If I recall correctly, this is also how Euler discovered the sums (pi^2/6, pi^4/90, ...). Warut On Tue, Dec 22, 2009 at 12:10 PM, Dan Asimov <dasimov@earthlink.net> wrote:
I recently learned from mathematician Mikael Passare a nice way to see Zeta(2), as long as you accept two ways to express sin(x) / x:
sin(x) / x = Sum{k=0..oo} (-1)^k x^(2k) / (2k+1)!
and
sin(x) / x = Prod{n=1..oo} (1 - x^2/(n pi)^2).
The value of Zeta(2) then falls out immediately by equating coefficients of x^2.
It soon became clear that equating coefficients of higher terms will, with a minor trick or two thrown in, reveal the sum for any Zeta(E) for E in 2Z+. (At least this also works for Zeta(4) and Zeta(6).) Along the way a few other interesting sums fall out.
--Dan
STAIRLIKE / TRISKELIA
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