40B05 Multiple sequences and series

Overview

40B05 develops multiple sequences and series in multiple sequences and series. Core themes include criteria for convergence behavior, structural inequalities, and summability principles used to characterize limiting behavior of sequences and series.

Related Wikipedia Page

Wikipedia search: Multiple sequences and series

Useful Links

• Wikipedia search: Multiple sequences and series
• Encyclopaedia of Mathematics
• arXiv: Classical Analysis and ODEs (math.CA)
• Analysis Mathematica
• Czechoslovak Mathematical Journal
• Mathematical Notes

Key Ideas

• Foundational definitions and model statements for multiple sequences and series
• Convergence, summability, and asymptotic behavior criteria
• Comparison, transform, and Tauberian-style techniques for proving direct results

Typical Uses

Used to prove convergence or divergence under weakened hypotheses, compare summation methods, and transfer results between sequence spaces and functional-analytic formulations.

Applications

• Asymptotic analysis of numerical and analytic approximation procedures
• Regularization and summation of series in analysis and applied mathematics
• Stability and convergence assessments in iterative and discretized models

References

• Pringsheim, A. (1900) "Zur Theorie der zweifach unendlichen Zahlenfolgen", Mathematische Annalen, 53, pp. 289-321.
• Hardy, G.H. (1949) Divergent Series. Oxford: Clarendon Press.
• Bojanic, R. and Seneta, E. (1974) "A unified theory of regularly varying sequences", Mathematische Zeitschrift, 134, pp. 91-106.

Recommended Textbooks

• Hardy, G.H. (1949) Divergent Series. Oxford: Clarendon Press.
• Zygmund, A. (2002) Trigonometric Series. 3rd edn. Cambridge: Cambridge University Press.
• Korner, T.W. (1988) Fourier Analysis. Cambridge: Cambridge University Press.