nLab divergent series

Idea

A series is called divergent if it is not convergent in the usual strong senses of pointwise or, sometimes, uniform convergence.

In asymptotic analysis, and in physics, especially in perturbative QFT etc., often a divergent series is still a very useful concept if regarded as an asymptotic series, as a sum of some such series can be defined in various weaker senses, like Borel summability.

Related $n$Lab entries

• resummation, regular summation method, regularization, renormalization, Borel summability, asymptotic series, Stokes phenomenon

References

• G. H. Hardy, Divergent series, Clarendon Press, Oxford, 1949.
• A. N. Kolmogorov, Sur la possibilité de la définition générale de la dérivée, de l’intégrale et de la sommation de séries divergentes, C. R. Acad. Sci. Paris 180 (1925), 362–364.
• Yu. I. Lyubich, Axiomatic theory of divergent series and cohomological equations, Fundamenta Math. 198 (2008) doi arxiv/0705.1578
• В. В. Козлов, Инвариантные меры гладких динамических систем, обобщенные функции и методы суммирования, Изв. РАН. Сер. матем., 80:2 (2016) 63–80 mathnet MR3507379 eng. transl.: V. V. Kozlov, Invariant measures of smooth dynamical systems, generalized functions and summation methods, Izv. Math. 80:2 (2016) 342–358
• Leonhard Euler, De seriebus divergentibus, Novi Commentarii academiae scientiarum Petropolitanae 5, 1760, pp. 205-237. Collected in Opera Omnia: Series 1, Volume 14, pp. 585 - 617. German translation in arXiv:1202.1506. See also the Euler archive
• Johann Boos, Classical and modern methods in summability, Oxford Mathematical Monographs, 2001

category: analysis