# Very quick question (1 Viewer)

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#### bomba923

$\forall n > 0,\;n \in \mathbb{R}$, does
$$\int\limits_n^\infty {\frac{{\sin x}} {x}dx}$$
converge? If so, to what?

#### dextercioby

Homework Helper
It's a function of "n".It converges for every possible "n".

$$\int_{n}^{\infty} \mbox{sinc} \ x \ dx =\frac{\pi}{2}-\mbox{Si}\ (x)$$.

Daniel.

#### steven187

hello there

well I have attached a plot of the intergrand, as you would see the function converges to 0 and should be integrable

Steven

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#### fourier jr

bomba923 said:
$\forall n > 0,\;n \in \mathbb{R}$, does
$$\int\limits_n^\infty {\frac{{\sin x}} {x}dx}$$
converge? If so, to what?
you can prove that it converges by multiplying the integrand by 1 (in this case pick x/x or x^2/x^2 or something) & use integration by parts, & then the comparison test. i don't think that helps find what it converges to but judging by the image it looks like it goes to 0.

#### bomba923

steven187 said:
...a plot of the intergrand Steven

#### dextercioby

Homework Helper
What program...?

"sinc" is graphed in a ton of books.It's even in physics books when discussing Fraunhofer diffraction from single or multiple slits.

Daniel.

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