Proof of expansion of a certain value

[dimitri151]

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.

 

[mathman]

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.

Notation is poor. Limits of summation? Summation term?

 

[lurflurf]

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