Overview in Summabilities: Summation Methods for Divergent Series, Ramanujan Summation and Fractional Finite Sums
Abstract
:1. Introduction
2. Divergent Series and Summation Formulae
2.1. About a General Summation Method
2.2. The Cesàro Summation Method
2.3. The Nörlund Means
2.4. The Abel Summation Method
2.5. The Euler Summation Method
2.6. The Borel Summation Methods
2.7. The Riesz Means
2.8. Some Examples
2.9. The Euler–Maclaurin Summation Formula
2.10. The Smoothed Sum Method
2.11. Additional Examples: Power Sums, Riemann Zeta Function, and Some Applications
3. Ramanujan Summation
3.1. Ramanujan Constant of a Series
3.2. The Definition of Ramanujan Summation
3.3. Some Properties of the Ramanujan Summation
3.4. About the Algebraic Framework
4. Fractional Finite Sums
4.1. Fractional Finite Sums, According to Ramanujan
4.2. Fractional Finite Sums, According to Müller and Schleicher
4.2.1. The Axioms for the Fractional Finite Sums
4.2.2. The Unique Possible Definition for the Fractional Finite Sums
4.2.3. Some Examples and Applications
4.3. Fractional Finite Sums for More General Functions
- Simple finite sum (SFS): sums of type
- Composite finite sum (CFS): sums of type
- Oscillatory simple finite sum (OSFS): sums of type
- Oscillatory composite finite sum (OCFS): sums of type
4.3.1. Simple Finite Sums
4.3.2. Composite Finite Sum
4.3.3. The Generalized Definition of Series
4.3.4. Oscillatory Simple Finite Sums
4.3.5. Oscillatory Composite Finite Sums
4.3.6. Methods to Evaluate Fractional Finite Sums
5. Discussion
6. Conclusions
Author Contributions
Funding
Institutional Review Board Statement
Informed Consent Statement
Acknowledgments
Conflicts of Interest
Abbreviations
CFS | Composite finite sum |
EMSF | Euler–Maclaurin summation formula |
EBSF | Euler–Boole summation formula |
FSF | Fractional summable function |
FFS | Fractional finite sum |
FFSF | Fundamental fractional summation formula |
OCFS | Oscillatory composite finite sum |
OSFS | Oscillatory simple finite sum |
RCS | Ramanujan constant of a series |
RS | Ramanujan summation |
SFS | Simple finite sum |
SM | Summation method |
WKB | Wentzel-Kramers-Brillouin |
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Chagas, J.Q.; Machado, J.A.T.; Lopes, A.M. Overview in Summabilities: Summation Methods for Divergent Series, Ramanujan Summation and Fractional Finite Sums. Mathematics 2021, 9, 2963. https://doi.org/10.3390/math9222963
Chagas JQ, Machado JAT, Lopes AM. Overview in Summabilities: Summation Methods for Divergent Series, Ramanujan Summation and Fractional Finite Sums. Mathematics. 2021; 9(22):2963. https://doi.org/10.3390/math9222963
Chicago/Turabian StyleChagas, Jocemar Q., José A. Tenreiro Machado, and António M. Lopes. 2021. "Overview in Summabilities: Summation Methods for Divergent Series, Ramanujan Summation and Fractional Finite Sums" Mathematics 9, no. 22: 2963. https://doi.org/10.3390/math9222963