Approximation of Sum of Harmonic Progression

Aryan Phadke

Abstract: Background: The harmonic sequence and the sum of infinite harmonic series are topics of great interest in mathematics. The sum of the infinite harmonic series has been linked to the Euler-Mascheroni constant. It has been demonstrated that, although the sum diverges, it can be expressed as the Euler-Mascheroni constant added to the natural log of infinity. By utilizing the Euler-Maclaurin method, we can extend the expression to approximate the sum of finite harmonic series with a fixed first term and a variable last term. However, natural extension is not possible for a variable value of the first term or the common difference of the reciprocals. Aim: The aim of this paper is to create a formula that generates an approximation of the sum of a harmonic progression for a variable first term and common difference. An objective remains that the resultant formula is fundamentally similar to Euler's equation of the constant and the result using the method. Method: The principal result of the paper is derived using approximation theory. The assertion that the graph of harmonic progression closely resembles the graph of y=1/x is key. The subsequent results come through a comparative view of Euler's expression and by using numerical manipulations on the Euler-Mascheroni Constant. Results: We created a general formula that approximates the sum of harmonic progression with variable components. Its fundamental nature is apparent because we can derive the results of the method from our results.

Keywords: Approximation theory, Harmonic progression, Euler-Mascheroni constant, Harmonic series

How to Cite?: Aryan Phadke, "Approximation of Sum of Harmonic Progression", Volume 13 Issue 1, January 2024, International Journal of Science and Research (IJSR), Pages: 1101-1107, https://www.ijsr.net/getabstract.php?paperid=SR24115160243, DOI: https://dx.doi.org/10.21275/SR24115160243