

A068985


Decimal expansion of 1/e.


52



3, 6, 7, 8, 7, 9, 4, 4, 1, 1, 7, 1, 4, 4, 2, 3, 2, 1, 5, 9, 5, 5, 2, 3, 7, 7, 0, 1, 6, 1, 4, 6, 0, 8, 6, 7, 4, 4, 5, 8, 1, 1, 1, 3, 1, 0, 3, 1, 7, 6, 7, 8, 3, 4, 5, 0, 7, 8, 3, 6, 8, 0, 1, 6, 9, 7, 4, 6, 1, 4, 9, 5, 7, 4, 4, 8, 9, 9, 8, 0, 3, 3, 5, 7, 1, 4, 7, 2, 7, 4, 3, 4, 5, 9, 1, 9, 6, 4, 3, 7, 4, 6, 6, 2, 7
(list;
constant;
graph;
refs;
listen;
history;
text;
internal format)



OFFSET

0,1


COMMENTS

From the "derangements" problem: this is the probability that if a large number of people are given their hats at random, nobody gets his own hat.
Also, decimal expansion of cosh(1)sinh(1).  Mohammad K. Azarian, Aug 15 2006
Also, this is lim_{n>inf}P(n), where P(n) is the probability that a random rooted forest on [n] be a tree. See image from Wikipedia link.  Washington Bomfim, Nov 01 2010
Lim_{x>infinity} (1  1/x)^x = 1/e.  Arkadiusz Wesolowski, Feb 17 2012
Also, location of the minimum of x^x.  Stanislav Sykora, May 18 2012
Also, 1/e is the global minimum of x*log(x) at x = 1/e and the global minimum of x*e^x at x = 1.  Rick L. Shepherd, Jan 11 2014


REFERENCES

A. Hald, A History of Probability and Statistics and Their Applications Before 1750, Wiley, NY, 1990 (Chapter 19).
Jolley, Summation of Series, Dover (1961) eq (103) on page 20.
J. Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, p. 65.


LINKS

Table of n, a(n) for n=0..104.
Mohammad K. Azarian, An Expansion of e, Problem # B765, Fibonacci Quarterly, Vol. 32, No. 2, May 1994, p. 181. Solution appeared in Vol. 33, No. 4, Aug. 1995, p. 377.  Mohammad K. Azarian, Feb 08 2009
Gerard P. Michon, Final Answers: InclusionExclusion
Eric Weisstein's World of Mathematics, Factorial Sums
Eric Weisstein's World of Mathematics, Spherical Bessel Function of the First Kind
Eric Weisstein's World of Mathematics, Sultan's Dowry Problem
Eric Weisstein's World of Mathematics, e
Wikipedia, Graph of probabilities  Washington Bomfim, Nov 01 2010


FORMULA

Equals 2*(1/3! + 2/5! + 3/7! + ...). [Jolley]
1  Sum_{i >= 1} (1)^(i  1)/i!. [Michon]
Equals j_1(i)/i = cos(i) + i*sin(i), where j_1(z) is the spherical Bessel function of the first kind and i is the imaginary unit in C.  Stanislav Sykora, Jan 11 2017


EXAMPLE

1/e = 0.3678794411714423215955237701614608674458111310317678... = A135005/5.


MATHEMATICA

RealDigits[N[1/E, 6! ]][[1]] (* Vladimir Joseph Stephan Orlovsky, Jun 18 2009 *)


PROG

(PARI)
default(realprecision, 110);
exp(1) \\ Rick L. Shepherd, Jan 11 2014


CROSSREFS

Cf. A000166, A001113, A068996, A092553.
Cf. A059193.
Sequence in context: A003458 A133339 A112267 * A081391 A073850 A048748
Adjacent sequences: A068982 A068983 A068984 * A068986 A068987 A068988


KEYWORD

nonn,cons,changed


AUTHOR

N. J. A. Sloane, Apr 08 2002


EXTENSIONS

More terms from Rick L. Shepherd, Jan 11 2014


STATUS

approved



