Summing series using recurrence relations

TL;DR

This paper presents a method for summing power series whose terms follow linear recurrence relations with polynomial coefficients by converting them into differential or algebraic equations, enabling closed-form solutions.

Contribution

It introduces a novel approach to sum such series by linking recurrence relations to differential or algebraic equations, solving them for closed-form expressions.

Findings

Successfully sums several series using the method

Solves two problems from the American Mathematical Monthly

Provides a systematic approach for series summation

Abstract

Power series in which the summand satisfies a linear recurrence relation with polynomial coefficients are shown to be the solution of a linear differential or algebraic equation. Solving the associated differential or algebraic equation yields a closed form for the series. This method is used to sum several series and to solve two {\it American Mathematical Monthly} problems.