A finite calculus approach to the partial sum and bounds of the harmonic series

Abstract

This study develops a new general partial summation formula for harmonic series, utilizing finite calculus techniques. The formula provides highly accurate upper and lower bounds without a correction term within which the exact partial sum lies. Additionally, an improved approximation formula for the summation of the harmonic series is introduced. The proposed general formula offers a straightforward and precise method for summing harmonic series, but has an unsolved term. The derived bounds are the closest to the exact partial sum, marking a significant advancement in the field. Comparisons with Euler's formula, based on general and RMS errors, demonstrate that the proposed approximation formula achieves superior accuracy. These findings bring a novel approach to harmonic series analysis, with potential applications in numerical analysis and theoretical research.