Approximation Of Logarithm, Factorial And Euler Mascheroni Constant Using Odd Harmonic Series

Abstract

We have proved in this paper that natural logarithm of consecutive number ratio, x/(x-1) approximates to 2/(2x - 1) where x is a real number except 1. Using this relation, we, then proved, x approximates to double the sum of odd harmonic series having first and last terms 1/3 and 1/(2x - 1) respectively. Thereafter, not limiting to consecutive number ratios, we extended its applicability to all the real numbers. Based on these relations, we, then derived a formula for approximating the value of Factorial x. We could also approximate the value of Euler-Mascheroni constant. In these derivations, we used only and only elementary functions, thus this paper is easily comprehensible to students and scholars.