# Greetings from The On-Line Encyclopedia of Integer Sequences! http://oeis.org/ Search: id:a073333 Showing 1-1 of 1 %I A073333 #117 Jul 01 2024 19:21:16 %S A073333 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 A073333 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 A073333 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 A073333 Decimal expansion of 1/(e - 1) = Sum_{k >= 1} exp(-k). %C A073333 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 A073333 Calvin C. Clawson, Mathematical Mysteries: The Beauty and Magic of Numbers, Springer, 2013. See p. 225. %D A073333 Wolfram Research, Mathematica, Version 4.1.0.0, Help Browser, under the function NSumExtraTerms %H A073333 G. C. Greubel, Table of n, a(n) for n = 0..20000 %H A073333 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. %H A073333 Mohammad K. Azarian, Euler's Number Via Difference Equations, International Journal of Contemporary Mathematical Sciences, Vol. 7, 2012, No. 22, pp. 1095-1102. %H A073333 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. %H A073333 Don Redmond, The Evaluation of Integral_{x=0..1} floor(-ln(x)) dx, Problem #153, Advanced Problem Archive, Missouri State University. %H A073333 Michel Waldschmidt, Continued fractions, Ecole de recherche CIMPA-Oujda, Théorie des Nombres et ses Applications, 18-29 mai 2015: Oujda (Maroc). %H A073333 Eric Weisstein's World of Mathematics, Continued Fraction Constants. %H A073333 Eric Weisstein's World of Mathematics, Generalized Continued Fraction. %H A073333 Index entries for transcendental numbers. %F A073333 Equals 1/(exp(1)-1). - Joseph Biberstine (jrbibers(AT)indiana.edu), Nov 03 2004 %F A073333 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 A073333 Equals Sum_{n>=0} B_n/n!, where B_n is a Bernoulli number. - _Fredrik Johansson_, Oct 18 2006 %F A073333 1/(e-1) = 1/(1+2/(2+3/(3+4/(4+5/(5+...(continued fraction)))))). - _Philippe Deléham_, Mar 09 2013 %F A073333 Equals Integral_{x=0..oo} floor(x)*exp(-x). - _Jean-François Alcover_, Mar 20 2013 %F A073333 From _Peter Bala_, Oct 09 2013: (Start) %F A073333 Equals (1/2)*Sum_{n >= 0} 1/sinh(2^n). (Gould, equation 22). %F A073333 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 A073333 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 A073333 From _Vaclav Kotesovec_, Oct 13 2018: (Start) %F A073333 Equals A185393 - 1. %F A073333 Equals -LambertW(exp(1/(1 - exp(1))) / (1 - exp(1))). %F A073333 Equals -1 - LambertW(-1, exp(1/(1 - exp(1))) / (1 - exp(1))). (End) %F A073333 From _Gleb Koloskov_, Sep 03 2021: (Start) %F A073333 Equals (coth(1/2)-1)/2 = (A307178-1)/2. %F A073333 Equals 1/2 + 2*Integral_{x=0..oo} sin(x)/(exp(2*Pi*x)-1) dx. %F A073333 Equals 1/2 + (1/Pi)*Integral_{x=0..1} sin(log(x)/(2*Pi))/(x-1) dx. (End) %F A073333 Equals -lim_{n->oo} zeta(1-n, n)*n^(1 - n). - _Vaclav Kotesovec_ and _Peter Luschny_, Nov 05 2021 %F A073333 Equals Integral_{x=0..1} floor(-log(x)) dx (see Redmond link). - _Amiram Eldar_, Oct 03 2023 %F A073333 Equals 1/2 + Sum_{k>=2} tanh(1/2^k)/2^k. - _Antonio Graciá Llorente_, Jan 21 2024 %e A073333 0.581976706869326424385002005109011558546869301075396136266787059648... %p A073333 h:=x->sum(1/exp(n),n=1..x); evalf[110](h(1500)); evalf[110](h(4000)); %t A073333 RealDigits[N[Sum[Exp[-n], {n, 1, Infinity}], 120]][[1]] %t A073333 RealDigits[1/(E - 1), 10, 120][[1]] (* _Eric W. Weisstein_, May 08 2013 *) %o A073333 (PARI) suminf(k=1,exp(-k)) \\ _Charles R Greathouse IV_, Oct 04 2011 %o A073333 (PARI) 1/(exp(1)-1) \\ _Charles R Greathouse IV_, Oct 04 2011 %o A073333 (Magma) 1/(Exp(1) - 1); // _G. C. Greubel_, Apr 09 2018 %Y A073333 Cf. A001113, A000027, A000166, A002467, A185393, A194807, A307178. %K A073333 cons,nonn %O A073333 0,1 %A A073333 _Robert G. Wilson v_, Aug 22 2002 %E A073333 Entry revised by _N. J. A. Sloane_, Apr 07 2006 # Content is available under The OEIS End-User License Agreement: http://oeis.org/LICENSE