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Search: id:A073003
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| A073003 |
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Decimal expansion of -exp(1)*Ei(-1), also called Gompertz's constant. |
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+0
7
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| 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, 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, 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
(list; cons; graph; listen)
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OFFSET
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0,1
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COMMENT
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0! - 1! + 2! - 3! + 4! - 5! + ... = (Borel) sum_{n=0}^infinity (-y)^n n! = KummerU(1,1,1/y)/y
Decimal expansion of phi(1) where phi(x)=integral(t=0,infinity, e^-t/(x+t) dt ). - Benoit Cloitre (benoit7848c(AT)orange.fr), Apr 11 2003
Contribution from Johannes W. Meijer (meijgia(AT)hotmail.com), Oct 16 2009: (Start)
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.
(End)
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REFERENCES
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Bruce C. Berndt, Ramanujan's notebooks Part II, Springer, p. 171
Bruce C. Berndt, Ramanujan's notebooks Part I, Springer, p. 144-145.
Walls, H. S., Analytic Theory of Continued Fractions, Van Nostrand, New York, 1948, p. 356
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LINKS
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Simon Plouffe, -exp(1)*Ei(-1)
Eric Weisstein's World of Mathematics, Link to a section of The World of Mathematics
Eric Weisstein's World of Mathematics, Exponential Integral
A. I. Aptekarev, On linear forms containing the Euler constant, arXiv:0902.1768 [math.NT]. [From R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Feb 14 2009]
G.H. Hardy, Divergent Series , Oxford University Press, 1949. p. 29. [From Johannes W. Meijer (meijgia(AT)hotmail.com), Oct 16 2009]
Ed Sandifer, Divergent Series, How Euler Did It, MAA Online, June 2006. [From Johannes W. Meijer (meijgia(AT)hotmail.com), Oct 16 2009]
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FORMULA
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phi(1)=e*(sum(k>=1, (-1)^(k-1)/k/k!) - Gamma)=0.596347362323194... where Gamma is the Euler constant.
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 DELEHAM (kolotoko(AT)wanadoo.fr), Aug 14 2005
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EXAMPLE
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0.59634736232319407434...
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MATHEMATICA
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RealDigits[ N[ -Exp[1]*ExpIntegralEi[ -1], 110]] [[1]]
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CROSSREFS
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Contribution from Johannes W. Meijer (meijgia(AT)hotmail.com), Oct 16 2009: (Start)
Equals A001113*A099285.
(End)
Sequence in context: A134879 A051158 A117605 this_sequence A087498 A121060 A021630
Adjacent sequences: A073000 A073001 A073002 this_sequence A073004 A073005 A073006
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KEYWORD
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cons,nonn
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AUTHOR
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Robert G. Wilson v (rgwv(AT)rgwv.com), Aug 03 2002
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EXTENSIONS
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Additional references from Gerald McGarvey (Gerald.McGarvey(AT)comcast.net), Oct 10 2005
Link corrected by Johannes W. Meijer (meijgia(AT)hotmail.com), Aug 01 2009
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