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%I #114 Feb 10 2024 15:13:17
%S 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,
%T 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,
%U 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
%N Decimal expansion of 1/(e - 1) = Sum_{k >= 1} exp(-k).
%C 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
%D Wolfram Research, Mathematica, Version 4.1.0.0, Help Browser, under the function NSumExtraTerms
%H G. C. Greubel, <a href="/A073333/b073333.txt">Table of n, a(n) for n = 0..20000</a>
%H Mohammad K. Azarian, <a href="http://www.jstor.org/stable/30044845">A Limit Expression of 1/(e-1), Problem # 799</a>, College Mathematics Journal, Vol. 36, No. 2, March 2005, p. 161. <a href="http://www.jstor.org/stable/27646306">Solution</a> appeared in Vol. 37, No. 2, March 2006, pp. 147-148.
%H Mohammad K. Azarian, <a href="http://www.m-hikari.com/ijcms/ijcms-2012/21-24-2012/azarianIJCMS21-24-2012.pdf">Euler's Number Via Difference Equations</a>, International Journal of Contemporary Mathematical Sciences, Vol. 7, 2012, No. 22, pp. 1095-1102.
%H H. W. Gould, <a href="https://www.fq.math.ca/Scanned/15-1/gould2.pdf">A rearrangement of series based on a partition of the natural numbers</a>, The Fibonacci Quarterly, Vol. 15, No. 1 (1977), pp. 67-72.
%H Don Redmond, <a href="http://people.missouristate.edu/lesreid/advsol153_redmond.pdf">The Evaluation of Integral_{x=0..1} floor(-ln(x)) dx</a>, Problem #153, Advanced Problem Archive, Missouri State University.
%H Michel Waldschmidt, <a href="http://webusers.imj-prg.fr/~michel.waldschmidt/articles/pdf/ContinuedFractionsOujda2015.pdf">Continued fractions</a>, Ecole de recherche CIMPA-Oujda, Théorie des Nombres et ses Applications, 18-29 mai 2015: Oujda (Maroc).
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/ContinuedFractionConstants.html">Continued Fraction Constants</a>.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/GeneralizedContinuedFraction.html">Generalized Continued Fraction</a>.
%H <a href="/index/Tra#transcendental">Index entries for transcendental numbers</a>.
%F Equals 1/(exp(1)-1). - Joseph Biberstine (jrbibers(AT)indiana.edu), Nov 03 2004
%F 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
%F Equals Sum_{n>=0} B_n/n!, where B_n is a Bernoulli number. - _Fredrik Johansson_, Oct 18 2006
%F 1/(e-1) = 1/(1+2/(2+3/(3+4/(4+5/(5+...(continued fraction)))))). - _Philippe Deléham_, Mar 09 2013
%F Equals Integral_{x=0..oo} floor(x)*exp(-x). - _Jean-François Alcover_, Mar 20 2013
%F From _Peter Bala_, Oct 09 2013: (Start)
%F Equals (1/2)*Sum_{n >= 0} 1/sinh(2^n). (Gould, equation 22).
%F 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.
%F 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)
%F From _Vaclav Kotesovec_, Oct 13 2018: (Start)
%F Equals A185393 - 1.
%F Equals -LambertW(exp(1/(1 - exp(1))) / (1 - exp(1))).
%F Equals -1 - LambertW(-1, exp(1/(1 - exp(1))) / (1 - exp(1))). (End)
%F From _Gleb Koloskov_, Sep 03 2021: (Start)
%F Equals (coth(1/2)-1)/2 = (A307178-1)/2.
%F Equals 1/2 + 2*Integral_{x=0..oo} sin(x)/(exp(2*Pi*x)-1) dx.
%F Equals 1/2 + (1/Pi)*Integral_{x=0..1} sin(log(x)/(2*Pi))/(x-1) dx. (End)
%F Equals -lim_{n->oo} zeta(1-n, n)*n^(1 - n). - _Vaclav Kotesovec_ and _Peter Luschny_, Nov 05 2021
%F Equals Integral_{x=0..1} floor(-log(x)) dx (see Redmond link). - _Amiram Eldar_, Oct 03 2023
%F Equals 1/2 + Sum_{k>=2} tanh(1/2^k)/2^k. - _Antonio Graciá Llorente_, Jan 21 2024
%e 0.581976706869326424385002005109011558546869301075396136266787059648...
%p h:=x->sum(1/exp(n),n=1..x); evalf[110](h(1500)); evalf[110](h(4000));
%t RealDigits[N[Sum[Exp[-n], {n, 1, Infinity}], 120]][[1]]
%t RealDigits[1/(E - 1), 10, 120][[1]] (* _Eric W. Weisstein_, May 08 2013 *)
%o (PARI) suminf(k=1,exp(-k)) \\ _Charles R Greathouse IV_, Oct 04 2011
%o (PARI) 1/(exp(1)-1) \\ _Charles R Greathouse IV_, Oct 04 2011
%o (Magma) 1/(Exp(1) - 1); // _G. C. Greubel_, Apr 09 2018
%Y Cf. A001113, A000027, A000166, A002467, A185393, A194807, A307178.
%K cons,nonn
%O 0,1
%A _Robert G. Wilson v_, Aug 22 2002
%E Entry revised by _N. J. A. Sloane_, Apr 07 2006
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