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#### Alexmahone

##### Active member

- Jan 26, 2012

- 268

- Thread starter Alexmahone
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- Thread starter
- #1

- Jan 26, 2012

- 268

- Feb 9, 2012

- 118

\int_{0}^{n\pi }{\frac{\left| \sin x \right|}{x}\,dx}&=\sum\limits_{j=0}^{n-1}{\int_{j\pi }^{(j+1)\pi }{\frac{\left| \sin x \right|}{x}\,dx}} \\

& =\sum\limits_{j=0}^{n-1}{\int_{0}^{\pi }{\frac{\sin x}{x+j\pi }\,dx}} \\

& \ge \frac{1}{\pi }\sum\limits_{j=0}^{n-1}{\int_{0}^{\pi }{\frac{\sin x}{j+1}\,dx}} \\

& =\frac{2}{\pi }\sum\limits_{j=0}^{n-1}{\frac{1}{j+1}}.

\end{aligned}$

As $n\to\infty$ the magic appears.