## RIS ID

107537

## Abstract

A continued fraction is an expression of the form

f_{0}+ g_{0}

f_{1}+g_{1}

f_{2}+g_{2}

and we will denote it by the notation [*f*_{0}, (*g*_{0}, *f*_{1}), (*g*_{1}, *f*_{2}), (*g*_{2}, *f*_{3}), … ]. If the numerators *g*_{i} are all equal to 1 then we will use a shorter notation [*f*_{0}, *f*_{1}, *f*_{2}, *f*_{3}, … ]. 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 *f*_{0}.

The finite continued fraction [*f*_{0}, (*g*_{0}, *f*_{1}), (*g*_{1}, *f*_{2}),…, (*g*_{k –1}, *f*_{k} )] is called the *k*th convergent of the infinite continued fraction [*f*_{0}, (*g*_{0}, *f*_{1}), (*g*_{1}, *f*_{2}),…]. We define

[*f*_{0}, (*g*_{0}, *f*_{1}), (*g*_{1}, *f*_{2}), (*g*_{2},*f*_{3}),...] = lim [*f*_{0}, (*g*_{0}, *f*_{1}), (*g*_{1}, *f*_{2}),..., (*g*_{k-1},*f*_{k})]

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

## Publication Details

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