A073333 Decimal expansion of 1/(e - 1) = Sum_{k >= 1} exp(-k).

5, 8, 1, 9, 7, 6, 7, 0, 6, 8, 6, 9, 3, 2, 6, 4, 2, 4, 3, 8, 5, 0, 0, 2, 0, 0, 5, 1, 0, 9, 0, 1, 1, 5, 5, 8, 5, 4, 6, 8, 6, 9, 3, 0, 1, 0, 7, 5, 3, 9, 6, 1, 3, 6, 2, 6, 6, 7, 8, 7, 0, 5, 9, 6, 4, 8, 0, 4, 3, 8, 1, 7, 3, 9, 1, 6, 6, 9, 7, 4, 3, 2, 8, 7, 2, 0, 4, 7, 0, 9, 4, 0, 4, 8, 7, 5, 0, 5, 7, 6, 5, 4, 6, 2, 0

Offset: 0

Author

Robert G. Wilson v, Aug 22 2002

Keywords

• cons
• nonn

Comments

The value of the general continued fraction with the partial numerators (A000027) and the partial denominators (A000027). The value of the fractional limit of the numerators (A000166) and the denominators (A002467). Abs(A002467/(e-1)-A000166)->0. - Seiichi Kirikami, Oct 30 2011

Examples

			0.581976706869326424385002005109011558546869301075396136266787059648...
		

References

• Calvin C. Clawson, Mathematical Mysteries: The Beauty and Magic of Numbers, Springer, 2013. See p. 225.
• Wolfram Research, Mathematica, Version 4.1.0.0, Help Browser, under the function NSumExtraTerms

Links

• G. C. Greubel, Table of n, a(n) for n = 0..20000
• Mohammad K. Azarian, A Limit Expression of 1/(e-1), Problem # 799, College Mathematics Journal, Vol. 36, No. 2, March 2005, p. 161. Solution appeared in Vol. 37, No. 2, March 2006, pp. 147-148.
• Mohammad K. Azarian, Euler's Number Via Difference Equations, International Journal of Contemporary Mathematical Sciences, Vol. 7, 2012, No. 22, pp. 1095-1102.
• H. W. Gould, A rearrangement of series based on a partition of the natural numbers, The Fibonacci Quarterly, Vol. 15, No. 1 (1977), pp. 67-72.
• Don Redmond, The Evaluation of Integral_{x=0..1} floor(-ln(x)) dx, Problem #153, Advanced Problem Archive, Missouri State University.
• Michel Waldschmidt, Continued fractions, Ecole de recherche CIMPA-Oujda, Théorie des Nombres et ses Applications, 18-29 mai 2015: Oujda (Maroc).
• Eric Weisstein's World of Mathematics, Continued Fraction Constants.
• Eric Weisstein's World of Mathematics, Generalized Continued Fraction.
• Index entries for transcendental numbers.

Crossrefs

Cf. A001113, A000027, A000166, A002467, A185393, A194807, A307178.

Programs

Magma

1/(Exp(1) - 1); // G. C. Greubel, Apr 09 2018

Maple

h:=x->sum(1/exp(n),n=1..x); evalf[110](h(1500)); evalf[110](h(4000));

Mathematica

RealDigits[N[Sum[Exp[-n], {n, 1, Infinity}], 120]][[1]]
RealDigits[1/(E - 1), 10, 120][[1]] (* Eric W. Weisstein, May 08 2013 *)

PARI

suminf(k=1,exp(-k)) \\ Charles R Greathouse IV, Oct 04 2011

PARI

1/(exp(1)-1) \\ Charles R Greathouse IV, Oct 04 2011

Formula

Equals 1/(exp(1)-1). - Joseph Biberstine (jrbibers(AT)indiana.edu), Nov 03 2004

Also the unique real solution to log(1+x) - log(x) = 1. Equals 1-1/(1+1/(exp(1)-2)). Continued fraction is [0:1, 1, 2, 1, 1, 4, 1, 1, 6, 1, 1, 8, 1, 1, 10, ...]. - Gerald McGarvey, Aug 14 2004

Equals Sum_{n>=0} B_n/n!, where B_n is a Bernoulli number. - Fredrik Johansson, Oct 18 2006

1/(e-1) = 1/(1+2/(2+3/(3+4/(4+5/(5+...(continued fraction)))))). - Philippe Deléham, Mar 09 2013

Equals Integral_{x=0..oo} floor(x)*exp(-x). - Jean-François Alcover, Mar 20 2013

From Peter Bala, Oct 09 2013: (Start)

Equals (1/2)*Sum_{n >= 0} 1/sinh(2^n). (Gould, equation 22).

Define s(n) = Sum_{k = 1..n} 1/k! for n >= 1. Then 1/(e - 1) = 1 - Sum_{n >= 1} 1/( (n+1)!*s(n)*s(n+1) ) is a rapidly converging series of rationals (see A194807). Equivalently, 1/(e - 1) = 1 - 1!/(1*3) - 2!/(3*10) - 3!/(10*41) - ..., where [1, 3, 10, 41, ... ] is A002627.

We also have the alternating series 1/(e - 1) = 1!/(1*1) - 2!/(1*4) + 3!/(4*15) - 4!/(15*76) + ..., where [1, 1, 4, 15, 76, ...] is A002467. (End)

From Vaclav Kotesovec, Oct 13 2018: (Start)

Equals A185393 - 1.

Equals -LambertW(exp(1/(1 - exp(1))) / (1 - exp(1))).

Equals -1 - LambertW(-1, exp(1/(1 - exp(1))) / (1 - exp(1))). (End)

From Gleb Koloskov, Sep 03 2021: (Start)

Equals (coth(1/2)-1)/2 = (A307178-1)/2.

Equals 1/2 + 2*Integral_{x=0..oo} sin(x)/(exp(2*Pi*x)-1) dx.

Equals 1/2 + (1/Pi)*Integral_{x=0..1} sin(log(x)/(2*Pi))/(x-1) dx. (End)

Equals -lim_{n->oo} zeta(1-n, n)*n^(1 - n). - Vaclav Kotesovec and Peter Luschny, Nov 05 2021

Equals Integral_{x=0..1} floor(-log(x)) dx (see Redmond link). - Amiram Eldar, Oct 03 2023

Equals 1/2 + Sum_{k>=2} tanh(1/2^k)/2^k. - Antonio Graciá Llorente, Jan 21 2024

Extensions

Entry revised by N. J. A. Sloane, Apr 07 2006