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A185393 Decimal expansion of e/(e-1) = 1 + 1/e + 1/e^2 + ...

%I A185393 #49 Jul 20 2025 14:49:02
%S A185393 1,5,8,1,9,7,6,7,0,6,8,6,9,3,2,6,4,2,4,3,8,5,0,0,2,0,0,5,1,0,9,0,1,1,
%T A185393 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,
%U A185393 4,3,8,1,7,3,9,1,6,6,9,7,4,3,2,8,7,2,0
%N A185393 Decimal expansion of e/(e-1) = 1 + 1/e + 1/e^2 + ...
%D A185393 Jean-Marie Monier, Analyse, Exercices corrigés, 2ème année, MP, Dunod, 1997, Exercice 3.3.29.a) pp. 286 and 307.
%H A185393 G. C. Greubel, <a href="/A185393/b185393.txt">Table of n, a(n) for n = 1..10000</a>
%H A185393 Wikipedia, <a href="https://en.wikipedia.org/wiki/Tannery%27s_theorem">Tannery's theorem</a>.
%H A185393 <a href="/index/Tra#transcendental">Index entries for transcendental numbers</a>
%F A185393 Equals Sum_{n>=0} 1/exp(n). - _Vaclav Kotesovec_, Jan 30 2015
%F A185393 From _Vaclav Kotesovec_, Oct 13 2018: (Start)
%F A185393 Equals 1 - LambertW(exp(1/(1 - exp(1))) / (1 - exp(1))).
%F A185393 Equals -LambertW(-1, exp(1/(1 - exp(1))) / (1 - exp(1))).
%F A185393 (End)
%F A185393 Equals Sum_{k>=0} (-1)^k*B(k)/k!, where B(k) is the k-th Bernoulli number. - _Amiram Eldar_, May 08 2021
%F A185393 Equals Integral_{x=0..oo} exp(-floor(x)) dx (Monier). - _Bernard Schott_, May 08 2022
%F A185393 Equals lim_{n->oo} Sum_{k=1..n} (k/n)^n (via Tannery's theorem). - _Stoyan Apostolov_, May 24 2022
%e A185393 1.58197670686932642438500200510901155854686930107539613626678705964804...
%t A185393 RealDigits[E/(E - 1), 10, 100][[1]] (* _G. C. Greubel_, Jun 29 2017 *)
%o A185393 (PARI) exp(1)/(exp(1)-1)
%o A185393 (Python)
%o A185393 from sympy import E
%o A185393 print(list(map(int, str((E/(E-1)).n(88))[:-1].replace(".", "")))) # _Michael S. Branicky_, May 25 2022
%Y A185393 Cf. A254445, A254446, A320349, A320350.
%Y A185393 Apart from 1st digit the same as A073333.
%K A185393 nonn,cons,easy
%O A185393 1,2
%A A185393 _Charles R Greathouse IV_, Mar 20 2012