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Decimal expansion of Sum_{k>=0} 1/(6*k)!.
4

%I #30 Feb 23 2024 02:15:49

%S 1,0,0,1,3,8,8,8,9,0,9,7,6,5,6,4,7,4,3,8,6,7,7,7,0,0,8,4,4,0,9,7,3,7,

%T 4,0,9,2,7,8,6,5,6,1,7,3,5,5,5,7,8,1,1,4,2,0,0,6,7,9,3,1,7,0,3,1,8,8,

%U 5,3,1,1,5,4,2,0,9,6,3,8,9,7,8,4,4,0,8

%N Decimal expansion of Sum_{k>=0} 1/(6*k)!.

%C For q integer >= 1, Sum_{m>=0} 1/(q*m)! = (1/q) * Sum_{k=1..q} exp(X_k) where X_1, X_2, ..., X_q are the q-th roots of unity.

%D Serge Francinou, Hervé Gianella, Serge Nicolas, Exercices de Mathématiques, Oraux X-ENS, Analyse 2, problème 3.10, p. 182, Cassini, Paris, 2004

%F Equals (1/3) * (cosh(1) + 2*cosh(1/2)*cos((sqrt(3))/2)).

%F Sum_{k>=0} (-1)^k / (6*k)! = (cos(1) + 2*cos(1/2)*cosh(sqrt(3)/2))/3 = 0.9986111131987866537... - _Vaclav Kotesovec_, Mar 02 2020

%F Continued fraction: 1 + 1/(720 - 720/(665281 - 665280/(13366081 - ... - P(n-1)/((P(n) + 1) - ... )))), where P(n) = (6*n)*(6*n - 1)*(6*n - 2)*(6*n - 3)*(6*n - 4)*(6*n - 5) for n >= 1. Cf. A346441. - _Peter Bala_, Feb 22 2024

%e 1.001388890976564743867770084409737409278656173555781142...

%p evalf(sum(1/(6*n)!,n=0..infinity),150);

%t RealDigits[(1/3)*(Cosh[1] + 2*Cosh[1/2]*Cos[Sqrt[3]/2]), 10, 120][[1]] (* _Amiram Eldar_, May 31 2023 *)

%o (PARI) sumpos(k=0, 1/(6*k)!) \\ _Michel Marcus_, Mar 02 2020

%Y Cf. A001113 (Sum 1/k!), A073743 (Sum 1/(2k)!), A143819 (Sum 1/(3k)!), A332890 (Sum 1/(4k)!), A269296 (Sum 1/(5k)!), this sequence (Sum 1/(6k)!), A346441.

%K nonn,cons

%O 1,5

%A _Bernard Schott_, Mar 02 2020