An infinite series for the natural logarithm that converges throughout its domain and makes concavity transparent

TL;DR

This paper introduces an infinite series representation of the natural logarithm that converges everywhere on its domain and clearly demonstrates its concavity, enhancing understanding of its fundamental properties.

Contribution

It presents a new infinite series for the natural logarithm that converges throughout its domain and explicitly reveals its concavity, simplifying theoretical analysis.

Findings

Series converges for all positive real numbers

Concavity of the natural logarithm is made transparent

Provides a new perspective on the fundamental inequality

Abstract

The natural logarithm can be represented by an infinite series that converges for all positive real values of the variable, and which makes concavity patently obvious. Concavity of the natural logarithm is known to imply, among other things, the fundamental inequality between the arithmetic and geometric mean.