Finding Limit of an Integral: $\lim_{n\to\infty}\int_a^b f(x)\sin^3{(nx)} \:dx$

Parisina 07/31/2017. 3 answers, 492 views
integration analysis limits continuity uniform-convergence

Suppose $f:[a,b]\to\mathbb{R}$ is continuous. Determine if the following limit exists

$$\lim_{n\to\infty}\int_a^b f(x)\sin^3{(nx)} \:dx.$$

As $f(x)$ and $\sin^3{(nx)}$ are continuous, so their product is Riemann integrable. However $\lim_{n\to\infty} f(x)\sin^3{(nx)} $ does not exist, so it's not uniformly convergence and we cannot pass the limit inside the integral. It also doesn't satisfy in the conditions of Dini Theorem. I don't know how to make a valid argument for this problem, but I think by what I said the limit doesn't exist. I appreciate any help.

3 Answers


Robert Israel 07/31/2017.

Riemann-Lebesgue lemma. Note that $\sin^3(nx) = \frac{3}{4} \sin(nx) - \frac{1}{4} \sin(3nx)$.

2 comments
Parisina 07/31/2017
Thanks, I think, I can complete it now
Teepeemm 07/31/2017
That seems to be more advanced than the problem is calling for.

Sangchul Lee 07/31/2017.

A slightly different way of solving this is to use the following observation.

Proposition. If $f : [a, b] \to \mathbb{R}$ is continuous, $g : \mathbb{R} \to \mathbb{R}$ is continuous and $L$-periodic, then

$$ \lim_{n\to\infty} \int_{a}^{b} f(x)g(nx) \, dx = \left( \int_{a}^{b} f(x) \, dx \right)\left( \frac{1}{L}\int_{0}^{L} g(x) \, dx \right). $$

  1. Assuming this statement, the answer follows immediately since $x \mapsto \sin^3 x$ is $2\pi$-periodic and

    $$ \int_{0}^{2\pi} \sin^3 x \, dx = 0. $$

  2. The intuition is very clear: If $n$ is very large, then on subinterval $[c,c+\frac{L}{n}] \subset [a, b]$ we have

    $$ \int_{c}^{c+\frac{L}{n}} f(x)g(nx) \, dx \approx f(c) \int_{c}^{c+\frac{L}{n}} g(nx) \, dx = f(c) \cdot \frac{1}{n} \int_{0}^{L} g(x) \, dx. $$

    So ignoring details, we would have

    $$ \int_{a}^{b} f(x)g(nx) \, dx \approx \left( \sum_{k=1}^{\lfloor n(b-a)/L \rfloor} f\left(a+\frac{kL}{n}\right) \frac{1}{n} \right)\left( \int_{0}^{L} g(x) \, dx \right) $$

    and taking limit as $n\to\infty$, the right-hand side converges to the desired value. Filling in the details is quite routine.

  3. The assumption on continuity is just a technical setting for simple proof, and you can relax them to certain degrees by paying more effort.


Michael Hartley 07/31/2017.

You can't conclude $$\lim_{n\rightarrow\infty} \int_a^b g(x,n)dx$$ doesn't exist just because $$\lim_{n\rightarrow\infty} g(x,n)$$ doesn't. For example, $$\lim_{n\rightarrow\infty} \sin(nx)$$ doesn't exist, but $$\lim_{n\rightarrow\infty} \int_0^\pi \sin(nx) dx = 0,$$ since the integral is zero for all $n$.

I'm afraid my usefulness runs out at this point, though I think the limit exists: you should, if nothing else, be able to find some epsilon-delta argument expressing the integral as the sum of a bunch of integrals on intervals of length $\frac{2\pi}{n}$. This may be a very bad way to tackle the problem.


HighResolutionMusic.com - Download Hi-Res Songs

1 (G)I-DLE

POP/STARS flac

(G)I-DLE. 2018. Writer: Riot Music Team;Harloe.
2 The Chainsmokers

Beach House flac

The Chainsmokers. 2018. Writer: Andrew Taggart.
3 Ariana Grande

​Thank U, Next flac

Ariana Grande. 2018. Writer: Crazy Mike;Scootie;Victoria Monét;Tayla Parx;TBHits;Ariana Grande.
4 Nicki Minaj

No Candle No Light flac

Nicki Minaj. 2018. Writer: Denisia “Blu June” Andrews;Kathryn Ostenberg;Brittany "Chi" Coney;Brian Lee;TJ Routon;Tushar Apte;ZAYN;Nicki Minaj.
5 Clean Bandit

Baby flac

Clean Bandit. 2018. Writer: Jack Patterson;Kamille;Jason Evigan;Matthew Knott;Marina;Luis Fonsi.
6 Imagine Dragons

Bad Liar flac

Imagine Dragons. 2018. Writer: Jorgen Odegard;Daniel Platzman;Ben McKee;Wayne Sermon;Aja Volkman;Dan Reynolds.
7 Halsey

Without Me flac

Halsey. 2018. Writer: Halsey;Delacey;Louis Bell;Amy Allen;Justin Timberlake;Timbaland;Scott Storch.
8 BTS

Waste It On Me flac

BTS. 2018. Writer: Steve Aoki;Jeff Halavacs;Ryan Ogren;Michael Gazzo;Nate Cyphert;Sean Foreman;RM.
9 BlackPink

Kiss And Make Up flac

BlackPink. 2018. Writer: Soke;Kny Factory;Billboard;Chelcee Grimes;Teddy Park;Marc Vincent;Dua Lipa.
10 Fitz And The Tantrums

HandClap flac

Fitz And The Tantrums. 2017. Writer: Fitz And The Tantrums;Eric Frederic;Sam Hollander.
11 Backstreet Boys

Chances flac

Backstreet Boys. 2018.
12 Kelly Clarkson

Never Enough flac

Kelly Clarkson. 2018. Writer: Benj Pasek;Justin Paul.
13 Diplo

Close To Me flac

Diplo. 2018. Writer: Ellie Goulding;Savan Kotecha;Peter Svensson;Ilya;Swae Lee;Diplo.
14 Anne-Marie

Rewrite The Stars flac

Anne-Marie. 2018. Writer: Benj Pasek;Justin Paul.
15 Little Mix

Woman Like Me flac

Little Mix. 2018. Writer: Nicki Minaj;Steve Mac;Ed Sheeran;Jess Glynne.
16 Imagine Dragons

Machine flac

Imagine Dragons. 2018. Writer: Wayne Sermon;Daniel Platzman;Dan Reynolds;Ben McKee;Alex Da Kid.
17 Little Mix

The Cure flac

Little Mix. 2018.
18 Bradley Cooper

Always Remember Us This Way flac

Bradley Cooper. 2018. Writer: Lady Gaga;Dave Cobb.
19 Rita Ora

Velvet Rope flac

Rita Ora. 2018.
20 Lady Gaga

I'll Never Love Again flac

Lady Gaga. 2018. Writer: Benjamin Rice;Lady Gaga.

Related questions

Hot questions

Language

Popular Tags