Concerning [an integral] and a Taylor series method
33. R. S. Anderssen, R. P. Brent, D. J. Daley and P. A. P. Moran,
and a Taylor series method,
SIAM J. Applied Mathematics 30 (1976), 22-30.
The integral of the title equals the mean distance
mk from the origin
of a point uniformly distributed over the k-dimensional
unit hypercube Ik.
Closed form expressions are given for
k = 1, 2 and 3,
while for general k,
mk is asymptotic to
Using inter alia methods from geometry,
and Taylor series expansions, several inequalities and an asymptotic series
for mk are established.
The Taylor series method also yields a slowly
converging infinite series for mk
and can be applied to more general
problems including the mean distance between two points independently
distributed at random in Ik.
For recent references, see the MathWorld
"Hypercube Line Picking" page.
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