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A073333
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Decimal expansion of 1/(e - 1) = Sum_{k >= 1} exp(-k).
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13
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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
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OFFSET
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0,1
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COMMENTS
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REFERENCES
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Wolfram Research, Mathematica, Version 4.1.0.0, Help Browser, under the function NSumExtraTerms
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LINKS
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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.
Michel Waldschmidt, Continued fractions, Ecole de recherche CIMPA-Oujda, Théorie des Nombres et ses Applications, 18-29 mai 2015: Oujda (Maroc).
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FORMULA
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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 (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)
Equals -LambertW(exp(1/(1 - exp(1))) / (1 - exp(1))).
Equals -1 - LambertW(-1, exp(1/(1 - exp(1))) / (1 - exp(1))). (End)
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 Integral_{x=0..1} floor(-log(x)) dx (see Redmond link). - Amiram Eldar, Oct 03 2023
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EXAMPLE
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0.581976706869326424385002005109011558546869301075396136266787059648...
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MAPLE
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h:=x->sum(1/exp(n), n=1..x); evalf[110](h(1500)); evalf[110](h(4000));
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MATHEMATICA
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RealDigits[N[Sum[Exp[-n], {n, 1, Infinity}], 120]][[1]]
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PROG
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CROSSREFS
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KEYWORD
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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