Approximation of Sum of Harmonic Progression using Approximation Theory

Background : Sum of Harmonic Progression is an old problem. While a few complex approximations have surfaced, a simple and efficient formula hasn't. Most of the previous formulas use summation as a tool for the approximation. Also, most notable formulas are only applicable to the harmonic progression where the first term and common difference of the corresponding Arithmetic progression is one. Aim of this paper is to create a formula without using 'summation' as a tool. On the contrary to previous formulas, the resultant formula is applicable to every harmonic progression regardless of the first term and common difference of its corresponding arithmetic progression. The resultant formula is applicable for numerical values in a harmonic progression. Algebraic terms in a harmonic progression do not value to any numeric terms and for that reason cannot be accounted for the approximation. The resultant formula has a variability in terms of accuracy and complexity to suit different users with different priorities. The Formula was obtained by using the fundamentals of approximation theory by equating the area enclosed in the graph of harmonic progression and area under the curve of equation (y=1/x). The logic being that the harmonic progression is completely inclusive in the function (y=1/x) is used to formulate the answer.