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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.,...

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Question 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...

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How 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...

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Finding 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...

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Pedagogical 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...

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Exponential 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...

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A 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...

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Approximation 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...

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Approximating 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...

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A 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...

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Finding 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...

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Proof: 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...

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Doubt 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...

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How 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...

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A 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...

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Separating 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...

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Clarification 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...

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How 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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