[math-fun] Method for summing Zeta(even)
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 _____________________________________________________________________ "It don't mean a thing if it ain't got that certain je ne sais quoi." --Peter Schickele
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
_____________________________________________________________________ "It don't mean a thing if it ain't got that certain je ne sais quoi." --Peter Schickele
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Yes, this was a long standing problem that Euler solved : http://en.wikipedia.org/wiki/Basel_problem the basel problem, but if I may, I think that he guessed the first result for Pi^2/6 and managed a way to make a proof in the general case. This is typical from Euler... many of his results are based on computation first then eventually a general result with a ~proof another example is the Euler-Maclaurin formula. Simon Plouffe
participants (3)
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Dan Asimov -
Simon Plouffe -
Warut Roonguthai