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40Jxx Other sequence and series methods

This subtopic studies other sequence and series methods, gathering complementary approaches that do not fit neatly into the standard summability classifications.

Specific topics

40J05 Summability in abstract structures

Overview

40J05 investigates summability in abstract structures as part of other sequence and series methods. The focus is on convergence acceleration, summability equivalences, and inversion principles that extend classical limits and characterize asymptotic behavior of series and transforms.

Related Wikipedia Page

Wikipedia search: Summability in abstract structures

Useful Links

Key Ideas

  • Core definitions and canonical constructions for summability in abstract structures
  • Relations among regular, strong, and absolute summation procedures
  • Direct/inverse principles connecting transforms with convergence properties

Typical Uses

Used to recover analytic information from transformed sequences, compare summation frameworks, and prove convergence statements under weaker hypotheses than ordinary limits.

Applications

  • Asymptotic expansions and transform-based analysis in pure and applied mathematics
  • Stability checks for computational procedures involving slowly convergent series
  • Signal/approximation workflows where generalized summation improves interpretability

References

Recommended Textbooks