While reading "Gauss, Landen, Ramanujan, the Arithmetic-Geometric Mean, Ellipses, π, and the Ladies Diary " [p.602] from
$F\left( -\dfrac {1}{2},-\dfrac {1}{2};1;\lambda^{2}\right)=1+\dfrac {\lambda ^{2}}{4\left( 1+w \right) }$ ,
It is said that "it can be shown that $w$ has the continued fraction expansion"
$w=\dfrac {1}{3}\left\{ \dfrac {-\dfrac {3}{16}\lambda ^{2}}{1}+\dfrac {-\dfrac {3}{16}\lambda ^{2}}{1}+\dfrac {-\dfrac {3}{16}\lambda ^{2}}{1}+\dfrac {-\dfrac {11}{48}\lambda ^{2}}{1}+. . .\right\}$.
I haven't had much experience with hypergeometric functions or continued fractions so I would greatly appreciate it if someone could explain how this step is possible in detail.