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%I #24 Nov 29 2024 21:04:09
%S 6,0,3,7,8,2,8,6,2,7,9,1,4,8,7,9,8,8,4,1,6,1,8,3,8,1,0,9,8,2,4,5,0,5,
%T 4,8,3,0,4,1,7,0,1,5,3,1,6,4,9,9,1,0,2,1,7,7,2,4,1,3,2,1,1,3,8,2,2,7,
%U 2,2,8,4,1,0,0,5,2,5,5,6,9,4,7,8,2,1,3,7,5,0,2,4,6,4,9,7,1,0,8,8
%N Decimal expansion of Sum_{n>=2} log(n)/n!.
%H István Mező, <a href="http://www.jstor.org/stable/10.4169/amer.math.monthly.121.10.946">Problem 11806</a>, Problems and Solutions, The American Mathematical Monthly, Vol. 121, No. 10 (2014), p. 947; <a href="https://www.jstor.org/stable/10.4169/amer.math.monthly.123.9.941">Parseval and Kummer</a>, Solution to Problem 11806 by Omran Kouba, ibid., Vol. 123, No. 9 (2016), pp. 943-944.
%F Equal to log(exp(1/2*log(2*exp(1/3*log(3*exp(1/4*log(4*exp(...)))))))).
%F Equals log(A296301). - _Vaclav Kotesovec_, Jun 22 2023
%F Equals Integral_{x=0..2*Pi} log(Gamma(x/(2*Pi))) * exp(cos(x)) * sin(x + sin(x)) dx - (e-1)*(log(2*Pi)+gamma), where gamma is Euler's constant (A001620) (Mező, 2014). - _Amiram Eldar_, Jan 25 2024
%F Equals Integral_{x=0..1} (exp(x) - 1)/(x*log(x)) - (exp(1) - 1)/log(x) dx. - _Velin Yanev_, Nov 29 2024
%e 0.6037828627914879884...
%t NSum[Log[n]/n!, {n, 2, Infinity}, WorkingPrecision -> 110,
%t NSumTerms -> 100] // RealDigits[#, 10, 100] &
%o (PARI) suminf(n=2, log(n)/n!) \\ _Michel Marcus_, Jan 31 2019
%Y Cf. A001620, A061444, A193424, A296301, A336730.
%K nonn,cons
%O 0,1
%A _Rok Cestnik_, Jan 31 2019