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 have∑n≥1(ζ(2n)−ζ(2n+1))=∑n≥1(∑k≥11k2n−∑k≥11k2n+1) =∑n≥1∑k≥1k−1k2n+1=∑k≥2∑n≥1k−1k2n+1 =∑k≥21k(k+1)=∑k≥2(1k−1k+1)=12.
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