‎Exploring the Harmonic Series by ‎$L$-Summing ‎Method and a Telescoping Summation Method

Document Type : Original Article

Author

Mehdi Hassani

‎Department of Mathematics‎, ‎Faculty of Sciences‎, ‎University of Zanjan‎, ‎‎Zanjan‎, ‎Iran

10.22034/kjm.2025.541247.3647

Abstract

In this note, we discuss the origin and significance of the ‎$‎L‎$‎-summing method and introduce a telescoping summation method based on an identity due to Anastase and D'{i}az-Barrero. A key feature of these approaches is their implementation in Maple, enabling the derivation of several nontrivial summation identities that Maple cannot compute directly. Of particular interest are the examples involving the classical series
\[‎
‎\zeta(s)=\sum_{k=1}^\infty\frac{1}{k^s},\quad‎
‎\zeta_n(s)=\sum_{k=1}^n\frac{1}{k^s},\quad‎
‎H_n=\zeta_n(1)=\sum_{k=1}^n\frac{1}{k}‎,
‎\]‎
for which we establish several new identities.‎

Keywords

$L$-summing ‎method

telescoping summation

harmonic number

Riemann zeta function

Subjects

• 05 Combinatorics

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