A209059 Decimal expansion of the triple integral Integral_{z = 0..1} Integral_{y = 0..1} Integral_{x = 0..1} (x*y*z)^(x*y*z) dx dy dz.

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

Offset: 0

Author

Peter Bala, Mar 04 2012

Keywords

• nonn
• easy
• cons

Comments

The double integral Integral_{y = 0..1} Integral_{x = 0..1} (x*y)^(x*y) dx dy equals Integral_{x = 0..1} x^x dx, which is listed as A083648.

Examples

			0.83493011063622359351...
		

Crossrefs

Cf. A073009, A083648, A209060.

Programs

Mathematica

digits = 103; 1/2*NSum[ (-1)^(n+1)*(1/n^n + 1/n^(n+1)), {n, 1, Infinity}, WorkingPrecision -> digits+10, NSumTerms -> 100] // RealDigits[#, 10, digits]& // First (* Jean-François Alcover, Feb 15 2013, from formula *)

Formula

The triple integral is most conveniently estimated from the identity Integral_{z = 0..1} Integral_{y = 0..1} Integral_{z = 0..1} (x*y*z)^(x*y*z) dx dy dz = (1/2)*Sum_{n >= 1} (-1)^(n+1)*(1/n^n + 1/n^(n+1)).