On inequalities and Integrals, again

My first post on Integrals and Inequalities wasn’t very popular, but I tried to make second one. I hope you’ll find it moe interesting than previous one. I’m looking forward to receiving any critics and feedback.

The second post may be found here.

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On a beautiful integral

Yesterday I tried to solve a very beautiful problem proposed to me by Gabriel Stuart Romon. The problem asked to find the following limit

\displaystyle\lim_{n\to\infty}\int^b_a f(x)|\sin(nx)|\,dx, \forall f(x)\subset C^0 [a,b].

The detailed solution to this problem, written by Gabriel may be found here. However, here I want to discuss another problem, which I found on math.stackexchange. The key idea of it helped me to elaborate the solution to Gabriel’s problem. I hope you’ll find it beautiful and interesting.

Problem. If g(x):\mathbb{R}\rightarrow\mathbb{R} is a continuous periodic function with period 1 and f(x):[0,1]\rightarrow\mathbb{R} is a continuous function, prove that

\displaystyle\lim_{n\to\infty}\int^1_0 f(x)g(nx)\,dx=\left(\int^1_0 f(x)\,dx\right)\left(\int^1_0 g(x)\,dx\right).

Solution. First of all we’ll break our single integral into the sum of integrals over the shorter intervals.

\displaystyle\int^1_0 f(x)g(nx)\,dx=\sum^{n-1}_{i=0}\int^{\frac{i+1}{n}}_{\frac{i}{n}} f(x)g(nx)\,dx=\frac{1}{n}\left(\sum^{n-1}_{i=0}\int^{i+1}_i f\left(\frac{u}{n}\right)g(u)\,du\right)

By applying mean-value theorem we’ll transform our sum into something similar to a Riemann sum and we’ll use the fact that g(x) is periodic in our favor.

\displaystyle\frac{1}{n}\left(\sum^{n-1}_{i=0}\int^{i+1}_{i} f\left(\frac{u}{n}\right)g(u)\,du\right)=\frac{1}{n}\sum^{n-1}_{i=0}\left(f(\xi_i)\int^{i+1}_{i} g(u)\,du\right)=\frac{1}{n}\left(\sum^{n-1}_{i=0}f(\xi_i)\right)\left(\int^1_0 g(u)\,du\right)

Because every \xi_i\in\left[\dfrac{i}{n},\dfrac{i+1}{n}\right], we have that the limit of our sum is \displaystyle \int^1_0 f(x)\, dx, so

\displaystyle\lim_{n\to\infty}\int^1_0 f(x)g(nx)\,dx=\lim_{n\to\infty}\left(\frac{1}{n}\sum^{n-1}_{i=0}f(\xi_i)\right)\left(\int^1_0 g(x)\,dx\right)=\left(\int^1_0 f(x)\,dx\right)\left(\int^1_0 g(x)\,dx\right).

P.S. For those of you, who wish to view and discuss this note on Brilliant, here is the link. A slight clarification on this post appeared. To resolve any misunderstanding please consult this comment.

Are you fond of Geometry?

Hey guys, there are two more days to complete the Geometry round of Proofathon! For those of you, who haven’t heard about the contest, here is the Facebook page and Contest page. I would like to remind that every round is a standalone competition, so you’ll be awarded for solving the current round problems, even if you haven’t participated in previous rounds. At Proofathon! problems are created and edited by organizers for every round, so we don’t simply repost the work of other’s. I’m sure you’ll find both easy or better to say motivating and challenging questions. Also, because we reached 500 likes on Facebook, for this round prize winners will be awarded with awesome Proofathon! T-shirts. Give it a try, I’m sure you’ll like it.

As the bonus I invite you to solve this easy geometry problem.

I’m also curious have you ever participated in online contests before? Please use the poll provided below, but don’t feel limited to it. You can provide additional details in comments

Inequalities and Integrals

I just satrted a new series of short articles on Brilliant.org. This time the topic will revolve around solving inequalities involving integrals. The topic was highly inspired by the book “Inqualities, Methods of proving” by N.M. Sedrakyan and A.M. Avoyan and problems encountred at Moldova and Romania National Mathematics Olympiads. I hope you’ll enjoy the following posts.

First post may be found here.

Limits and Riemann Sums

Some concepts in high-school mathematics are usually overlooked and studied without great implication into the process. However, mathematics has a very delicate inner structure, so by skiping some topics we may skip some beatiful problems and solutions. In the following series of posts I tried to demistify Riemann Sums and their relationship to limits. This topic is often treated as unpractical during the school Calculus course, but it has very neat applications in some problems. I learned the discussed technique during preparation for National Mathematics Olympiad. I hope you’ll find this material interesting and helpful.

Introduction, Part 1, Part 2, Part 3.