Formulas for the Sums of the Series of Reciprocals of the Polynomial of Degree Two with Non-zero Integer Roots

Abstract

AbstractThis chapter deals with the sums of the series of reciprocals of the quadratic polynomials with non-zero integer roots. It is a follow-up and a completion to the previous author’s papers dealing with the sums of these series, where the quadratic polynomials have all possible types of positive and negative integer roots. First, the conditions for the coefficients of a reduced quadratic equation are given so that this equation has only integer roots. Further, summary formulas for the sums of the series of reciprocals of the quadratic polynomials with non-zero integer roots are stated and derived. These formulas are verified by some examples using the basic programming language of the computer algebra system Maple. The series we deal with so belong to special types of infinite series which sums are given analytically by simple formulas.KeywordsSum of the seriesHarmonic numberGeneralized harmonic numberRoots of the reduced quadratic equationTelescoping seriesComputer algebra system maple