%I #71 Feb 18 2024 08:21:41
%S 1,2,9,1,2,8,5,9,9,7,0,6,2,6,6,3,5,4,0,4,0,7,2,8,2,5,9,0,5,9,5,6,0,0,
%T 5,4,1,4,9,8,6,1,9,3,6,8,2,7,4,5,2,2,3,1,7,3,1,0,0,0,2,4,4,5,1,3,6,9,
%U 4,4,5,3,8,7,6,5,2,3,4,4,5,5,5,5,8,8,1,7,0,4,1,1,2,9,4,2,9,7,0,8,9,8,4,9,9
%N Decimal expansion of Sum_{n >= 1} 1/n^n.
%H Kenny Lau, <a href="/A073009/b073009.txt">Table of n, a(n) for n = 1..10001</a>
%H Johan Bernoulli, Demonstratio Methodi Analyticae, qua usus est pro determinanda aliqua Quadratura exponentiali per seriem, Actis Eruditorum A (1697), p. 131. Collected in <a href="https://archive.org/details/johannisbernoul00berngoog">Opera Omnia, vol. 3</a>, 1742. See <a href="https://archive.org/stream/johannisbernoul00berngoog#page/n406/mode/2up">p. 376ff</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 Jaroslav HanĨl and Simon Kristensen, <a href="https://arxiv.org/abs/1802.03946">Metrical irrationality results related to values of the Riemann zeta-function</a>, arXiv:1802.03946 [math.NT], 2018.
%H Randall Munroe, <a href="http://xkcd.com/1047/">Approximations</a>, xkcd Web Comic #1047, Apr 25 2012.
%H Simon Plouffe, <a href="https://web.archive.org/web/20150913023027/http://www.worldwideschool.org/library/books/sci/math/MiscellaneousMathematicalConstants/chap87.html">Sum(1/n^n, n=1..infinity)</a>. [internet archive]
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/PowerTower.html">Power Tower</a>.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/SophomoresDream.html">Sophomore's Dream</a>.
%F Equals Integral_{x = 0..1} dx/x^x.
%F Constant also equals the double integral Integral_{y = 0..1} Integral_{x = 0..1} 1/(x*y)^(x*y) dx dy. - _Peter Bala_, Mar 04 2012
%F Approximately log(3)^e, see Munroe link. - _Charles R Greathouse IV_, Apr 25 2012
%F Another approximation is A + A^(-19), where A is Glaisher-Kinkelin constant (A074962). - _Noam Shalev_, Jan 16 2015
%F From _Petros Hadjicostas_, Jun 29 2020: (Start)
%F Equals -Integral_{x=0..1, y=0..1} dx dy/((x*y)^(x*y)*log(x*y)). (Apply Theorem 1 or Theorem 2 of Glasser (2019) to the integral Integral_{x = 0..1} dx/x^x.)
%F Equals -Integral_{x=0..1} log(x)/x^x dx. (Apply Theorem 1 or Theorem 2 of Glasser (2019) to the double integral of _Peter Bala_ above.) (End)
%e 1.291285997062663540407282590595600541498619368...
%p evalf(Sum(1/n^n, n=1..infinity), 120); # _Vaclav Kotesovec_, Jun 24 2016
%t RealDigits[N[Sum[1/n^n, {n, 1, Infinity}], 110]] [[1]]
%o (PARI) suminf(n=1,n^-n) \\ _Charles R Greathouse IV_, Apr 25 2012
%Y Cf. A077178 (continued fraction expansion).
%Y Cf. A083648, A229191, A245637.
%K cons,nonn
%O 1,2
%A _Robert G. Wilson v_, Aug 03 2002