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

TL;DR

This paper introduces new elementary-function-based formulas to approximate logarithms, factorials, and the Euler-Mascheroni constant using odd harmonic series, extending their applicability to all real numbers.

Contribution

It presents novel approximations for logarithms, factorials, and Euler-Mascheroni constant derived from odd harmonic series, with broad applicability to real numbers.

Findings

Derived a new approximation for logarithm ratios

Formulated a method to estimate factorials using harmonic series

Estimated the Euler-Mascheroni constant with elementary functions

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.