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Math Help - Infinite series (5)

  1. #1
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    Infinite series (5)

    By proving \lim_{n\to\infty}(\cos n)^n \neq 0, or any other method you like, show that \sum_{n=1}^{\infty} (\cos n)^n is divergent.
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  2. #2
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    Far from a rigorous proof but seeing as there hasn't been a reply I thought I might as well say something because I thought this limit was quite interesting.

    If we first define  p_n \in \mathbb{N} such that  p_n = \lfloor {(\pi \times 10^n)} \rfloor

    Then I think the reason the limit doesn't converge to zero is that as  n \to \infty, \quad p_n \to 314159265358979\ldots and so  \lim_{n \to \infty} \cos(p_n) = \pm 1

    Like I said, far from rigorous, but nice question anyway.


    Thanks
    pomp.
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  3. #3
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    How about this:

    The set \{|cos(n)|\},n\in\mathbb{N}^{+} is dense in (0,1) and therefore their exist an \epsilon>0 such that 1-|\cos(n)|<\epsilon for some n\in \mathbb{N}^{+} and therefore |\cos(n)|^n >0 for some n\in \mathbb{N}^{+} no matter how large n.
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