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

Author

Listed:
  • Jocemar Q. Chagas

    (Department of Mathematics and Statistics, State University of Ponta Grossa, Ponta Grossa 84030-900, PR, Brazil
    Department of Mechanical Engineering, Faculty of Engineering, University of Porto, 4200-465 Porto, Portugal
    These authors contributed equally to this work.)

  • José A. Tenreiro Machado

    (Department of Electrical Engineering, Institute of Engineering, Polytechnic of Porto, 4249-015 Porto, Portugal
    These authors contributed equally to this work.)

  • António M. Lopes

    (LAETA/INEGI, Faculty of Engineering, University of Porto, 4200-465 Porto, Portugal
    These authors contributed equally to this work.)

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.

Suggested Citation

  • Jocemar Q. Chagas & José A. Tenreiro Machado & António M. Lopes, 2021. "Overview in Summabilities: Summation Methods for Divergent Series, Ramanujan Summation and Fractional Finite Sums," Mathematics, MDPI, vol. 9(22), pages 1-38, November.
    • Handle: RePEc:gam:jmathe:v:9:y:2021:i:22:p:2963-:d:683848
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    Cited by:

    1. Jocemar Q. Chagas & José A. Tenreiro Machado & António M. Lopes, 2022. "Revisiting the Formula for the Ramanujan Constant of a Series," Mathematics, MDPI, vol. 10(9), pages 1-15, May.

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