Observations on continued fractions

TL;DR

This paper presents Euler's foundational work on continued fractions, including transformations of series, solutions to differential equations, and notable formulas like the continued fraction for hypergeometric series quotients.

Contribution

It provides a translation of Euler's original work, highlighting his methods and discoveries in the theory of continued fractions.

Findings

Transformation techniques for series into continued fractions

Solution of Riccati-Differential equation via continued fractions

Derivation of the continued fraction for hypergeometric series quotient

Abstract

This is a translation of Euler's Latin paper "De fractionibus continuis observationes" into English. In this paper Euler describes his theory of continued fractions. He teaches, how to transform series into continued fractions, solves the Riccati-Differential equation by means of continued fractions and finds many other interesting formulas and results (e.g, the continued fraction for the quotient of two hypergeometric series usually attributed to Gau{\ss})