Generalized Harmonic Progression

Abstract

This paper presents formulae for the sum of the terms of a harmonic progression of order $k$ with integer parameters, $\mathrm{HP}_k(n)$, and for the partial sums of its two associated Fourier series, $C^z_{k}(a,b,n)$ and $S^z_{k}(a,b,n)$. $\mathrm{HP}_k(n)$ is built from the ground up, with a power series for $1/(aj+b)^k$ that is summed over $j$ using Faulhaber's formula. These new formulae are a generalization of the formulae created in a previous paper and were achieved using a slightly modified version of the reasoning employed before.