New Decompositions of Rational Fractions and Their Applications to Infinite Series

In this paper, we establish and prove a general theoretical framework for evaluating both non-alternating and alternating infinite series involving rational components of the form 1 / (x^n (b x^m + a)) and (-1)^(k-1) / (x^n (b x^m + a)) for parameters a, b in R+ and n, m, k in N+. By applying Saif Raad's Partial Fraction Identity under the structural condition k m - n = 0, we provide closed-form representations for these infinite sums in terms of special functions, specifically the Riemann Zeta function zeta(s), Dirichlet's Eta function eta(s), and the Digamma function psi(z). Furthermore, several explicit corollaries and special cases are deduced, demonstrating the analytical utility and efficiency of the proposed general identity in evaluating complex infinite series.