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A Proof of expansion of a certain value

  1. Oct 29, 2016 #1
    How do I begin proving:

    I got this from Mathematica.

    Thanks in advance for any help.
  2. jcsd
  3. Oct 29, 2016 #2


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    Science Advisor

    Notation is poor. Limits of summation? Summation term?
  4. Oct 29, 2016 #3


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    The usual method is to observe that pi*coth(pi*x) has simple poles at the integers

    $$\sum_{k=1}^\infty \frac{8}{k^4+4}=-1+\frac{1}{2}\sum_{k=-\infty}^\infty \frac{8}{k^4+4}$$
    then use
    $$\sum_{k=-\infty}^\infty f(k)=-\sum_{\text{z is a pole of f}}\mathrm{Res} \, \pi \cot(\pi z)f(z) $$
    see for example
    Last edited: Oct 29, 2016
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