A229191 -id:A229191 - cp's OEIS frontend

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

Robert G. Wilson v, Aug 03 2002

			1.291285997062663540407282590595600541498619368...

Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Section 6.11, p. 449.

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.

Cf. A077178 (continued fraction expansion).

Cf. A083648, A229191, A245637.

evalf(Sum(1/n^n, n=1..infinity), 120); # Vaclav Kotesovec, Jun 24 2016

RealDigits[N[Sum[1/n^n, {n, 1, Infinity}], 110]] [[1]]

suminf(n=1,n^-n) \\ Charles R Greathouse IV, Apr 25 2012

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)

A245637 Decimal expansion of Integral_{x = 1..infinity} 1/x^x dx.

Original entry on oeis.org

7, 0, 4, 1, 6, 9, 9, 6, 0, 4, 3, 7, 4, 7, 4, 4, 6, 0, 0, 1, 1, 4, 4, 2, 1, 0, 7, 8, 5, 7, 1, 2, 3, 8, 1, 0, 5, 8, 7, 5, 9, 7, 2, 6, 8, 6, 9, 3, 4, 5, 6, 5, 5, 5, 4, 7, 8, 2, 9, 7, 6, 1, 5, 8, 4, 6, 0, 8, 7, 0, 7, 8, 3, 8, 1, 3, 3, 1, 9, 0, 7, 5, 0, 8, 1, 3, 7, 8, 8, 6, 6, 6, 0, 0, 3, 4, 1, 6, 8, 0, 7, 3, 1, 7

Offset: 0

Jean-François Alcover, Jul 28 2014

			0.704169960437474460011442107857123810587597268693456555478297615846...

Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 4.6 Fransén-Robinson Constant, p. 263.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Peter Bala, Borel summation of a family of divergent series
M. L. Glasser, A note on Beukers's and related integrals, Amer. Math. Monthly 126(4) (2019), 361-363.
Math Overflow, Value of divergent sum Sum_{n >= 0} (-1)^n*n^n.
G. N. Watson, Theorems stated by Ramanujan (VIII): Theorems on Divergent Series, Journal of the London Mathematical Society, Volume s1-4, Issue 2, April 1929, Pages 82-86.

Cf. A073009, A083648, A229191.

NIntegrate[1/x^x, {x, 1, Infinity}, WorkingPrecision -> 104] // RealDigits // First

Equals A229191 - A073009. - Vaclav Kotesovec, Jul 28 2014
From Peter Bala, Nov 10 2019: (Start)
Equals Integral_{x = 1..oo} x*(1 + log(x))/x^x dx - 1.
Equals Integral_{x = 1..oo} x*(1 - log^2(x))/x^x dx.
Conjecturally, equals 1 - Integral_{x = 1..oo, y = 1..oo} 1/(x*y)^(x*y) dx dy. [added Dec 21 2022: follows from Glasser's Theorem 1.] (End)
From Peter Bala, Dec 21 2022: (Start)
Equals 1 - Integral_{x = 1..oo} log(x)/x^x dx (since d/d(1/x^x) = -(1 + log(x))/x^x).
Equals the Borel sum of the divergent series 1 - 1^1 + 2^2 - 3^3 + 4^4 - .... See Watson, Section 5. Compare with the convergent series 1/1^1 - 1/2^2 + 1/3^3 - 1/4^4 + ... = Integral_{x = 0..1} x^x dx. See A083648.
More generally, for nonnegative integers a and b, the divergent series Sum_{n >= 0} (-1)^n*(a*n + b)^n is Borel summable to Integral_{x = 1..oo} x^(a-b-1)/x^(x^a) dx. (End)

A319830 Decimal expansion of Integral_{0..oo} (x^(1/x-x)) dx.

Original entry on oeis.org

1, 3, 2, 0, 7, 3, 0, 4, 0, 0, 8, 6, 9, 6, 3, 6, 6, 6, 5, 4, 8, 8, 6, 1, 4, 8, 2, 7, 7, 8, 0, 7, 2, 6, 2, 0, 7, 5, 2, 3, 2, 4, 4, 7, 9, 5, 1, 8, 2, 5, 9, 6, 0, 7, 0, 6, 6, 7, 8, 7, 8, 5, 8, 5, 8, 6, 6, 3, 0, 3, 4, 9, 7, 2, 9, 7, 7, 2, 4, 3, 7, 4, 8, 1, 2, 5, 0, 3, 8, 5, 9, 2, 1, 9, 6, 6, 7, 2, 1, 7, 3, 9, 1, 7, 4

Offset: 1

Sam Coutteau, Sep 28 2018

			1.3207304008696366654886148277807262075232447951825960706678785858663...

Cf. A229191.

evalf(Int(x^(1/x-x), x = 0..infinity), 120);

RealDigits[NIntegrate[x^(1/x - x), {x, 0, Infinity}, WorkingPrecision -> 120]][[1]] (* Vaclav Kotesovec, Jan 15 2019 *)

Equals Integral_{0..oo} (x^(1/x-x)) dx (definition).
Equals Integral_{0..1} (x^(1/x-x) * (1 + 1/x^2) ) dx.
Equals Integral_{0..oo} ( (x + sqrt(x^2 + 4))/2 )^(-x) dx.

More terms from Vaclav Kotesovec, Jan 15 2019

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A073009 Decimal expansion of Sum_{n >= 1} 1/n^n.

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A245637 Decimal expansion of Integral_{x = 1..infinity} 1/x^x dx.

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A319830 Decimal expansion of Integral_{0..oo} (x^(1/x-x)) dx.

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