For $1\le p
I have some doubts about the proof of this theorem. From time to time I will put my justification.For $1\le p < +\infty$, $L^p$ is a Banach spaceLet $(f_n)_n$ be a Cauchy sequence in $L^p$, i.e.,...
View ArticleQuestion on the Change of Base-Point Theorem for Power Series
I am self-studying “An Introduction to Multivariable Mathematics” by Leon Simon. While studying the section on power and Taylor series, I have encountered a question on the “consequence” of the...
View ArticleHow to show this function is decreasing?
The function is $$g(y) = \frac{\int_0^\pi e^{y\cos\theta} \cos\theta \sin^{n-1}\theta d\theta}{y\int_0^\pi e^{y\cos\theta} \sin^{n-1}\theta d\theta} $$ Its defined on positive reals. I want to show...
View ArticleFinding a limit of a recurrently given sequence
I am struggling with one example of limit required for recurrently given sequence. There are 3 examples in total, I will also post the ones I did.$ x_1>0$ and recursively given sequence:...
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A question of limits/series/sequences
Can anyone help me prove the following theorem:$\lim_{n\to\infty} \frac{1^p+2^p+...+n^p}{n^{p+1}}=\frac{1}{p+1}$note: p is any real number.About my stating p is any real number, it was correctly...
View ArticlePedagogical Approach to Limits
Working through Rudin's PMA, I found it very strange that following his terse yet rigorous treatise on metric spaces in Chapter 2, the author decided to introduce sequential limits in the typical...
View ArticleExponential series with fractional exponents $\sum_{n=0}^{\infty}...
I´m trying to prove that $\sum_{n=0}^{\infty} \frac{x^{\delta n}}{\Gamma(\delta n+1)}$ converges on $\mathbb{R}_{\geq 0}$. For $\delta=1$ we would simpy have $\exp(x)$ but I´m mostly curious about the...
View ArticleA difficult example to show that limits and integrals can't always be switched
I wanted an interesting one, so I started with$\lim\limits_{N\to\infty}\int\limits_{-\infty}^{\infty} \operatorname{sech}\left(x-\sum\limits_{n=1}^{N}\frac{1}{x+n^{2}}\right)dx$The idea was that, by...
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Make a non-smooth function smooth
I am dealing with a piecewise affine function $f$ defined as follows: $f(x)=0$ if $x<1$, $f(x)=1-x$ if $x\in [1,2]$ and $f(x)=-1$ otherwise.I want to make it smooth. I looked at sigmoid functions of...
View ArticleApproximation of Absolute Error of Babylonian Method as a function of n
The original problem is from Zorich Mathematical Analysis I 3.1 Exercises:a) Show that if $a>0$, the sequence $x_{n+1}=\frac{1}{2}(x_n+\frac{a}{x_n})$ converges to the squareroot of $a$ for any...
View ArticleApproximating integral of product of gaussian and cosines
I am trying to evaluate the following integral\begin{equation}I=\int_{-\infty}^{\infty}\exp\left(-\frac{x^2}{b}\right)\prod_i^N\cos{\left(a_ix\right)}dx,\end{equation}where $b>0$ and $a_i$ are...
View ArticleA simpler proof that $\sqrt{2}$ is not a rational number? [duplicate]
Assuming that $\sqrt{2}=\frac{p}{q}$ where $p,q\in\mathbb{N}$.That implies that $p^2=2q^2$, and since $p$ is a natural number then the right hand side $2q^2$ must be a square of natural number and...
View ArticleFinding an exponential equation solution (general trinomial problem)
I have an equation of the form $$ax^{1-p}+bx^{-p}=c$$where $x,a,b,c,p\in \mathbb{R}_+$ and $p\ge 1$. $a,b,c,p$ are given constants. I'm looking for clues for solving it. Does it even permit any closed...
View ArticleProof: If $\mu$ is $\sigma$-finite and $\mathscr{A}$ is countably generated,...
BackgroundI have some trouble understanding a step of the proof of the following proposition:Proposition$\quad$Let $X$ be a measure space, and let $p$ satisfy $1\leq p<+\infty$. If $\mu$ is...
View ArticleDoubt regarding limits tending to infinity [closed]
In my previous question quite related to this I got into an argument with another user regarding this, so just wanted to clarify whether I am incorrect in my reasoning somewhere.We have to evaluate...
View ArticleHow to solve the integral $\int_0^\infty\frac{e^{-x} \sin x}{(e^{3 x} + 1)...
$$\mbox{How to solve the following integral ?}:\quad\int_{0}^{\infty}\frac{{\rm e}^{-x}\sin\left(x\right)}{\left({\rm e}^{3 x} + 1\right)x^{3/10}}{\rm d}x$$I think it cannot be solved using elementary...
View ArticleA function is convex if and only if its gradient is monotone.
Let a convex $ U \subset_{op} \mathbb{R^n} , n \geq 2$, with the usual inner product. A function $F: U \rightarrow \mathbb{R^n} $ is monotone if $ \langle F(x) - F(y), x-y \rangle \geq 0, \forall x,y...
View ArticleSeparating domains by an algebraic variety
Suppose $\Omega_1$ and $\Omega_2$ are two disjoint domains in $\mathbb{R}^{2n}$, $n \in \mathbb{N}$. Can there be conditions on $\Omega_1$ and $\Omega_2$ so that these two domains can be separated by...
View ArticleClarification of "second fundamental theorem of calculus"
A corollary of the fundamental theorem of calculus states:If $f$ is continuous on $[a, b]$ and $f=g'$ for some $g$, then$$\int_a^b f = g(b) - g(a)$$A "second fundamental theorem of calculus" replaces...
View ArticleHow to prove measure theoretically that $\int_U u_{x_i} dx = \int_{\delta U}...
I've been studying PDEs and one important technique is integration by parts, due to which, given functions $u, v \in C^1(\overline{U})$, we have$$\int_U u_{x_i} v dx = -\int_U uv_{x_i} dx +...
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