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%I #26 Nov 04 2022 14:43:29
%S 1,6,2,8,4,7,3,7,1,2,9,0,1,5,8,4,4,4,7,0,5,5,8,8,9,1,4,3,2,6,1,8,8,3,
%T 0,3,1,6,5,0,5,4,0,3,1,0,9,5,4,6,2,1,4,1,6,4,7,4,1,3,6,4,3,0,0,9,2,3,
%U 8,5,9,7,0,5,1,8,1,1,9,8,0,4,8,6,4,3,2,6,4,4,0,3,1,2,9,6,2,0,5,3,4,3,6,5,2
%N Decimal expansion of Sum_{n >= 1} n/(n^n).
%C From _Peter Bala_, Oct 17 2019: (Start)
%C Equals 1 + Integral_{x = 0..1} x/x^x dx. More generally, for k = 0,1,2,..., Sum_{n >= k+1} n^k/n^n = Integral_{x = 0..1} x^k/x^x dx.
%C Also equals the double integral Integral_{x = 0..1, y = 0..1} (1 + x*y)/ (x*y)^(x*y) dx dy. Cf. A073009. (End)
%C Equals Integral_{x = 0..1} (1 - x*log(x))/x^x dx. - _Peter Bala_, Jul 21 2022
%C From _Peter Bala_, Nov 02 2022: (Start)
%C Equals Integral_{x = 0..1} (1 + x*log(x)^2)/x^x dx.
%C Equals the double integral Integral_{x = 0..1, y = 0..1} (x*y*log(x*y) - 1)/( (x*y)^(x*y) * log(x*y) ) dx dy and also equals 1 - Integral_{x = 0..1, y = 0..1} x*y/( (x*y)^(x*y) * log(x*y) ) dx dy by Glasser, Theorem 1. (End)
%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.
%e 1.62847371290158444705588914326188303165054031095462141647413643009...
%p evalf(add(n/(n^n), n = 0..65), 100); # _Peter Bala_, Nov 02 2022
%t s = 0; Do[s = N[s + n/n^n, 128], {n, 62}]; RealDigits[s, 10, 111][[1]] (* _Robert G. Wilson v_, Nov 03 2004 *)
%o (PARI) suminf(n=1, 1/n^(n-1)) \\ _Michel Marcus_, Oct 21 2019
%Y Cf. A001113, A013661, A073009.
%K cons,nonn
%O 1,2
%A Joseph Biberstine (jrbibers(AT)indiana.edu), Oct 27 2004
%E More terms from _Robert G. Wilson v_, Nov 03 2004
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