Untypical methods of convergence acceleration

Abstract

Iterated Aitken’s method is one of classical procedures which permit to accelerate series or sequences convergence. It may be a starting point of constructing better methods in some classes of series whose important parameters are known. Such untypical modifications are here proposed and investigated. They based on a common idea and refer to two kinds of series; cf. Section 2 (series with rational coefficients, hypergeometric series and many others) and Section 3 (so-called quasi-geometric series). The second kind of series is associated with a class of infinite products whose convergence may be also accelerated. Behaviour of Levin’s and Weniger’s methods depends on a parameter β. In Section 4 its role is investigated and possibility of an improvement of their initial steps is showed.