whats the n-th derivative of $\frac{e^x}{x}$ [duplicate]
I need this for finding a general solution to a family of integrals, plus it just seems a cool problem.if any of my efforts so far have been in the right direction it should come out as a partial sum...
View ArticleRudin's Real and Complex analysis - Exercise 5 Chapter 1
Point (a) of Exercise 5 Chapter 1 in Rudin's "Real and complex analysis" requires to prove the following:If $f,g:X \to [-\infty, +\infty]$ are measurable then $\{x:f(x)<g(x)\}$ and $\{x:f(x)=g(x)\}$...
View ArticleIs there a path-connected, "anti-convex" subset of $\mathbb R^2$ containing...
For a vector space $V$ over $\mathbb R$, I say a subset $S$ of $V$ is "anti-convex" if $\forall a,b\in S (a\ne b)$, $\exists t\in ]0,1[$, $b+t(a-b)\not\in S$. For example, all hollow circles...
View ArticleProof regarding absolutely convergent series
Assume that $\sum_{n=1}^{\infty} c_n$ is absolutely convergent. Let $\phi$ : $\mathbb{N}$ → $\mathbb{N}$ be a bijection. Set $d_n = c_{\phi(n)}$ for $n\in \mathbb{N}.$ Show that $\sum_{n=1}^{\infty}...
View ArticleDecay of Fourier coefficients of a function on the disk
What can be said about the decay of the Fourier coefficients of a smooth function on the unit disk $B_1(0)$? More precisely, I'm in the following setting:Let $v \in \mathcal{C}^\infty(B_1(0),...
View ArticleZeros of two equations
Consider the equations$$ 1+\frac{1}{z^k}=0 \quad\mbox{and}\quad 1+\frac{1}{z^k}+\frac{1}{(z+1)^k}=0,$$where $k$ is a positive integer $\ge 4$. I would like to show for instance that the number of zeros...
View ArticlePartial derivatives "split" over $\mathbb R$
Kaplansky's book Fields and Rings, page 30, implicitly contains the following question.For which fields $K$ does the following statement hold true for every polynomial $f \in K[X]$: If $g$ splits over...
View ArticleLimit of sequence using integral test and L'Hopital's rule
$$\lim_{n \to \infty}\frac{1}{\log n}\sum_{k=1}^{n^2}1/k$$My attempt:$$\lim_{n \to \infty}\sum_{k=1}^{n^2}1/k=\lim_{n \to\infty} \int_1^{n^2}1/xdx$$Then this converges to infinity as it is harmonic...
View ArticleLim supremum of sin
$$\limsup_{n\to\infty} \sin(√3n\pi)$$I know that sin function is bounded between -1 and 1. But does is straight away imply that the limsup is 1 or is there any different way to solve this.Also if...
View ArticleProving cluster points for a sequence
I have the sequence $a_n = ⌊cos(\sqrt{n})⌋, n \in ℕ_0 $, and I have to determine the amount of cluster points. Of course, cos(x) oscillates between -1 and 1, as x varies, and therefore...
View ArticleInterchanging integral and sum
I have the following dilemma - I wish to prove rigourously the following:$$\int_{0}^{1} \int_{0}^{1} \frac{1}{1-xy} dxdy = \sum_{n\geq1} \frac{1}{n^2} $$Here the obvious strategy is to use the power...
View ArticleDoes $x\in f(A)$ implies $f^{-1}(x)\in A$ and why it does or does not?
I encountered this problem while proving$$A\subset B \implies (f(A)\subset f(B))\ne (A\subset B)$$on Zorich Mathematical Analysis. While A and B are subsets of $X$ and $f:X\to Y$
View ArticleProving Second Mean Value Theorem for Integrals
There seem to be a number of theorems that call themselves "the Second Mean Value Theorem for Integrals," but the statement my textbook has is the following:Let $f$ and $g$ be continuous on $[a,b]$ and...
View ArticleRigorous proof that sequence converges
I have been given the sequence $x_n=\frac{\log(3n+2)}{\log(n^2+2)}$ which has the limit $\frac{1}{2}$, and I have been tasked with proving that.This is the first time I have ever proven a sequence, and...
View ArticleApproximation of a function f on $\mathbb{T}$
First of all, I apologize if this has been asked before, but I could not find it.Let $f : \mathbb{T} \to \mathbb{C}$ be an $n$-times differentiable function such $f^{(n)}$ is bounded. I must specify...
View ArticleSufficient conditions on Fourier coefficients that imply smoothness
Given a square-integrable function $f$, let $\hat{f}(n)$ denote its Fourier coefficients. It is well-known that differentiability of $f$ implies that the $\hat{f}(n)$ satisfy certain bounds, i.e. they...
View ArticleNull function and Lebesgue measure on $\mathbb{R}^N$
We know that some measurable function $f:(0,T)\times\Omega\to\mathbb{R}$, where $\Omega\subset\mathbb{R}^N$ has the following property:For almost all $t\in (0,T)$ (Lebesgue measure in $\mathbb{R}$) we...
View Article$\displaystyle\lim_{x\to\infty}f(x)=0$ and $f'(x)\geq0$, then...
Is it correct to say that if a function $f$ of class $C^1$$\displaystyle\lim_{x\to\infty}f(x)=0$ and $f'(x)\geq0$,then $\displaystyle\lim_{x\to\infty}f'(x)=0$?I attempted to demonstrate this problem...
View ArticleWhy do we consider Borel sets instead of (Lebesgue) measurable sets?
Dumb/Challenging conventional wisdom question possibly related to my previous question.Why do we sometimes consider a measure space $(S, \Sigma, \mu) = (\mathbb{R}, \mathscr{B}(\mathbb{R}), \lambda)$...
View ArticleA step in the proof of Fubini theorem (Theorem 2.36, Folland)
This is a first case of the proof of the Fubini-Tonelli theorem, given in Folland's Real Analysis. I'm confused with the line underlined in blue at the end (namely, 'the preceding argument applies to'...
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