Saturday, 28 May 2016

calculus - Polygamma function series: suminftyk=1left(Psi(1)(k)right)2



Applying the Copson's inequality, I found:

S=k=1(Ψ(1)(k))2<23π2 where
Ψ(1)(k) is the polygamma function.
Is it known any sharper bound for the sum S?
Thanks.


Answer



The upper bound can be improved using asymptofic series :



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