Does $\int\limits_n^\infty {\frac{{\sin x}} {x}dx}$ Converge?

[bomba923]

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

 

[dextercioby]

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

 

[fourier jr]

$\forall n > 0,\;n \in \mathbb{R}$, 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]

...a plot of the intergrand Steven

Hey um, where can I download/get this program?

 

[dextercioby]

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.