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Math Help - a convergence/divergence question (i've been going at it for two days now . . . .)

  1. #1
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    Unhappy a convergence/divergence question (i've been going at it for two days now . . . .)

    i've been thinking about this problem for almost two days, and i havent made much headway


    1.) does the following series converge?

    [epsilon]infiniti,n=1 (sin(1/n))/n
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  2. #2
    MHF Contributor chisigma's Avatar
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    From geometry it is well known that for 0<x<\frac{\pi}{2}...

    \sin x < x < \tan x (1)

    Now from (1) we derive that...

    \frac{\sin \frac{1}{n}}{n}<\frac {1}{n^{2}}

    The series...

    \sum_{n=1}^{\infty} \frac{1}{n^{2}}

    ... converges so that also converges the series...

    \sum_{n=1}^{\infty} \frac{\sin \frac{1}{n}}{n}

    Kind regards

    \chi \sigma
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