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Today's calculation of Integral 168

Source: Nagasaki university entrance exam 1980

January 15, 2007
calculusintegrationtrigonometrycalculus computations

Problem Statement

Prove that ∑n=1∞∫1n+11n∣1xsin⁡πx∣dx\sum_{n=1}^{\infty}\int_{\frac{1}{n+1}}^{\frac{1}{n}}{\left|\frac{1}{x}\sin \frac{\pi}{x}\right| dx} diverge for x>0.x>0.
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