Aitken's delta-squared process

Remarks

Equivalently, $A_n = \dfrac{s_n s_{n+2} - s_{n+1}^2}{s_{n+2} - 2 s_{n+1} + s_n}$. Example: if $s_n = s + c \lambda^n$ with $c \ne 0$ and $\lambda \ne 0, 1$, then $\Delta s_n = c\lambda^n(\lambda - 1)$ and $\Delta^2 s_n = c \lambda^n (\lambda - 1)^2$, so $A_n = s_n - c\lambda^n = s$ for every $n$. Non-example: for an arithmetic sequence such as $s_n = n$ one has $\Delta^2 s_n = 0$, so $A_n$ is not defined. A classical result: if $s_n \to s$, $s_n \ne s$, and $\frac{s_{n+1} - s}{s_n - s} \to \lambda$ with $\lambda \ne 1$, then $(A_n)$ accelerates the convergence of $(s_n)$.