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%I #32 Dec 27 2022 14:53:19
%S 7,0,4,1,6,9,9,6,0,4,3,7,4,7,4,4,6,0,0,1,1,4,4,2,1,0,7,8,5,7,1,2,3,8,
%T 1,0,5,8,7,5,9,7,2,6,8,6,9,3,4,5,6,5,5,5,4,7,8,2,9,7,6,1,5,8,4,6,0,8,
%U 7,0,7,8,3,8,1,3,3,1,9,0,7,5,0,8,1,3,7,8,8,6,6,6,0,0,3,4,1,6,8,0,7,3,1,7
%N Decimal expansion of Integral_{x = 1..infinity} 1/x^x dx.
%D Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 4.6 Fransén-Robinson Constant, p. 263.
%H Vincenzo Librandi, <a href="/A245637/b245637.txt">Table of n, a(n) for n = 0..1000</a>
%H Peter Bala, <a href="/A245637/a245637.pdf">Borel summation of a family of divergent series</a>
%H M. L. Glasser, <a href="https://doi.org/10.1080/00029890.2019.1565856">A note on Beukers's and related integrals</a>, Amer. Math. Monthly 126(4) (2019), 361-363.
%H Math Overflow, <a href="https://mathoverflow.net/questions/403347/value-of-divergent-sum-sum-n-0-infty-1n-nn?rq=1">Value of divergent sum Sum_{n >= 0} (-1)^n*n^n</a>.
%H G. N. Watson, <a href="https://doi.org/10.1112/jlms/s1-4.14.82">Theorems stated by Ramanujan (VIII): Theorems on Divergent Series</a>, Journal of the London Mathematical Society, Volume s1-4, Issue 2, April 1929, Pages 82-86.
%F Equals A229191 - A073009. - _Vaclav Kotesovec_, Jul 28 2014
%F From _Peter Bala_, Nov 10 2019: (Start)
%F Equals Integral_{x = 1..oo} x*(1 + log(x))/x^x dx - 1.
%F Equals Integral_{x = 1..oo} x*(1 - log^2(x))/x^x dx.
%F Conjecturally, equals 1 - Integral_{x = 1..oo, y = 1..oo} 1/(x*y)^(x*y) dx dy. [added Dec 21 2022: follows from Glasser's Theorem 1.] (End)
%F From _Peter Bala_, Dec 21 2022: (Start)
%F Equals 1 - Integral_{x = 1..oo} log(x)/x^x dx (since d/d(1/x^x) = - (1 + log(x)/x^x).
%F Equals the Borel sum of the divergent series 1 - 1^1 + 2^2 - 3^3 + 4^4 - .... See Watson, Section 5. Compare with the convergent series 1/1^1 - 1/2^2 + 1/3^3 - 1/4^4 + ... = Integral_{x = 0..1} x^x dx. See A083648.
%F More generally, for nonnegative integers a and b, the divergent series Sum_{n >= 0} (-1)^n*(a*n + b)^n is Borel summable to Integral_{x = 1..oo} x^(a-b-1)/x^(x^a) dx. (End)
%e 0.704169960437474460011442107857123810587597268693456555478297615846...
%t NIntegrate[1/x^x, {x, 1, Infinity}, WorkingPrecision -> 104] // RealDigits // First
%Y Cf. A073009, A083648, A229191.
%K nonn,cons,easy
%O 0,1
%A _Jean-François Alcover_, Jul 28 2014
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