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.

Robert Israel 07/31/2017.

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

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

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

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

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.

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.

- Limit at infinity of a uniformly continuous integrable function
- Let $f:[a,b]\to\mathbb R$. Evaluate $\lim_{n\to\infty}\int_a^bf(x)\sin(nx)\,dx$
- Approximation Lemma for Riemann-integrable functions
- When does interchangibility of limit and Riemann integral imply uniform convergence?
- If $f_n \to f$ , $f , f_n \in \mathcal R[a,b] $ , then is it true that $\lim_{n \to \infty} \int_a^bf_n=\int_a^b f$ ?
- $\int_{a}^{b}f'=f(b)-f(a)=>\int_{a}^{b}\lim_{n->\infty}Diff_{\frac{1}{n}}f=\lim_{n->\infty}\int_{a}^{b}Diff_{\frac{1}{n}}f$
- Uniform convergence of integral and derivatives of $f_n(x)=\frac{\sin nx}{n}$
- Establish if $g_n (\alpha)=\int_a^b \ \alpha(x) \ \sin (nx) \ \cos(nx) $ converges uniformly
- Find $\lim_{n \to \infty} \int_{0}^{1} f\left(\frac{nx}{1+nx}\right)\,dx$
- Existence of the limit of a certain integral

- How to avoid wishing my coworker on her birthday?
- How did Google know I looked something up?
- How to approach my spouse about their weight gain?
- A standalone SVG editor?
- Term for being unable to see glaring errors after working for some time on a task?
- Eldritch Knight and Arcane focus
- Is it bad practice to write code that relies on compiler optimizations?
- What is this strange copy constructor error complaining about?
- Find a polynomial in two or three queries
- How skilled does a wizard need to be to successfully perform the three Unforgivable Curses?
- How can a desert have high humidity?
- In the US why is nationalism equated with racism?
- Algebraic Geometry: What am I doing wrong?
- How to alternatively display different names for even and odd entries in enumerate
- Plotting pressure as a function of altitude
- Did Muslim states hire Western European knights as mercenaries before the Crusades?
- Should one avoid using the phrase "note that" in math writings?
- Why were Captain America and Black Widow in such a hurry to close the portal?
- Mental condition that spreads through social contact
- What is the Alt-Left?
- Are attacks on characters stuck in a web at advantage?
- What English words contain the vowel letters, a, e, i, o, and u, with fewest consonant letters?
- What If I don't want it to layer
- Drop voltage a bit