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

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 discussi...

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Autores principales:	Jocemar Q. Chagas, José A. Tenreiro Machado, António M. Lopes
Formato:	article
Lenguaje:	EN
Publicado:	MDPI AG 2021
Materias:	divergent series summation methods Euler–Maclaurin summation formula Ramanujan summation fractional finite sum Mathematics QA1-939
Acceso en línea:	https://doaj.org/article/8c4aa85df2094ef99a20f62dfad2d231
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spelling	oai:doaj.org-article:8c4aa85df2094ef99a20f62dfad2d2312021-11-25T18:17:37ZOverview in Summabilities: Summation Methods for Divergent Series, Ramanujan Summation and Fractional Finite Sums10.3390/math92229632227-7390https://doaj.org/article/8c4aa85df2094ef99a20f62dfad2d2312021-11-01T00:00:00Zhttps://www.mdpi.com/2227-7390/9/22/2963https://doaj.org/toc/2227-7390This 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.Jocemar Q. ChagasJosé A. Tenreiro MachadoAntónio M. LopesMDPI AGarticledivergent seriessummation methodsEuler–Maclaurin summation formulaRamanujan summationfractional finite sumMathematicsQA1-939ENMathematics, Vol 9, Iss 2963, p 2963 (2021)
institution	DOAJ
collection	DOAJ
language	EN
topic	divergent series summation methods Euler–Maclaurin summation formula Ramanujan summation fractional finite sum Mathematics QA1-939
spellingShingle	divergent series summation methods Euler–Maclaurin summation formula Ramanujan summation fractional finite sum Mathematics QA1-939 Jocemar Q. Chagas José A. Tenreiro Machado António M. Lopes Overview in Summabilities: Summation Methods for Divergent Series, Ramanujan Summation and Fractional Finite Sums
description	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.
format	article
author	Jocemar Q. Chagas José A. Tenreiro Machado António M. Lopes
author_facet	Jocemar Q. Chagas José A. Tenreiro Machado António M. Lopes
author_sort	Jocemar Q. Chagas
title	Overview in Summabilities: Summation Methods for Divergent Series, Ramanujan Summation and Fractional Finite Sums
title_short	Overview in Summabilities: Summation Methods for Divergent Series, Ramanujan Summation and Fractional Finite Sums
title_full	Overview in Summabilities: Summation Methods for Divergent Series, Ramanujan Summation and Fractional Finite Sums
title_fullStr	Overview in Summabilities: Summation Methods for Divergent Series, Ramanujan Summation and Fractional Finite Sums
title_full_unstemmed	Overview in Summabilities: Summation Methods for Divergent Series, Ramanujan Summation and Fractional Finite Sums
title_sort	overview in summabilities: summation methods for divergent series, ramanujan summation and fractional finite sums
publisher	MDPI AG
publishDate	2021
url	https://doaj.org/article/8c4aa85df2094ef99a20f62dfad2d231
work_keys_str_mv	AT jocemarqchagas overviewinsummabilitiessummationmethodsfordivergentseriesramanujansummationandfractionalfinitesums AT joseatenreiromachado overviewinsummabilitiessummationmethodsfordivergentseriesramanujansummationandfractionalfinitesums AT antoniomlopes overviewinsummabilitiessummationmethodsfordivergentseriesramanujansummationandfractionalfinitesums
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Overview in Summabilities: Summation Methods for Divergent Series, Ramanujan Summation and Fractional Finite Sums

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