A068985 Decimal expansion of 1/e.

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

Offset: 0

Author

N. J. A. Sloane, Apr 08 2002

Keywords

• nonn
• cons

Comments

From the "derangements" problem: this is the probability that if a large number of people are given their hats at random, nobody gets their 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] is a tree. See linked file. - Washington Bomfim, Nov 01 2010

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

Also, the asymptotic probability of success in the secretary problem (also known as the sultan's dowry problem). - Andrey Zabolotskiy, Sep 14 2019

The asymptotic density of numbers with an odd number of trailing zeros in their factorial base representation (A232745). - Amiram Eldar, Feb 26 2021

For large range size s where numbers are chosen randomly r times, the probability when r = s that a number is randomly chosen exactly 1 time. Also the chance that a number was not chosen at all. The general case for the probability of being chosen n times is (r/s)^n / (n! * e^(r/s)). - Mark Andreas, Oct 25 2022

Examples

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

References

• Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, Sections 1.3 and 5,23,3, pp. 14, 409.
• Anders Hald, A History of Probability and Statistics and Their Applications Before 1750, Wiley, NY, 1990 (Chapter 19).
• John Harris, Jeffry L. Hirst, and Michael Mossinghoff, Combinatorics and Graph Theory, Springer Science & Business Media, 2009, p. 161.
• L. B. W. Jolley, Summation of Series, Dover, 1961, eq. (103) on page 20.
• Traian Lalescu, Problem 579, Gazeta Matematică, Vol. 6 (1900-1901), p. 148.
• John Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, p. 65.
• Manfred R. Schroeder, Number Theory in Science and Communication, Springer Science & Business Media, 2008, ch. 9.5 Derangements.
• Jerome Spanier and Keith B. Oldham, "Atlas of Functions", Hemisphere Publishing Corp., 1987, chapter 26, page 233.
• Walter D. Wallis and John C. George, Introduction to Combinatorics, CRC Press, 2nd ed. 2016, theorem 5.2 (The Derangement Series).
• David Wells, The Penguin Dictionary of Curious and Interesting Numbers. Penguin Books, NY, 1986, Revised edition 1987, p. 27.

Links

• G. C. Greubel, Table of n, a(n) for n = 0..10000
• Washington Bomfim, Probabilities of connected random forests and derangements, Oct 31 2010.
• James Grime and Brady Haran, Derangements, Numberphile video, 2017.
• Peter J. Larcombe, Jack Sutton, and James Stanton, A note on the constant 1/e, Palest. J. Math. (2023) Vol. 12, No. 2, 609-619.
• Gérard P. Michon, Final Answers: Inclusion-Exclusion.
• Michael Penn, A cool, quick limit, YouTube video, 2022.
• Eric Weisstein's World of Mathematics, Derangement.
• 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.
• OEIS Wiki, Number of derangements.
• Index entries for transcendental numbers.

Crossrefs

Cf. A000166, A001113, A068996, A092553, A232745.

Cf. A059193.

Cf. asymptotic probabilities of success for other "nothing but the best" variants of the secretary problem: A325905, A242674, A246665.

Cf. A049470, A346441, A346440, A346439, A346438, A346437, A346436, A346435, A196498.

Programs

Mathematica

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

PARI

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

Formula

Equals 2*(1/3! + 2/5! + 3/7! + ...). [Jolley]

Equals 1 - Sum_{i >= 1} (-1)^(i - 1)/i!. [Michon]

Equals lim_{x->infinity} (1 - 1/x)^x. - Arkadiusz Wesolowski, Feb 17 2012

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 = sqrt(-1). - Stanislav Sykora, Jan 11 2017

Equals Sum_{i>=0} ((-1)^i)/i!. - Maciej Kaniewski, Sep 10 2017

Equals Sum_{i>=0} ((-1)^i)(i^2+1)/i!. - Maciej Kaniewski, Sep 12 2017

From Peter Bala, Oct 23 2019: (Start)

The series representation 1/e = Sum_{k >= 0} (-1)^k/k! is the case n = 0 of the following series acceleration formulas:

1/e = n!*Sum_{k >= 0} (-1)^k/(k!*R(n,k)*R(n,k+1)), n = 0,1,2,..., where R(n,x) = Sum_{k = 0..n} (-1)^k*binomial(n,k)*k!*binomial(-x,k) are the row polynomials of A094816. (End)

1/e = 1 - Sum_{n >= 0} n!/(A(n)*A(n+1)), where A(n) = A000522(n). - Peter Bala, Nov 13 2019

Equals Integral_{x=0..1} x * sinh(x) dx. - Amiram Eldar, Aug 14 2020

Equals lim_{x->oo} (x!)^(1/x)/x. - L. Joris Perrenet, Dec 08 2020

Equals lim_{n->oo} (n+1)!^(1/(n+1)) - n!^(1/n) (Lalescu, 1900-1901). - Amiram Eldar, Mar 29 2022

Extensions

More terms from Rick L. Shepherd, Jan 11 2014