The alternate congruo-harmonic series Part 2 - Accelerations of convergence

Les séries congruo-harmoniques alternées Partie 2 -Accélérations de convergence

David Pouvreau (1)

1 FST - Université de La Réunion - Faculté des Sciences et Technologies

For every couple (p;q) of strictly positive integers, the « alternate congruo-harmonic » series parametrized by (p;q), whose general term is (-1)^k/(pk+q), converges infra-linearly and very slowly. On the basis of a generalized continued fraction expansion of the partial rest of the series, this paper elaborates a family of algorithms which accelerate its convergence. The convergence speed of the sequences generated by these algorithms are compared. A precise asymptotic analysis is conducted, which reveals the possibility to accelerate the convergence either infra-linearly (but with an infinite diversity of possible speeds), or linearly (with a convergence rate that appears universal relatively to (p;q)), or super-linearly, by means of sequences extractions. Several open problems are also discussed, which concern the relative « performance » of the algorithms thus built and the possible optimality of some of them.

Mots clés: alternate congruo-harmonic series; generalized continued fraction expansion; convergence acceleration

Domaines: Analyse classique [math.CA]

David Pouvreau. Les séries congruo-harmoniques alternées Partie 2 -Accélérations de convergence. Quadrature, 2022, 124, pp.15-24. ⟨hal-03712401⟩