The Harmonic Series and the nth Term Test for Divergence

TL;DR

This paper demonstrates the divergence of the harmonic series using a comparison with a telescoping series and a classical inequality, providing a clear proof of divergence.

Contribution

It offers a straightforward proof of the harmonic series divergence using the inequality x>=log(1+x) and comparison with a telescoping series.

Findings

Proves harmonic series diverges using comparison with a telescoping series.

Utilizes the inequality x>=log(1+x) for the proof.

Establishes the divergence through natural logarithm growth.

Abstract

The divergence of the harmonic series is proved by direct comparison with a series whose nth partial sum telescopes to the natural logarithm of n. The key idea is to apply the classical inequality x>=log(1+x) (valid for x>-1) with x=1/k and sum over k, 1<=k<=n-1.