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Math Help - Uniform Convergence of a series

  1. #1
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    Uniform Convergence of a series

    Hello!

    I have the following series:

    \sum _{n=1}^{\infty} 3^{n}\sin\frac{1}{4^{n}x}  ,  x\in(0,\infty)

    I need to prove that it converges, and then check if the convergence is uniformly.

    So, I said that:

    3^{n}\sin\frac{1}{4^{n}x} \le \frac{3^n}{4^{n}x}

    Because for every t : \sin(t) \le t

    First of all, is this way right? (For every x in range I think that \sum_{n=1}^\infty \frac{3^n}{4^{n}x} converges...)

    I also don't have any idea how to check if it converges uniformly (because I can't seem to find the limit-function of this series), and since I can't find a maximum for this series (for the variable x), I can't find a converging series that is larger than my series.

    Thank you very much!
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  2. #2
    Senior Member roninpro's Avatar
    Joined
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    Your series certainly converges uniformly on any interval \left[\delta, \infty\right) by the Weierstrass M-test, since the term \frac{3^n}{4^n x} would be bounded. However, I am extremely suspicious about including the 0 as the (open) endpoint, because \sin(\frac{1}{4^nx}) has a nasty oscillation around 0.

    Maybe you can show that the Cauchy Criterion fails?
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