OFFSET
1,2
LINKS
Kenny Lau, Table of n, a(n) for n = 1..10001
Johan Bernoulli, Demonstratio Methodi Analyticae, qua usus est pro determinanda aliqua Quadratura exponentiali per seriem, Actis Eruditorum A (1697), p. 131. Collected in Opera Omnia, vol. 3, 1742. See p. 376ff.
M. L. Glasser, A note on Beukers's and related integrals, Amer. Math. Monthly 126(4) (2019), 361-363.
Jaroslav HanĨl and Simon Kristensen, Metrical irrationality results related to values of the Riemann zeta-function, arXiv:1802.03946 [math.NT], 2018.
Randall Munroe, Approximations, xkcd Web Comic #1047, Apr 25 2012.
Simon Plouffe, Sum(1/n^n, n=1..infinity). [internet archive]
Eric Weisstein's World of Mathematics, Power Tower.
Eric Weisstein's World of Mathematics, Sophomore's Dream.
FORMULA
Equals Integral_{x = 0..1} dx/x^x.
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
Approximately log(3)^e, see Munroe link. - Charles R Greathouse IV, Apr 25 2012
Another approximation is A + A^(-19), where A is Glaisher-Kinkelin constant (A074962). - Noam Shalev, Jan 16 2015
From Petros Hadjicostas, Jun 29 2020: (Start)
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.)
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)
EXAMPLE
1.291285997062663540407282590595600541498619368...
MAPLE
evalf(Sum(1/n^n, n=1..infinity), 120); # Vaclav Kotesovec, Jun 24 2016
MATHEMATICA
RealDigits[N[Sum[1/n^n, {n, 1, Infinity}], 110]] [[1]]
PROG
(PARI) suminf(n=1, n^-n) \\ Charles R Greathouse IV, Apr 25 2012
CROSSREFS
KEYWORD
cons,nonn
AUTHOR
Robert G. Wilson v, Aug 03 2002
STATUS
approved