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),…, (gk –1, fk )] is called the k th 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),..., (gk1,fk)]
if this limit exists and in this case we say that the infinite continued fraction converges.