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A354014 Decimal expansion of Sum_{n>0} u(n) where u(n) is the unique positive solution to the equation Integral_{u(n)..1} e^t/t dt = n.

%I A354014 #23 May 22 2022 12:25:58
%S A354014 1,2,4,9,0,0,7,7,3,2,9,5,7,8,2,0,5,6,7,8,4,9,7,7,1,8,4,9,8,3,1,5,4,1,
%T A354014 4,5,5,2,5,9,2,5,9,6,9,9,3,7,5,6,6,4,4,0,4,4,0,6,9,3,7,2,1,2,3,2,3,5,
%U A354014 4,5,1,0,7,8,5,7,5,7,7,2,6,9,2,3,7,1,9,2,0,9,3,8,4,3,9,4,8,5,5,9,5,6,7,7,8
%N A354014 Decimal expansion of Sum_{n>0} u(n) where u(n) is the unique positive solution to the equation Integral_{u(n)..1} e^t/t dt = n.
%C A354014 Near infinity, u(n) ~ e^(lambda)/e^n, with lambda = A229837 = Integral_{t=0..1} (e^t-1)/t dt, so this series Sum_{n>0} u(n) is convergent.
%D A354014 Jean-Marie Monier, Analyse, Tome 3, 2ème année, MP.PSI.PC.PT, Dunod, 1997, Exercice 3.2.30 pp. 252 and 450-451.
%H A354014 Amiram Eldar, <a href="/A354014/a354014.jpg">Plot of u(n) for n=1..10</a>
%H A354014 Amiram Eldar, <a href="/A354014/a354014_1.jpg">Logarithmic plot of u(n) for n=1..10</a>
%H A354014 Amiram Eldar, <a href="/A354014/a354014_2.jpg">Plot of Sum_{i=1..n} u(i) for n=1..10</a>
%e A354014 1.24900773295782056784977184983154145525925969937566...
%o A354014 (PARI) N = 100;
%o A354014 default(realprecision, N);
%o A354014 u(n) = {my(integ = intformal(sum(k=1, N, x^(k-1)/k!), x)); solve(y=1./10^N, 1, subst(integ, x, 1) - log(y) - subst(integ, x, y) - n);}
%o A354014 sum(k=1, N, u(k)) \\ _Michel Marcus_, May 20 2022
%Y A354014 Cf. A229837.
%K A354014 nonn,cons
%O A354014 1,2
%A A354014 _Bernard Schott_, May 14 2022
%E A354014 More terms from _Amiram Eldar_, May 14 2022