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How do I show that for $\alpha > 1$ the integral $\displaystyle \int_1^\infty \! \sin^\alpha(1/x) \, \mathrm{d}x$ converges?

I am given the hint:

Compare with the integral $\displaystyle \int_1^\infty \! x^{-\alpha} \, \mathrm{d}x$.

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    And how did you try to use the hint?2011-05-08
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    A related question: http://math.stackexchange.com/questions/9867/convergence-divergence-of-sum-n-1-infty-sin1-n/9869#98692011-05-08

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