VOOZH about

URL: https://oeis.org/A332890

⇱ A332890 - OEIS

login
A332890
Decimal expansion of Sum_{k>=0} 1/(4*k)!.
9
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
1,3
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.
This constant is irrational (proof: mimic the standard proof that e is irrational). Is it transcendental? - Charles R Greathouse IV, Oct 10 2025
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.
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
EXAMPLE
1.0416914703416917479394211141000191431669197664918929...
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 *)
PROG
(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
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.
Sequence in context: A205137 A223534 A222391 * A154905 A124739 A122662
KEYWORD
nonn,cons
AUTHOR
Bernard Schott, Mar 01 2020
EXTENSIONS
More terms from Hugo Pfoertner, Mar 02 2020
STATUS
approved