Multiple-correction and summation of the rational series

TL;DR

This paper develops a systematic method to find closed forms and continued fractions for rational series, applying it to Mathieu series and others using Ramanujan's formulas.

Contribution

Introduces a novel systematic approach for deriving closed forms and continued fractions of rational series, utilizing multiple-correction and Ramanujan's continued fraction formulas.

Findings

Derived continued fraction representations for the alternating Mathieu series.

Obtained closed form expressions for specific rational series.

Demonstrated the effectiveness of the method on various series.

Abstract

The goal of this work is to formulate a systematical method for looking for the simple closed form or continued fraction representation of a class of rational series. As applications, we obtain the continued fraction representations for the alternating Mathieu series and some rational series. The main tools are multiple-correction and two of Ramanujan's continued fraction formulae involving the quotient of the gamma functions.