Geometrically convergent sequences of upper and lower bounds on the Wallis ratio and related expressions

Abstract

Sequences of algebraic upper and lower bounds on the Wallis ratio ( x + 1)/( x + 1 ) are given with the relative errors that converge to 0 geometrically and uniformly on any interval of the form (x0,1) for x0 > 1 ; moreover, the relative and absolute errors converge to 0 as x ! 1. These conclusions are based on corresponding results for the digamma function := 0 /. Relations with other relevant results are discussed, as well as the corresponding computational aspects. This work was motivated by studies of exact bounds involving the Student probability distribution. AMS 2000 subject classifications: Primary 33B15, 26D07, 26D15, 41A17; secondary 33F05, 65D20, 60E15, 62E15, 62E17. Keywords and phrases: Gamma function, digamma function, Wallis ra- tio, upper bounds, lower bounds, exact bounds, inequalities in approxima- tion, Student's distribution, Student's statistic, probability inequalities.