Overview in Summabilities: Summation Methods for Divergent Series, Ramanujan Summation and Fractional Finite Sums

Jocemar Q Chagas,José A Tenreiro Machado,António M Lopes

doi:10.3390/math9222963

Abstract

This work presents an overview of the summability of divergent series and fractional finite sums, including their connections. Several summation methods listed, including the smoothed sum, permit obtaining an algebraic constant related to a divergent series. The first goal is to revisit the discussion about the existence of an algebraic constant related to a divergent series, which does not contradict the divergence of the series in the classical sense. The well-known Euler–Maclaurin summation formula is presented as an important tool. Throughout a systematic discussion, we seek to promote the Ramanujan summation method for divergent series and the methods recently developed for fractional finite sums.

Highlights

We present some sums evaluated under specific summation method (SM) for series that are divergent in the classical sense
This work presented an overview covering a wide range of summability theories
The work started by presenting the classical summation methods for divergent series and went up to the most recent advances in the fractional summability theory

Summary

Introduction

Academic Editor: Christopher GoodrichReceived: 4 October 2021Accepted: 15 November 2021Published: 20 November 2021Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations.Licensee MDPI, Basel, Switzerland.Attribution (CC BY) license (https://creativecommons.org/licenses/by/ 4.0/).The sum was probably the first mathematical operation that humans performed and abstracted, and the properties of finite sums are well known. The symbol used to shortly represent a sum is due to L. Euler, who introduced the symbol Σ. Indeed, in 1755, he wrote:

Methods

Discussion

Conclusion