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Math Help - uniform convergence

  1. #1
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    uniform convergence

    Show that the series {\Sigma_{n=1}^{\infty}} \frac {sin nx}{n^2} uniformly converges on \mathbb{R}. Verify that

    \int_{0}^{\pi} ({\Sigma_{n=1}^{\infty}}\frac {sin nx}{n^2}) dx = {\Sigma_{n=1}^{\infty}}\frac {2}{(2n-1)^3}
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  2. #2
    MHF Contributor chisigma's Avatar
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    Because is...

    \displaystyle \int_{0}^{\pi} \sin nx\ dx = \frac{1-\cos n \pi}{n} (1)

    ... if the series \displaystyle \sum_{n=1}^{\infty} \frac{\sin nx}{n^{2}} converges uniformely, then is...

    \displaystyle \int_{0}^{\pi}\ \sum_{n=1}^{\infty} \frac{\sin nx}{n^{2}}\ dx =  \sum_{n=1}^{\infty}\ \int_{0}^{\pi} \frac{\sin nx}{n^{2}}\ dx =  2\ \sum_{n=1}^{\infty} \frac{1}{(2n-1)^{3}} (2)

    At this point it remains to demonstrate the first part of Your statement...

    Kind regards

    \chi \sigma
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  3. #3
    MHF Contributor chisigma's Avatar
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    Quote Originally Posted by chisigma View Post
    At this point it remains to demonstrate the first part of Your statement...
    Because in all \mathcal {R} is...

    \displaystyle \frac{|\sin nx|}{n^{2}} \le \frac{1}{n^{2}} (1)

    ... and the series \displaystyle \sum_{n=1}^{\infty} \frac{1}{n^{2}} converges absolutely, the the series...

    \displaystyle f(x) = \sum_{n=1}^{\infty} \frac{\sin nx}{n^{2}} (2)

    ... converges uniformely in all \mathcal{R}...

    Kind regards

    \chi \sigma
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