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anonymous

  • one year ago

Can we find a closed form? \[\lim_{k\to\infty}\sum_{n=1}^\infty \tan\frac{1}{n^k}\] Numerically, the limit approaches \(1.5574078\).

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  1. freckles
    • one year ago
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    *

  2. Zarkon
    • one year ago
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    \[\tan(1)\]

  3. anonymous
    • one year ago
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    Hmm, for \(n>1\) terms, you have \(\tan(0) = 0\).

  4. anonymous
    • one year ago
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    the behavior is a consequence of the fact that \(1/n^k\to 0\) very quickly as \(k\to\infty\) for \(n>1\), so quickly in fact that we can observe that only the \(n=1\) term will 'survive' and we get \(1/1^k=1\) and thus we get \(\tan(1)\)

  5. anonymous
    • one year ago
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    Right, this question seemed much more difficult at first glance when I was typing random series into Mathematica :P I'll try to find something more challenging.

  6. Zarkon
    • one year ago
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    you can't just bring the limit past the sum...you need justification for that

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