OFFSET
0,1
COMMENTS
The definition of hypervolume for a ball of radius r, generalized to continuous dimension d, is given by ((Pi^(d/2))*(r^d))/Gamma((d/2) + 1). Assigning r = 1/2, the d > 0 which maximizes this formula is the non-integral real number 0.4765825... whose digits form this sequence.
FORMULA
Maximizing ((Pi^(d/2))*((1/2)^d))/Gamma((d/2) + 1) for d>0 we obtain a volume of 1.0386933280526... when d equals the positive real root of the derivative: ((2^(-1-d))*(Pi^(d/2))*((log(4*Pi) + PolyGamma(0, 1+d/2))))/(Gamma(1+d/2)). - Corrected by Eric R. Carter, May 09 2019
EXAMPLE
d = 0.47658258230608529520761576885882324030164...
MATHEMATICA
RealDigits[d/.FindRoot[Log[4/Pi] + PolyGamma[0, 1 + d/2], {d, 1}, WorkingPrecision -> 200]][[1]]
CROSSREFS
KEYWORD
nonn,cons
AUTHOR
Eric R. Carter, Nov 13 2016
STATUS
approved