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

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Discussion Overview

The discussion centers on the convergence of the integral $\int\limits_n^\infty {\frac{{\sin x}} {x}dx}$ for all real numbers \( n > 0 \). Participants explore whether the integral converges and, if so, to what value, including considerations of the integrand's behavior and potential methods of proof.

Discussion Character

  • Exploratory
  • Mathematical reasoning
  • Debate/contested

Main Points Raised

  • One participant asserts that the integral converges for every possible \( n \) and provides a formula involving the sine integral function.
  • Another participant presents a plot of the integrand, suggesting that it converges to 0 and is integrable.
  • A different viewpoint mentions using integration by parts and the comparison test to prove convergence, but notes that this does not determine the value to which it converges.
  • There is a request for information about the program used to generate the plot, indicating interest in the tools for visualizing the function.
  • One participant references the "sinc" function's appearance in various physics texts, implying its relevance in broader contexts.

Areas of Agreement / Disagreement

Participants express differing views on the convergence of the integral and its value. While some suggest it converges to 0, others focus on the methods of proving convergence without reaching a consensus on the specific value.

Contextual Notes

Some assumptions about the behavior of the integrand and the conditions for convergence are not fully explored, and the discussion includes references to mathematical techniques that may not be universally agreed upon.

Who May Find This Useful

Readers interested in mathematical analysis, particularly in the study of improper integrals and convergence behavior, may find this discussion relevant.

bomba923
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[itex]\forall n > 0,\;n \in \mathbb{R}[/itex], does
[tex]\int\limits_n^\infty {\frac{{\sin x}} {x}dx}[/tex]
converge? If so, to what?
 
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It's a function of "n".It converges for every possible "n".

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

Daniel.
 
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
 

Attachments

bomba923 said:
[itex]\forall n > 0,\;n \in \mathbb{R}[/itex], does
[tex]\int\limits_n^\infty {\frac{{\sin x}} {x}dx}[/tex]
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.
 
steven187 said:
...a plot of the intergrand Steven

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