|
|
A332892
|
|
Decimal expansion of Sum_{k>=0} 1/(6*k)!.
|
|
4
|
|
|
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
(list;
constant;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
OFFSET
|
1,5
|
|
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.
|
|
REFERENCES
|
Serge Francinou, Hervé Gianella, Serge Nicolas, Exercices de Mathématiques, Oraux X-ENS, Analyse 2, problème 3.10, p. 182, Cassini, Paris, 2004
|
|
LINKS
|
|
|
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
|
|
EXAMPLE
|
1.001388890976564743867770084409737409278656173555781142...
|
|
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 *)
|
|
PROG
|
|
|
CROSSREFS
|
|
|
KEYWORD
|
|
|
AUTHOR
|
|
|
STATUS
|
approved
|
|
|
|