{N}, A\subseteq A_n$) and ($B \subseteq A_n ,\forall n \in \mathbb{N} \implies B...
View ArticleHow do we rigorously eliminate $r^n$ and $\log r$ terms in a Fourier series...
In my PDE module, the general solution to Laplace's equation $\nabla^2 T=0$ in the plane (in polar coordinates) was shown to be $$T(r,\theta)=A_0+B_0\log...
View ArticleEvery positive rational number $x$ can be expressed in the form $\sum_{k=1}^n...
I have this theorem which I can't prove.Please help."Show that every positive rational number $x$ can be expressed in the form $\sum_{k=1}^n \frac{a_k}{k!}$ in one and only one way where each $a_k$ is...
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An infinite interval can not be covered by a sequence of intervals of finite...
This is about the construction of the Lebesgue measure. This is from a book I am reading, I have highlighted the sentence in read:Since we have not yet created the Lebegue measure when using this...
View ArticleShowing a lower bound for the square root of a function
Suppose that we have a function $f$ that is of exponential type, and that the function $g(x) = |f(x-i)|^2$ is such that it satisfies $$\left(\frac{1}{|B|} \int_B g(x)\right)\left(\frac{1}{|B|} \int_B...
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Example of a pointwise convergent sequence not convergent in $L^1$.
I'm trying to solve the following problem and I have this solution. So the solution shows that $f_n-f$ is not even in $L^1$ space, when we need $\|f_n-f\|_1\to 0$ to show that the sequence converges to...
View ArticleHow can I prove the existence of multiplicative inverses for the complex...
If the complex number system is defined to be the set $\mathbb C = \mathbb R \times \mathbb R$ (called the complex numbers) together with the two functions from $\mathbb C \times \mathbb C $ into...
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