Math Is Fun Forum

Kurre

Member

Registered: 2006-07-18

Posts: 280

Sum 1/(n^2+x^2)

Inspired by Eulers "proof" of Zeta(2)=pi^2/6, I came up with the following:
Factoring Sinz just as Euler did as a product of the roots:

Using conjugate rule:

Now, differentiating and using product rule yields:

Dividing by Sinz yields:

or as:

letting z=pi*x and multiplying by pi^2 we obtain:

Cool eh?
We could also let x=iw, which would reduce to:

Plugging the sum into mathematica immediately gives the answer, so it wasnt really anything new, only a brutal math of rigorous mathematics;)
But still kind of cool

and yeah, taking the limit as z->0, we obtain zeta(2)=pi^2/6

Last edited by Kurre (2009-02-19 01:21:35)