Continuants and convergence of certain continued fractions

TL;DR

This paper introduces the theory of continuants and explores their application in proving the convergence of certain infinite continued fractions, including semi-regular and purely periodic types, based on Perron's work.

Contribution

It provides a concise overview of continuants and demonstrates their use in convergence proofs for specific classes of continued fractions, highlighting historical and mathematical insights.

Findings

Continuants are effective tools for analyzing continued fraction convergence.

Perron's methods are applicable to semi-regular and purely periodic continued fractions.

The paper clarifies the role of continuants in convergence proofs.

Abstract

We give a concise introduction to the theory of continuants and show how Perron used them in his proof of Tietze theorem on the convergence of infinite semi-regular continued fractions, as well as for the study of the convergence of purely periodic continued fractions.