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Show that,$$\int_0^\pi \bigg|\dfrac{\sin nx}{x}\bigg|\mathrm{d}x \ge \dfrac{2}{\pi}\bigg(1+\dfrac12+\cdots+\dfrac{1}{n}\bigg)$$

I could not approach the problem at all. Please help.

Ayman Hourieh
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Hawk
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    The zeros of $\sin nx$, where it changes sign, are $\dfrac{k\pi}{n},; k = 0,,1,,\dotsc,,n$. Splitting the integral and looking at each part alone ought to help. – Daniel Fischer Feb 20 '14 at 14:35
  • @DanielFischer Let me try...thank you for the hint... – Hawk Feb 20 '14 at 14:35
  • @DanielFischer I got the representation as you said...but, would I need to evaluate each and every integral or I would need to get another function which looks similar to the above function but is less than or equal to the above expression to get the inequality? – Hawk Feb 20 '14 at 14:44
  • See http://math.stackexchange.com/a/390841/4583 – Ayman Hourieh Feb 20 '14 at 14:56
  • @AymanHourieh I would really appreciate it a lot if you could post a solution here without using the summation notion. I face very much difficulty to conceive the representation with summation. – Hawk Feb 20 '14 at 15:02
  • @Hawk Did you try to expand the sums yourself? Converting from one notation to another is purely mechanical. – Ayman Hourieh Feb 20 '14 at 17:06

1 Answers1

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$$ \begin{align} I &=\sum_{i=0}^{n-1} \int_{i\pi/n}^{(i+1)\pi/n} \frac{|\sin{nx}|}{|x|}\,dx \\ &=\sum_{i=0}^{n-1} \int_{0}^{\pi/n} \frac{|\sin{nx}|}{i\pi/n+ x}\,dx \\ &\gt \sum_{i=1}^{n} \int_{0}^{\pi/n} \frac{|\sin{nx}|}{i\pi/n}\,dx \\ &= \sum_{i=1}^{n} \frac{2/n}{i\pi/n} \\ &= \frac{2}{\pi}\sum_{i=1}^{n} \frac{1}{i} \end{align} $$

Anant
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viplov_jain
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