A099285 Decimal expansion of -Ei(-1), negated exponential integral at -1.

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

Author

Robert G. Wilson v, Oct 08 2004

Keywords

• cons
• nonn

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

Examples

			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
		

Links

• G. C. Greubel, Table of n, a(n) for n = 0..10000
• Eric Weisstein's World of Mathematics, Exponential Integral

Crossrefs

Cf. A073003, A091725, A245780.

Programs

Maple

Digits:=105: evalf(-Ei(-1)); evalf(Ei(1,1)); # Johannes W. Meijer, Oct 16 2009

Mathematica

RealDigits[ ExpIntegralE[1, 1], 10, 105][[1]]

PARI

eint1(1, 1) \\ Michel Marcus, Aug 01 2020

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)

Extensions

Definition corrected by Johannes W. Meijer, Jul 26 2009

Corrected Name (minus 1, not 1), Stanislav Sykora, May 18 2012