Geodesic
A finite calculus approach to the partial sum and bounds of the harmonic series
Mohammad Abdur Rob1 ; Mohammad Kamal Hossen1 ; Mohammad Zakir Hossen1 ; Motiur Rahman1 ; Mohammad Abdur Rob ; Mohammad Kamal Hossen ; Mohammad Zakir Hossen ; Motiur Rahman
1Department of Computer Science and Engineering, Dhaka University of Engineering and Technology, Gazipur, Dhaka, Bangladesh
Acta mathematica Universitatis Comenianae, Tome 94 (2025) no. 3, pp. 209-224 
Mohammad Abdur Rob; Mohammad Kamal Hossen; Mohammad Zakir Hossen; Motiur Rahman; Mohammad Abdur Rob; Mohammad Kamal Hossen; Mohammad Zakir Hossen; Motiur Rahman. A finite calculus approach to the partial sum and bounds of the harmonic series. Acta mathematica Universitatis Comenianae, Tome 94 (2025) no. 3, pp. 209-224. http://geodesic.mathdoc.fr/item/AMUC_2025_94_3_a5/
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     title = { A finite calculus approach to the partial sum and bounds of the harmonic series},
     journal = {Acta mathematica Universitatis Comenianae},
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}
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Voir la notice de l'article provenant de la source Comenius University

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.