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A099285
Decimal expansion of -Ei(-1), negated exponential integral at -1.
45
2, 1, 9, 3, 8, 3, 9, 3, 4, 3, 9, 5, 5, 2, 0, 2, 7, 3, 6, 7, 7, 1, 6, 3, 7, 7, 5, 4, 6, 0, 1, 2, 1, 6, 4, 9, 0, 3, 1, 0, 4, 7, 2, 9, 3, 4, 0, 6, 9, 0, 8, 2, 0, 7, 5, 7, 7, 9, 7, 8, 6, 1, 3, 0, 7, 3, 5, 6, 8, 6, 9, 8, 5, 5, 9, 1, 4, 1, 5, 4, 4, 7, 2, 2, 2, 1, 0, 2, 5, 1, 0, 3, 5, 1, 3, 7, 2, 4, 9, 9, 5, 4, 7, 5, 8
OFFSET
0,1
COMMENTS
The divergent series g(x=1,m) = 1^m*1! - 2^m*2! + 3^m*3! - 4^m*4! + ..., m=>-1, is closely related to the value of -Ei(-1). We discovered that g(x=1,m) = (-1)^m*(A040027(m) - A000110(m+1)*Ei(1,1)*exp(1)), see A163940. We observe that Ei(1,1) = E(1,1,1) = -Ei(-1) is the constant given above and that Ei(1,1)*exp(1) = A073003 is Gompertz's constant. - Johannes W. Meijer, Oct 16 2009
LINKS
Eric Weisstein's World of Mathematics, Exponential Integral
FORMULA
-Ei(-n) = Integral_{a=n..oo} ( Integral_{b=1..oo} 1/e^(a*b) db ) da , n>0 (According to Mathematica). - Mats Granvik, May 25 2013
Equals the difference between the absolute values of A239069 and A001620. - R. J. Mathar, Mar 07 2016
From Amiram Eldar, Aug 01 2020: (Start)
Equals Integral_{x=1..oo} log(x)/exp(x) dx.
Equals Integral_{x=0..oo} exp(-exp(x)) dx.
Equals Integral_{x=0..oo} x*exp(x-exp(x)) dx. (End)
From Peter Bala, Jun 17 2024: (Start)
Equals lim_{n -> oo} Integral_{x = 0..n} x^(n-1)/(1 + x)^n dx = lim_{n -> oo} ( log(n+1) + Sum_{k = 0..n-2} (-1)^(n-k-1)* binomial(n-1, k)*((n + 1)^(k+1-n) - 1)/(k + 1 - n) ).
Alternatively, equals lim_{n -> oo} Sum_{k >= n} (n/(n + 1))^k/k = lim_{n -> oo} ( log(1/(1 - x)) - Sum_{k = 1..n-1} x^k/k ), where x = n/(n+1).
More generally, for alpha > 0, -Ei(-alpha) = lim_{n -> oo} Integral_{x = 0..n/alpha} x^(n-1)/(1 + x)^n dx. (End)
EXAMPLE
0.219383934395520273677163775460121649031047293406908207577978613...
With n := 10^6, Integral_{x = 0..n} x^(n-1)/(1 + x)^n dx = 0.21938(43...). - Peter Bala, Jun 19 2024
MAPLE
Digits:=105: evalf(-Ei(-1)); evalf(Ei(1, 1)); # Johannes W. Meijer, Oct 16 2009
MATHEMATICA
RealDigits[ ExpIntegralE[1, 1], 10, 105][[1]]
PROG
(PARI) eint1(1, 1) \\ Michel Marcus, Aug 01 2020
CROSSREFS
KEYWORD
cons,nonn
AUTHOR
Robert G. Wilson v, Oct 08 2004
EXTENSIONS
Definition corrected by Johannes W. Meijer, Jul 26 2009
Corrected Name (minus 1, not 1), Stanislav Sykora, May 18 2012
STATUS
approved