A test-ratio test for continued fractions
Walter Leighton · Bulletin of the American Mathematical Society · 1939
Introduction.The general question of convergence of continued fractions of the form l+i£f [&n/l] remains in a large measure unanswered, even though continued fractions of this type are of especial importance from a function-theoretic point of view.Valuable contributions have been made by E. B. Van Vleck, A. Pringsheim, O. Szâsz, O. Perron, and others.Leighton and Wall [7] recently gave new types of convergence criteria for continued fractions of this kind.Jordan and Leighton in a paper to be published soon give a large number of new sets of sufficient conditions for convergence.The purpose of the present paper is to establish the first test-ratio test for continued fractions and a very general theorem on convergence, which is also believed to be the first of its kind.This test leads to a class of continued fractions, the precise region of convergence of which is the interior of a circle.This is a new phenomenon. A test-ratio test. Let