|
%I #20 Nov 03 2022 07:45:07
%S 1,2,1,4,8,3,7,9,9,6,0,1,7,1,6,2,7,0,0,6,9,1,1,2,0,5,2,4,8,0,2,4,2,1,
%T 2,2,2,2,3,8,2,7,3,8,8,4,9,0,5,5,6,1,1,9,9,9,4,6,1,4,2,2,9,5,2,1,1,1,
%U 4,1,3,7,5,2,4,0,0,3,7,7,1,0,5,9,1,2,1,2,4,0,0,7,7,8,8,7,4,2,1,8,3,8,1
%N Decimal expansion of the triple integral Integral_{z = 0..1} Integral_{y = 0..1} Integral_{x = 0..1} 1/(x*y*z)^(x*y*z) dx dy dz.
%C Cf. A209059. The double integral Integral_{y = 0..1} Integral_{x = 0..1} 1/(x*y)^(x*y) dx dy equals Integral_{x = 0..1} 1/x^x dx, which is listed as A073009.
%F The triple integral is most conveniently estimated from the identity Integral_{z = 0..1} Integral_{y = 0..1} Integral_{x = 0..1} 1/(x*y*z)^(x*y*z) dx dy dz = 1/2*Sum_{n = 1..oo} (1/n^n + 1/n^(n+1)).
%e 1.21483799601716270069...
%t digits = 103; 1/2*NSum[ (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 *)
%o (PARI) default( realprecision, 105); v = Vec( Str( suminf( n=1, n^-n + n^-(n+1)) / 20)); for( n=3, 105, print1( v[n],",")); /* _Michael Somos_, Mar 07 2012 */
%Y Cf. A073009, A083648, A135608, A209059.
%K nonn,easy,cons
%O 1,2
%A _Peter Bala_, Mar 04 2012
|
