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%I #14 Aug 05 2024 14:22:34
%S 0,5,2,9,6,1,0,2,7,7,8,6,5,5,7,2,8,5,5,0,1,1,3,0,9,0,9,5,8,3,0,1,9,8,
%T 0,2,8,1,7,6,6,6,9,3,5,3,8,7,1,7,7,1,7,4,9,0,8,0,2,6,6,8,5,6,5,3,4,5,
%U 3,9,1,0,6,0,5,6,0,9,7,8,7,8,3,9,3,3,2,0,6,5,9,5,0,4
%N Decimal expansion of Sum_{k >= 1} (-1)^(k+1)*1/((k+2)*(k+3)).
%H Michael I. Shamos, <a href="https://citeseerx.ist.psu.edu/pdf/ae33a269baba5e8b1038e719fb3209e8a00abec5">A catalog of the real numbers</a> (2011) p.100.
%F Equals Sum_{k >= 1} (-1)^(k+1)*/((k+2)*(k+3)).
%F Equals 2*log(2) - 4/3 = Sum_{k >= 2} 1/(4*k^3 - k) = Sum_{k >= 1} (zeta(2*k + 1) - 1)/(4^k). [from the Shamos reference]
%F Equals Sum_{k >= 1} 1/((2^k)*(4*k + 12)). [from the Shamos reference]
%F Equals Sum_{k>=3} (-1)^(k+1)/A002378(k). - _Amiram Eldar_, Nov 21 2022
%e 0.0529610277865572855011309095830198028176669353...
%t Join[{0}, RealDigits[2*Log[2] - 4/3, 10, 120][[1]]] (* _Amiram Eldar_, Nov 21 2022 *)
%o (PARI) 2*log(2) - 4/3
%Y Cf. A002378, A242023, A242024, A358517, A358519.
%Y Cf. A002162.
%K nonn,cons
%O 0,2
%A _Claude H. R. Dequatre_, Nov 20 2022