A073009 Decimal expansion of Sum_{n >= 1} 1/n^n.
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, 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, 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
Offset: 1
Examples
1.291285997062663540407282590595600541498619368...
References
- Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Section 6.11, p. 449.
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 Hančl 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.
Programs
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Maple
evalf(Sum(1/n^n, n=1..infinity), 120); # Vaclav Kotesovec, Jun 24 2016
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Mathematica
RealDigits[N[Sum[1/n^n, {n, 1, Infinity}], 110]] [[1]] -
PARI
suminf(n=1,n^-n) \\ Charles R Greathouse IV, Apr 25 2012
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)