{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,2]],"date-time":"2026-02-02T05:08:56Z","timestamp":1770008936203,"version":"3.49.0"},"reference-count":142,"publisher":"MDPI AG","issue":"22","license":[{"start":{"date-parts":[[2021,11,20]],"date-time":"2021-11-20T00:00:00Z","timestamp":1637366400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Mathematics"],"abstract":"<jats:p>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\u2013Maclaurin 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.<\/jats:p>","DOI":"10.3390\/math9222963","type":"journal-article","created":{"date-parts":[[2021,11,21]],"date-time":"2021-11-21T21:00:50Z","timestamp":1637528450000},"page":"2963","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":2,"title":["Overview in Summabilities: Summation Methods for Divergent Series, Ramanujan Summation and Fractional Finite Sums"],"prefix":"10.3390","volume":"9","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-6828-3340","authenticated-orcid":false,"given":"Jocemar Q.","family":"Chagas","sequence":"first","affiliation":[{"name":"Department of Mathematics and Statistics, State University of Ponta Grossa, Ponta Grossa 84030-900, PR, Brazil"},{"name":"Department of Mechanical Engineering, Faculty of Engineering, University of Porto, 4200-465 Porto, Portugal"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-4274-4879","authenticated-orcid":false,"given":"Jos\u00e9 A. Tenreiro","family":"Machado","sequence":"additional","affiliation":[{"name":"Department of Electrical Engineering, Institute of Engineering, Polytechnic of Porto, 4249-015 Porto, Portugal"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-7359-4370","authenticated-orcid":false,"given":"Ant\u00f3nio M.","family":"Lopes","sequence":"additional","affiliation":[{"name":"LAETA\/INEGI, Faculty of Engineering, University of Porto, 4200-465 Porto, Portugal"}]}],"member":"1968","published-online":{"date-parts":[[2021,11,20]]},"reference":[{"key":"ref_1","unstructured":"Euler, L. (1755). Institutiones Calculi Differentialis cum eius usu in Analysi Finitorum ac Doctrina Serierum, Academiae Imperialis Scientiarum Petropolitanae."},{"key":"ref_2","doi-asserted-by":"crossref","unstructured":"Euler, L. (2000). Fondations of Differential Calculus, Springer.","DOI":"10.1007\/b97699"},{"key":"ref_3","unstructured":"Cajori, F. (1993). 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