A332890 Decimal expansion of Sum_{k>=0} 1/(4*k)!.

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

Offset: 1

Author

Bernard Schott, Mar 01 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

			1.0416914703416917479394211141000191431669197664918929...
		

References

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

Links

• Michael I. Shamos, A catalog of the real numbers, (2011). See p. 76.

Crossrefs

Cf. A001113 (Sum 1/k!), A073743 (Sum 1/(2k)!), A143819 (Sum 1/(3k)!), this sequence (Sum 1/(4k)!), A269296 (Sum 1/(5k)!), A332892 (Sum 1/(6k)!), A346441.

Programs

Maple

evalf(1/2 * (cos(1) + cosh(1)), 100);

Mathematica

RealDigits[Sum[1/(4n)!,{n,0,\[Infinity]}],10,120][[1]] (* Harvey P. Dale, Apr 18 2023 *)

PARI

suminf(k=0,(1 + (-1)^k)/((2*k)!))/2 \\ Hugo Pfoertner, Mar 01 2020

PARI

suminf(k=0, 1/(4*k)!) \\ Michel Marcus, Mar 02 2020

Formula

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

Equals (1/2) * Sum_{k>=0} (1 + (-1)^k)/((2*k)!). - Peter Luschny, Mar 01 2020

Sum_{k>=0} (-1)^k / (4*k)! = cos(1/sqrt(2)) * cosh(1/sqrt(2)) = 0.958358132833... - Vaclav Kotesovec, Mar 02 2020

Continued fraction: 1 + 1/(24 - 24/(1681 - 1680/(11881 - ... - P(n-1)/((P(n) + 1) - ... )))), where P(n) = (4*n)*(4*n - 1)*(4*n - 2)*(4*n - 3) for n >= 1. Cf. A346441. - Peter Bala, Feb 22 2024

Extensions

More terms from Hugo Pfoertner, Mar 02 2020