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A068985
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Decimal expansion of 1/e.
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93
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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
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OFFSET
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0,1
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COMMENTS
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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
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REFERENCES
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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.
Walter D. Wallis and John C. George, Introduction to Combinatorics, CRC Press, 2nd ed. 2016, theorem 5.2 (The Derangement Series).
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LINKS
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Mohammad K. Azarian, An Expansion of e, Problem # B-765, 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.
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FORMULA
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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, 1900-1901). - Amiram Eldar, Mar 29 2022
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EXAMPLE
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1/e = 0.3678794411714423215955237701614608674458111310317678... = A135005/5.
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MATHEMATICA
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PROG
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(PARI)
default(realprecision, 110);
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CROSSREFS
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Cf. asymptotic probabilities of success for other "nothing but the best" variants of the secretary problem: A325905, A242674, A246665.
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KEYWORD
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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