A note on a theorem of Perron

Abstract:

Given an infinite simple continued fraction $\left [ {{a_0},{a_1}, \ldots ,{a_n}, \ldots } \right ]$, let ${M_n}$ denote $\left [ {0,{a_n},{a_{n - 1}}, \ldots ,{a_1}} \right ] + \left [ {{a_{n + 1}},{a_{n + 2}}, \ldots } \right ]$. A well-known result due to Perron [1, III, 212] states: If ${a_{n + 2}} = m$, then there is a $k$ in $\left \{ {n,n + 1,n + 2} \right \}$ for which ${M_k} > \sqrt {{m^2} + 4}$. In this note we give a new proof for this result and add that there is a $j$ in $\left \{ {n,n + 1,n + 2} \right \}$ for which ${M_j} < \sqrt {{m^2} + 4}$.