$\lim_{n\rightarrow\infty} \int_n^{n+1}\frac{\sin x}x \, dx$

Question

I am asked to calculate $\displaystyle \lim_{n\rightarrow\infty} \int_n^{n+1}\frac{\sin x} x \, dx$.

Before letting the $\lim$ to confuse me I used integration by parts, but it didn't get me far.

Any hint?

And with that, look at the limit of the absolute value of your integral — Alan, Oct 03 '15 at 02:51
So... I get the absolute value of the integral is between 0 and $\ln(\frac{n+1}n)$ which goes to 0 when n goes to $\infty$, so by the Sandwich theorem we get the limit of the integral is zero? — Whyka, Oct 03 '15 at 03:08
You don't even need to go that far - just note that $|\sin x/x|<1/n$ on $(1/n,1/(n+1))$. — Thomas Andrews, Oct 03 '15 at 03:14

score 3 · Answer 1 · answered Oct 03 '15 at 04:39

3

There are a couple of answers already but I'll provide the simple one that was alluded to in comments. For all $x\in[n,n+1]$, $|\frac{\sin x}x|\le\frac1n$ and so $|\int_n^{n+1}\frac{\sin x}x\,\mathrm dx|\le\frac1n\to0$.

answered Oct 03 '15 at 04:39

Jason

15,438

score 0 · Answer 2 · edited Apr 13 '17 at 12:19

0

If you permit special functions, $\int \frac{\sin x}{x}dx = \text{Si}(x)$. Further, there is the well known identity $\displaystyle\lim_{x \to \infty} \text{Si}(x) = \frac{\pi}{2}$. From this, we get $$\lim_{n\rightarrow\infty} \int_n^{n+1}\frac{\sin x} x \, dx = \lim_{n \to \infty} [\text{Si}(n+1)-\text{Si}(n)]$$ We can see that the growth of $n$ in the first sine integral overtakes the addition by 1, so we get $$= \lim_{n \to \infty} [\text{Si}(n)-\text{Si}(n)] = \color{red}{0}$$

For proofs of the above identity, see here. While the proofs are not that hard to grasp, they are often long in length and are great in number; as such I leave the proof-checking to you.

edited Apr 13 '17 at 12:19

Community

1

answered Oct 03 '15 at 04:28

Brevan Ellefsen

14,793

The "well known identity" you quote implies the result immediately, but requires a fair bit more to prove. For this reason I don't think using the function $\text{Si}(x)$ is appropriate here. – Jason Oct 03 '15 at 04:41
@Jason I have encountered the identity often, and I thus consider it fairly "well known", especially compared to other identities of trig integrals. Nevertheless, you can probably show this integral without needed show that Si$(\infty) = \frac{\pi}{2}$ as long as you can show that the integral converges to a finite value. I'll agree your solution is probably better as an elementary answer however, and is simpler to show a full proof of. – Brevan Ellefsen Oct 03 '15 at 04:48

score 0 · Answer 3 · answered Oct 03 '15 at 04:47

Let $y = x-n \Rightarrow dx = dy \Rightarrow \displaystyle \int_{n}^{n+1} \dfrac{\sin x}{x} dx = \displaystyle \int_{0}^1 \dfrac{\sin(y+n)}{y+n}dy$. We treat this as Lebesgue integral, and the function $\dfrac{\sin(y+n)}{y+n}$ is dominated by $1$ which is integrable over $[0,1]$ because $\sin(y+n) \leq 1 \leq y+n, y \geq 0 \Rightarrow \displaystyle \lim_{n\to \infty} \int_{n}^{n+1}\dfrac{\sin x}{x}dx = \displaystyle \lim_{n\to \infty} \int_{0}^1 \dfrac{\sin(y+n)}{y+n}dy=\displaystyle \int_{0}^1 \lim_{n\to \infty} \dfrac{\sin(y+n)}{y+n}dy=\displaystyle \int_{0}^1 0 dy = 0$

$\lim_{n\rightarrow\infty} \int_n^{n+1}\frac{\sin x}x \, dx$

3 Answers3