|
%I #126 Apr 04 2024 10:30:46
%S 5,9,6,3,4,7,3,6,2,3,2,3,1,9,4,0,7,4,3,4,1,0,7,8,4,9,9,3,6,9,2,7,9,3,
%T 7,6,0,7,4,1,7,7,8,6,0,1,5,2,5,4,8,7,8,1,5,7,3,4,8,4,9,1,0,4,8,2,3,2,
%U 7,2,1,9,1,1,4,8,7,4,4,1,7,4,7,0,4,3,0,4,9,7,0,9,3,6,1,2,7,6,0,3,4,4,2,3,7
%N Decimal expansion of -exp(1)*Ei(-1), also called Gompertz's constant, or the Euler-Gompertz constant.
%C 0! - 1! + 2! - 3! + 4! - 5! + ... = (Borel) Sum_{n>=0} (-y)^n n! = KummerU(1,1,1/y)/y.
%C Decimal expansion of phi(1) where phi(x) = Integral_{t>=0} e^-t/(x+t) dt. - _Benoit Cloitre_, Apr 11 2003
%C The divergent series g(x=1,m) = 1^m*1! - 2^m*2! + 3^m*3! - 4^m*4! + ..., m => -1, is intimately related to Gompertz's constant. We discovered that g(x=1,m) = (-1)^m * (A040027(m) - A000110(m+1) * A073003) with A000110 the Bell numbers and A040027 a sequence that was published by Gould, see for more information A163940. - _Johannes W. Meijer_, Oct 16 2009
%C Named by Le Lionnais (1983) after the English self-educated mathematician and actuary Benjamin Gompertz (1779 - 1865). It was named the Euler-Gompertz constant by Finch (2003). Lagarias (2013) noted that he has not located this constant in Gompertz's writings. - _Amiram Eldar_, Aug 15 2020
%D Bruce C. Berndt, Ramanujan's notebooks Part II, Springer, p. 171
%D Bruce C. Berndt, Ramanujan's notebooks Part I, Springer, p. 144-145.
%D S. R. Finch, Mathematical Constants, Cambridge, 2003, pp. 424-425.
%D Francois Le Lionnais, Les nombres remarquables, Paris: Hermann, 1983. See p. 29.
%D H. S. Wall, Analytic Theory of Continued Fractions, Van Nostrand, New York, 1948, p. 356.
%H Robert Price, <a href="/A073003/b073003.txt">Table of n, a(n) for n = 0..10000</a>
%H A. I. Aptekarev, <a href="http://arxiv.org/abs/0902.1768">On linear forms containing the Euler constant</a>, arXiv:0902.1768 [math.NT], 2009. - _R. J. Mathar_, Feb 14 2009
%H Richard P. Brent, M. L. Glasser, and Anthony J. Guttmann, <a href="https://arxiv.org/abs/1812.00316">A Conjectured Integer Sequence Arising From the Exponential Integral</a>, arXiv:1812.00316 [math.NT], 2018.
%H G. H. Hardy, <a href="http://www.archive.org/texts/flipbook/flippy.php?id=divergentseries033523mbp">Divergent Series</a>, Oxford University Press, 1949. p. 29. - _Johannes W. Meijer_, Oct 16 2009
%H Jeffrey C. Lagarias, <a href="https://doi.org/10.1090/S0273-0979-2013-01423-X">Euler's constant: Euler's work and modern developments</a>, Bull. Amer. Math. Soc., Vol. 50, No. 4 (2013), pp. 527-628, <a href="http://arxiv.org/abs/1303.1856">preprint</a>, arXiv:1303.1856 [math.NT], 2013.
%H István Mezo, <a href="http://www.naturalspublishing.com/files/published/j18jp677r69ri8.pdf">Gompertz constant, Gregory coefficients and a series of the logarithm function</a>, Journal of Analysis & Number Theory, Vol. 2, No. 2 (2014), pp. 33-36.
%H Michael Penn, <a href="https://www.youtube.com/watch?v=W-FBMPS-fJg">why some series are "regularizable"</a>, YouTube video (2023).
%H Simon Plouffe, <a href="https://web.archive.org/web/20150911101850/http://www.worldwideschool.org/library/books/sci/math/MiscellaneousMathematicalConstants/chap20.html">-exp(1)*Ei(-1)</a>
%H Tanguy Rivoal, <a href="http://doi.org/10.1307/mmj/1339011525">On the arithmetic nature of the values of the gamma function, Euler's constant, and Gompertz's constant</a>, Michigan Math. J., Vol. 61, No. 2 (2012), pp. 239-254.
