Authors: Aryan Phadke
Background: The harmonic sequence and the infinite harmonic series have been a topic of great interest to mathematicians for many years. The sum of the infinite harmonic series has been linked to the Euler-Mascheroni constant. It has been demonstrated by Euler that, although the sum diverges, it can be expressed as the Euler-Mascheroni constant added to the natural log of infinity. 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, a 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. The objective remains that the resultant formula is fundamentally similar to Euler's equation of the constant and the result using the Maclaurin method. Method: The principle 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 Maclaurin method from our results.
Comments: 21 Pages.
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