Harmonic numbers as the summation of integrals

TL;DR

This paper explores harmonic numbers, presenting a novel integral representation involving exponential and hyperbolic secant functions, with proofs based on induction and integration by parts.

Contribution

It introduces a new integral-based perspective on harmonic numbers, expanding their analytical understanding.

Findings

Harmonic numbers can be expressed as integrals involving exponential and hyperbolic secant functions.

The proof of this representation uses simple techniques like induction and integration by parts.

The approach provides a new analytical tool for studying harmonic numbers.

Abstract

Harmonic numbers arise from the truncation of the harmonic series. The $n^\text{th}$ harmonic number is the sum of the reciprocals of each positive integer up to $n$. In addition to briefly introducing the properties of harmonic numbers, we cover harmonic numbers as the summation of integrals that involve the product of exponential and hyperbolic secant functions. The proof is relatively simple since it only comprises the Principle of Mathematical Induction and integration by parts.