Wednesday, 23 October 2013

convergence divergence - Is this :suminftyn=1zeta(2n)zeta(2n+1)=frac12?



I would like to know the behavior of the Riemann zeta function values at even
and odd integers for studying irrationality between those values. I have tried using wolfram alpha to check the value of this sum:
n=1[ζ(2n)ζ(2n+1)].
It tells me it equals 12 .



Note: I don't have any method to show if the titled sum is true . Maybe I find who is help me here for evaluating the titled sum.



Thanks for any help.


Answer



Note that , due to the absolute convergence, we haven1(ζ(2n)ζ(2n+1))=n1(k11k2nk11k2n+1) =n1k1k1k2n+1=k2n1k1k2n+1 =k21k(k+1)=k2(1k1k+1)=12.


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