%H Ed Sandifer, <a href="https://www.maa.org/sites/default/files/pdf/editorial/euler/How%20Euler%20Did%20It%2032%20divergent%20series.pdf">Divergent Series</a>, How Euler Did It, MAA Online, June 2006. - _Johannes W. Meijer_, Oct 16 2009
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/GompertzConstant.html">Gompertz Constant</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/ExponentialIntegral.html">Exponential Integral</a>
%F phi(1) = e*(Sum_{k>=1} (-1)^(k-1)/(k*k!) - Gamma) = 0.596347362323194... where Gamma is the Euler constant.
%F G = 0.596347... = 1/(1+1/(1+1/(1+2/(1+2/(1+3/(1+3/(1+4/(1+4/(1+5/(1+5/(1+6/(... - _Philippe Deléham_, Aug 14 2005
%F Equals A001113*A099285. - _Johannes W. Meijer_, Oct 16 2009
%F From _Peter Bala_, Oct 11 2012: (Start)
%F Stieltjes found the continued fraction representation G = 1/(2 - 1^2/(4 - 2^2/(6 - 3^2/(8 - ...)))). See [Wall, Chapter 18, (92.7) with a = 1]. The sequence of convergents to the continued fraction begins [1/2, 4/7, 20/34, 124/209, ...]. The numerators are in A002793 and the denominators in A002720.
%F Also, 1 - G has the continued fraction representation 1/(3 - 2/(5 - 6/(7 - ... -n*(n+1)/((2*n+3) - ...)))) with convergents beginning [1/3, 5/13, 29/73, 201/501, ...]. The numerators are in A201203 (unsigned) and the denominators are in A000262.
%F (End)
%F G = f(1) with f solution to the o.d.e. x^2*f'(x) + (x+1)*f(x)=1 such that f(0)=1. - _Jean-François Alcover_, May 28 2013
%F From _Amiram Eldar_, Aug 15 2020: (Start)
%F Equals Integral_{x=0..1} 1/(1-log(x)) dx.
%F Equals Integral_{x=1..oo} exp(1-x)/x dx.
%F Equals Integral_{x=0..oo} exp(-x)*log(x+1) dx.
%F Equals Integral_{x=0..oo} exp(-x)/(x+1) dx. (End)
%F From _Gleb Koloskov_, May 01 2021: (Start)
%F Equals Integral_{x=0..1} LambertW(e/x)-1 dx.
%F Equals Integral_{x=0..1} 1+1/LambertW(-1,-x/e) dx. (End)
%F Equals lim_{n->infinity} A040027(n)/A000110(n+1). - _Vaclav Kotesovec_, Feb 22 2021
%F G = lim_{n -> infinity} A321942(n)/A000262(n). - _Peter Bala_, Mar 21 2022
%F Equals Sum_{n >= 1} 1/(n*L(n, -1)*L(n-1, -1)), where L(n, x) denotes the n-th Laguerre polynomial. This is the case x = 1 of the identity Integral_{t >= 0} exp(-t)/(x + t) dt = Sum_{n >= 1} 1/(n*L(n, -x)*L(n-1, -x)) valid for Re(x) > 0. - _Peter Bala_, Mar 21 2024
%e 0.59634736232319407434107849936927937607417786015254878157348491...
%t RealDigits[N[-Exp[1]*ExpIntegralEi[-1], 105]][[1]]
%t (* Second program: *)
%t G = 1/Fold[Function[2*#2 - #2^2/#1], 2, Reverse[Range[10^4]]] // N[#, 105]&; RealDigits[G] // First (* _Jean-François Alcover_, Sep 19 2014 *)
%o (PARI) eint1(1)*exp(1) \\ _Charles R Greathouse IV_, Apr 23 2013
%o (Magma) SetDefaultRealField(RealField(100)); ExponentialIntegralE1(1)*Exp(1); // _G. C. Greubel_, Dec 04 2018
%o (Sage) numerical_approx(exp_integral_e(1,1)*exp(1), digits=100) # _G. C. Greubel_, Dec 04 2018
%Y Cf. A000522 (arrangements), A001620, A000262, A002720, A002793, A058006 (alternating factorial sums), A153229, A201203, A283743 (Ei(1)/e), A321942, A369883.
%K cons,nonn
%O 0,1
%A _Robert G. Wilson v_, Aug 03 2002
%E Additional references from _Gerald McGarvey_, Oct 10 2005
%E Link corrected by _Johannes W. Meijer_, Aug 01 2009
|
