Approximation of Harmonic Series

Background : Harmonic Series is the sum of Harmonic Progression. There have been multiple formulas to approximate the harmonic series, from Euler's formula to even a few in the 21st Century. Mathematicians have concluded that the sum cannot be calculated, however any approximation better than the previous others is always needed. In this paper we will discuss the flaws in Euler's formula for approximation of harmonic series and provide a better formula. We will also use the infinite harmonic series to determine the approximations of finite harmonic series using the Euler-Mascheroni constant. We will also look at the Leibniz series for Pi and determine the correction factor that Leibniz discussed in his paper which he found using Euler numbers. Each subsequent approximation we find in this paper is better than all previous ones. Different approximations for different types of harmonic series are calculated, best fit for the given type of harmonic series. The correction factor for Leibniz series might not provide any applied results but it is a great way to ponder some other infinite harmonic series.