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%I #9 Jun 27 2024 16:54:01
%S 3,7,7,1,0,0,9,4,9,1,4,0,0,9,2,3,2,2,6,0,7,9,0,8,1,1,3,7,6,7,7,3,3,8,
%T 4,1,2,4,3,5,0,9,3,6,9,9,8,4,2,2,3,1,9,0,7,3,0,0,0,9,4,4,5,9,5,9,1,8,
%U 9,2,3,5,5,0,5,6,2,1,7,4,2,9,2,2,9,0,5,2,2,9,5,7,1,7,9,9,3,6,0,5,6,7,4,6,3
%N Decimal expansion of Sum_{k >= 1} log(k)/(k*sqrt(k+1)).
%H Math StackExchange, <a href="https://math.stackexchange.com/questions/3221376">How do I test for convergence of log(n)/n/sqrt(n+1)</a>, (2019)
%F Equals sum_{l>=0} (-1)^(l+1) (2l-1)!! *Zeta'(3/2+l) /(2l)!!.
%e 3.77100949140092...
%p Digits := 120 ;
%p x := 0.0 ;
%p for l from 0 to 600 do
%p x := x+(-1)^(l+1)*doublefactorial(2*l-1)/doublefactorial(2*l)*Zeta(1,3/2+l) ;
%p x := evalf(x) ;
%p print(x) ;
%p end do: # _R. J. Mathar_, Jun 27 2024
%o (PARI) default(realprecision, 200); sumalt(k=0, (-1)^(k+1) * (2*k)! * zeta'(k+3/2) / (k!^2 * 4^k)) \\ _Vaclav Kotesovec_, Jun 27 2024
%Y Cf. A131688.
%K nonn,cons,new
%O 1,1
%A _R. J. Mathar_, Jun 19 2024
%E More terms from _Vaclav Kotesovec_, Jun 27 2024
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