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Math Help - Series convergence/divergence question

  1. #1
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    Series convergence/divergence question

    Hello, I'm having a bit of trouble with with testing if this series converges or diverges.

    \sum_1^\infty\frac{sin(\frac{1}{n})}{\sqrt{n}}

    I've tried multiple tests and I can't seem to find one that gets me anywhere.
    Any help is appreciated. Thank you.
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  2. #2
    MHF Contributor Bruno J.'s Avatar
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    Let b_n=\frac{1}{n^{3/2}}. We know \sum_{n=0}^\infty b_n converges, and we have \lim_{n\rightarrow \infty} \left(\frac{\sin(1/n)}{\sqrt{n}}\right)/b_n =\lim_{n\rightarrow \infty} {n\sin(1/n)}  = 1 and therefore, by the limit comparison test your series converges.
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