Log-Convex Trapezoidal Approximation of an Elementary Integral

摘要

The definite integral over the unit interval of X to the power S, S0 is approximated by the canonical trapezoidal rule T sub n which is a function of an integer n. When this integer ranges over all the positive integers, a sequence, whose log-convexity is studied, results. The method used is an integral representation of T sub n. The sequence T sub n, n variable, is proved to be log-convex in n if the parameter S belongs to the open interval defined either by the pair of integers 1,3 or by the pair 5,7.