## Concerning [an integral] and a Taylor series method

33. R. S. Anderssen, R. P. Brent, D. J. Daley and P. A. P. Moran, Concerning and a Taylor series method, SIAM J. Applied Mathematics 30 (1976), 22-30. MR 52#15773.

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## Abstract

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 (k/3)1/2. Using inter alia methods from geometry, Cauchy-Schwarz inequalities 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.