This volume studies convergence, divergence, and summation methods for sequences and series.
This volume studies convergence, divergence, and summation methods for sequences and series. It extends classical analysis with refined convergence concepts and summability techniques.
Part I. Sequences
Chapter 1. Basic Properties of Sequences
1.1 Definitions and examples 1.2 Convergence and divergence 1.3 Boundedness and monotonicity 1.4 Subsequences 1.5 Limit superior and inferior
Chapter 2. Cauchy Sequences
2.1 Definition 2.2 Completeness of ℝ 2.3 Equivalence with convergence 2.4 Examples 2.5 Applications
Chapter 3. Special Sequences
3.1 Arithmetic and geometric sequences 3.2 Recursively defined sequences 3.3 Oscillatory sequences 3.4 Examples 3.5 Applications
Part II. Series
Chapter 4. Infinite Series
4.1 Definitions 4.2 Partial sums 4.3 Convergence criteria 4.4 Divergence 4.5 Examples
Chapter 5. Tests for Convergence
5.1 Comparison tests 5.2 Ratio and root tests 5.3 Integral test 5.4 Alternating series test 5.5 Applications
Chapter 6. Absolute and Conditional Convergence
6.1 Definitions 6.2 Rearrangement of series 6.3 Riemann rearrangement theorem 6.4 Examples 6.5 Applications
Part III. Power Series
Chapter 7. Power Series
7.1 Definitions 7.2 Radius of convergence 7.3 Interval of convergence 7.4 Operations on series 7.5 Examples
Chapter 8. Analytic Functions via Series
8.1 Taylor series 8.2 Expansion of functions 8.3 Approximation 8.4 Applications 8.5 Examples
Chapter 9. Fourier Series (Overview)
9.1 Periodic functions 9.2 Fourier coefficients 9.3 Convergence issues 9.4 Applications 9.5 Examples
Part IV. Summability Theory
Chapter 10. Summability Methods
10.1 Motivation 10.2 Cesàro summation 10.3 Abel summation 10.4 Other methods 10.5 Examples
Chapter 11. Regular Summability Methods
11.1 Definitions 11.2 Matrix summability 11.3 Equivalence of methods 11.4 Applications 11.5 Examples
Chapter 12. Tauberian Theorems
12.1 Motivation 12.2 Basic results 12.3 Applications 12.4 Examples 12.5 Connections
Part V. Advanced Convergence Concepts
Chapter 13. Modes of Convergence
13.1 Pointwise convergence 13.2 Uniform convergence 13.3 Almost everywhere convergence 13.4 Convergence in measure 13.5 Examples
Chapter 14. Series of Functions
14.1 Convergence of function series 14.2 Uniform convergence tests 14.3 Interchange of limits 14.4 Applications 14.5 Examples
Chapter 15. Rearrangements and Stability
15.1 Rearrangement effects 15.2 Stability of convergence 15.3 Conditional convergence issues 15.4 Applications 15.5 Examples
Part VI. Functional and Analytical Methods
Chapter 16. Sequence Spaces
16.1 ℓᵖ spaces 16.2 Norms and metrics 16.3 Completeness 16.4 Dual spaces 16.5 Applications
Chapter 17. Transform Methods
17.1 Generating functions 17.2 Fourier transforms 17.3 Laplace transforms 17.4 Applications 17.5 Examples
Chapter 18. Operator Methods
18.1 Linear operators on sequences 18.2 Summation operators 18.3 Spectral properties 18.4 Applications 18.5 Examples
Part VII. Applications
Chapter 19. Analysis Applications
19.1 Approximation theory 19.2 Functional analysis links 19.3 Harmonic analysis 19.4 Applications 19.5 Examples
Chapter 20. Number Theory Applications
20.1 Dirichlet series 20.2 Zeta functions 20.3 Summation methods 20.4 Applications 20.5 Examples
Chapter 21. Computational Aspects
21.1 Numerical summation 21.2 Error estimation 21.3 Stability issues 21.4 Software tools 21.5 Applications
Part VIII. Research Directions
Chapter 22. Advanced Topics
22.1 Divergent series methods 22.2 Summability in functional analysis 22.3 Connections to probability 22.4 Modern developments 22.5 Emerging areas
Chapter 23. Open Problems
23.1 Convergence classification 23.2 Summability equivalence 23.3 Computational challenges 23.4 Analytical limits 23.5 Future directions
Chapter 24. Historical and Conceptual Notes
24.1 Development of series theory 24.2 Key contributors 24.3 Evolution of summability 24.4 Cross-disciplinary impact 24.5 Summary
Appendix
A. Convergence test summary B. Common series expansions C. Proof techniques checklist D. Example catalog E. Cross-reference to other MSC branches
This volume develops sequences and series as fundamental analytical tools. It emphasizes convergence behavior, summability methods, and applications across mathematics.