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

TL;DR

This paper develops and compares algorithms to accelerate the slow infra-linear convergence of the alternate congruo-harmonic series, revealing diverse convergence speeds and discussing potential optimality and open problems.

Contribution

It introduces a family of algorithms based on continued fractions to accelerate the series' convergence and analyzes their asymptotic behavior.

Findings

Algorithms can accelerate convergence infra-linearly with diverse speeds

Some methods achieve linear or super-linear convergence

The paper discusses potential optimality and open problems in convergence acceleration

Abstract

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.