A332892 Decimal expansion of Sum_{k>=0} 1/(6*k)!.

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

Offset: 1

Author

Bernard Schott, Mar 02 2020

Keywords

• nonn
• cons

Comments

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.

Examples

			1.001388890976564743867770084409737409278656173555781142...
		

References

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

Crossrefs

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.

Programs

Maple

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

Mathematica

RealDigits[(1/3)*(Cosh[1] + 2*Cosh[1/2]*Cos[Sqrt[3]/2]), 10, 120][[1]] (* Amiram Eldar, May 31 2023 *)

PARI

sumpos(k=0, 1/(6*k)!) \\ Michel Marcus, Mar 02 2020

Formula

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

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

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