A simple proof of Euler's continued fraction of e^{1/M}

Authors

Joseph Tonien, University of WollongongFollow

RIS ID

107537

Publication Details

Tonien, J. (2016). A simple proof of Euler's continued fraction of e^{1/M}. The Mathematical Gazette, 100 (548), 279-287.

Abstract

A continued fraction is an expression of the form

f₀+ g₀

f₁+g₁

f₂+g₂

and we will denote it by the notation [f₀, (g₀, f₁), (g₁, f₂), (g₂, f₃), … ]. If the numerators g_i are all equal to 1 then we will use a shorter notation [f₀, f₁, f₂, f₃, … ]. A simple continued fraction is a continued fraction with all the g_i coefficients equal to 1 and with all the f_i coefficients positive integers except perhaps f₀.

The finite continued fraction [f₀, (g₀, f₁), (g₁, f₂),…, (g_{k –1}, f_k )] is called the kth convergent of the infinite continued fraction [f₀, (g₀, f₁), (g₁, f₂),…]. We define

[f₀, (g₀, f₁), (g₁, f₂), (g₂,f₃),...] = lim [f₀, (g₀, f₁), (g₁, f₂),..., (g_k-1,f_k)]

if this limit exists and in this case we say that the infinite continued fraction converges.