Non‐linear Transformations of Divergent and Slowly Convergent Sequences

This paper discusses a family of non‐linear sequence‐to‐sequence transformations designated as e k , e k m , ẽ k , and e d . A brief history of the transforms is related and a simple motivation for the transforms is given. Examples are given of the application of these transformations to divergent and slowly convergent sequences. In particular the examples include numerical series, the power series of rational and meromorphic functions, and a wide variety of sequences drawn from continued fractions, integral equations, geometry, fluid mechanics, and number theory. Theorems are proven which show the effectiveness of the transformations both in accelerating the convergence of (some) slowly convergent sequences and in inducing convergence in (some) divergent sequences. The essential unity of these two motives is stressed. Theorems are proven which show that these transforms often duplicate the results of well‐known, but specialized techniques. These special algorithms include Newton's iterative process, Gauss's numerical integration, an identity of Euler, the Padé Table, and Thiele's reciprocal differences. Difficulties which sometimes arise in the use of these transforms such as irregularity, non‐uniform convergence to the wrong answer, and the ambiguity of multivalued functions are investigated. The concepts of antilimit and of the spectra of sequences are introduced and discussed. The contrast between discrete and continuous spectra and the consequent contrasting response of the corresponding sequences to the e 1 transformation is indicated. The characteristic behaviour of a semiconvergent (asymptotic) sequence is elucidated by an analysis of its spectrum into convergent components of large amplitude and divergent components of small amplitude.