Basics

Abstract

The transformations $S_n(w) := a_n/(b_n + w)$ are linear fractional transformations with $S_n(\infty) = 0$ Hence S_n := s₁ ο s₂ ο…ο s_n are linear fractional transformations with $S_n(\infty) = S_{n-1}(0)$ and a continued fraction is essentially a sequence of linear fractional transformations {S_n} with $S_n(\infty) = S_{n-1}(0)$ for all n It is therefore natural to define convergence of continued fractions as convergence of {S_n} in some sense.