A method of convergence acceleration of some continued fractions

A new method of convergence acceleration is proposed for continued fractions $b_0+K(a_n/b_n)$, where $a_n$ and $b_n$ are polynomials in $n$ ($\deg \,a_{n} = 2$, $\deg \,b_{n} \leqslant 1$) for $n$ sufficiently large. It uses the fact that the modified approximant $S_n(t_n')$ approaches the continued fraction value, if $t_n'$ is sufficiently close to the $n$th tail $t_n$. Presented method is of iterative character; in each step, by means of an approximation $t_n'$, it produces a new better approximation $t_n''$ of the $n$th tail $t_n$. Formula for $t_n''$ is very simple and contains only arithmetical operations. Hence described algorithm is fully rational.