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

1,2

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 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.

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

Cf. A077178 (continued fraction expansion).

Cf. A083648, A229191, A245637.

Sequence in context: A094242 A199381 A083649 * A011064 A086773 A176124

Adjacent sequences: A073006 A073007 A073008 * A073010 A073011 A073012

Keyword

cons,nonn

Author

Robert G. Wilson v, Aug 03 2002

Status

approved