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.
Prove or disprove the convergence of improper integral
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real-analysis
improper-integrals
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0Estimate $$\int_{k\pi}^{(k+1)\pi} \frac{\lvert \sin x\rvert}{x}\,dx.$$ – 2017-02-13
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0@DanielFischer Could you explain more specific, please? How could we estimate that integral? – 2017-02-13
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0Find 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]. – 2017-02-13
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0as 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 $\int_{k\pi}^{(k+1)\pi} \frac{\lvert \sin x\rvert}{(k+1)\pi}\,dx$ – 2017-02-13
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0BTW is it ok to say $X < \infty$ – 2017-02-13