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Math Help - series convegence

  1. #1
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    series convegence

    Show that (sum from 1 to infinity) sin(x/n^2) converges uniformly on any closed interval [a, b].

    I think I can use Weierstrass M-Test to prove this, but I can't find the limit for fn(x).

    Thx in advance...
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  2. #2
    MHF Contributor chisigma's Avatar
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    The Weiestrass M-test can be used in this case. If x \in [a,b] and we set c= max [|a|,|b|] is…

    |\sin (\frac{x}{n^{2}})|\le \frac{c}{n^{2}}

    But the series…

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

    … is absolutely convergent so that the series…

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

    … is uniformly convergent in [a,b]

    Kind regards

    \chi \sigma
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