Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

A Proof of expansion of a certain value

  1. Oct 29, 2016 #1
    How do I begin proving:
    sum(k>=1)8/(k^4+4)=pi*coth(pi)-1?

    I got this from Mathematica.

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

    mathman

    User Avatar
    Science Advisor
    Gold Member

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

    lurflurf

    User Avatar
    Homework Helper

    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
    http://www.supermath.info/InfiniteSeriesandtheResidueTheorem.pdf
     
    Last edited: Oct 29, 2016
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?
Draft saved Draft deleted