Continued Fraction of a Finite Conued Fraction
\[x \gt 1\]
have finite continued fraction \[[ a_1,a_2,...,a_n ]\]
. Then \[x=a_1+ \frac{1}{a_2+ \frac{1}{a_3+\frac{1}{\ddots + \frac{1}{a_{n-1} + \frac{1}{a_n}}}}}\]
.Then
\[\frac{1}{x} =0+ \frac{1}{a_1+ \frac{1}{a_2+ \frac{1}{a_3+\frac{1}{\ddots + \frac{1}{a_{n-1} + \frac{1}{a_n}}}}}} \]
.The finite continued fraction of
\[\frac{1}{x}\]
is \[[ 0,a_1,a_2,...,a_n ]\]
.Conversely if
\[0 \lt x \lt 1\]
and the finite continued fraction of \[x\]
is \[[ 0,a_1,a_2,...,a_n ]\]
, then the finite continued fraction of \[\frac{1}{x}\]
is \[[a_1,a_2,...,a_n ]\]
. 
