# A question about the limit of a Lebesgue integration

For arbitrary $$r_n\in\mathbb{R}$$,show that $$\lim_{n\rightarrow\infty} \int_{0}^{+\infty} e^{-x}\sin^{n}(x+r_n)\, dx=0$$ By Lebesgue Dominated Convergence Theorem, since $$|e^{-x}\sin^{n}(x+r_n)| We have $$\lim_{n\rightarrow\infty} \int_{0}^{+\infty} e^{-x}\sin^{n}(x+r_n) dx= \int_{0}^{+\infty} e^{-x}\lim_{n\rightarrow\infty} \left(\sin^{n}(x+r_n)\right) dx$$ However, the limit of $$\lim_{n\rightarrow\infty} \sin^{n}(x+r_n)$$ doesn't exist.

• Hint : The limit $\lim_{n\rightarrow\infty} sin^{n}(x+r_n)$ does not exist for all $x$, but it exists for almost every $x$. Nov 5, 2020 at 15:08
• @TheSilverDoe, I am skeptical that the limit exists for almost every $x$, if $r_n$ is chosen appropriately. Thinking that the peaks of $\sin^n x$ had with $\sim n^{-1/2}$, it will be enough to consider $r_n=c\sqrt{n}$ for some small $c$. Nov 5, 2020 at 17:31

The integral is bounded above in absolute value by

$$\int_0^\infty e^{-x}|\sin^n(x+r_n)|\,dx = \sum_{k=0}^{\infty} \int_{k\pi}^{(k+1)\pi}e^{-x}|\sin^n(x+r_n)|\,dx$$ $$\tag 1 \le \sum_{k=0}^{\infty} e^{-\pi k}\int_{k\pi}^{(k+1)\pi}|\sin^n(x+r_n)|\,dx.$$

But note $$|\sin^n x|$$ is $$\pi$$-periodic. Thus

$$\int_a^{a+\pi}|\sin^n x|\,dx =\int_0^\pi|\sin^nx|\,dx$$

for any $$a.$$ Therefore

$$\int_{k\pi}^{(k+1)\pi}|\sin^n(x+r_n)|\,dx = \int_{k\pi+r_n}^{(k+1)\pi+r_n}|\sin^n(y)|\,dy = \int_0^\pi |\sin^n(y)|\,dy$$

for all $$k,n.$$ Now the last integral $$\to 0$$ as $$n\to \infty$$ by the DCT. So the limit of $$(1)$$ equals

$$\lim_{n\to \infty}\left (\int_0^\pi |\sin^n(y)|\,dy\right)\cdot \sum_{k=0}^{\infty} e^{-\pi k}=0.$$

This gives the desired result.