

A068985


Decimal expansion of 1/e.


93



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 their own hat.
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, 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


REFERENCES

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 (19001901), 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.
Walter D. Wallis and John C. George, Introduction to Combinatorics, CRC Press, 2nd ed. 2016, theorem 5.2 (The Derangement Series).


LINKS

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
James Grime and Brady Haran, Derangements, Numberphile video, 2017.
Eric Weisstein's World of Mathematics, e.


FORMULA

Equals 2*(1/3! + 2/5! + 3/7! + ...). [Jolley]
Equals 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
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_{n>oo} (n+1)!^(1/(n+1))  n!^(1/n) (Lalescu, 19001901).  Amiram Eldar, Mar 29 2022


EXAMPLE

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


MATHEMATICA



PROG

(PARI)
default(realprecision, 110);


CROSSREFS

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


KEYWORD



AUTHOR



EXTENSIONS



STATUS

approved



