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Prove or disprove that $$\int_0^\infty \left|\frac{\sin{x}}{x}\right|dx < \infty.$$

I tried by Wolframalpha and this integral seems to satisfy the Cauchy property, but I do not know how to prove it.

Thank you very much.

Tien Kha Pham
  • 3,000
  • Estimate $$\int_{k\pi}^{(k+1)\pi} \frac{\lvert \sin x\rvert}{x},dx.$$ – Daniel Fischer Feb 13 '17 at 14:23
  • @DanielFischer Could you explain more specific, please? How could we estimate that integral? – Tien Kha Pham Feb 13 '17 at 14:30
  • Find a lower bound so that the series of lower bounds diverges [or an upper bound so that the series of upper bounds converges, but in this case, we have divergence]. – Daniel Fischer Feb 13 '17 at 14:34
  • as well as an estimate, you could try and show that the integral below is a lower bound, then show that it is possible to always collect a finite number of these in order to add the same value each time to the total integral - if you are familar with the summation $\Sigma^\infty \frac{1}{n}$
    • then consider this integral as lower bound $\int_{k\pi}^{(k+1)\pi} \frac{\lvert \sin x\rvert}{(k+1)\pi},dx$
     – Cato Feb 13 '17 at 14:38
  • BTW is it ok to say $X < \infty$ – Cato Feb 13 '17 at 14:44

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