Wynn's epsilon algorithm

Remarks

Check for $k = 1$: $\varepsilon_1^{(n)} = \frac{1}{\Delta A_n}$, where $\Delta A_n = A_{n+1} - A_n$, and $$\varepsilon_2^{(n)} = A_{n+1} + \frac{1}{\frac{1}{\Delta A_{n+1}} - \frac{1}{\Delta A_n}} = A_{n+1} - \frac{\Delta A_n \Delta A_{n+1}}{\Delta A_{n+1} - \Delta A_n},$$ which equals $S(A_{n+1}) = \frac{A_{n+2}A_n - A_{n+1}^2}{A_{n+2} - 2A_{n+1} + A_n}$. Example: for $A_0 = 1$, $A_1 = \frac23$, $A_2 = \frac{13}{15}$ one gets $\varepsilon_1^{(0)} = -3$, $\varepsilon_1^{(1)} = 5$ and $\varepsilon_2^{(0)} = \frac23 + \frac18 = \frac{19}{24}$. Non-example: for a constant sequence $A_n = c$ already $\varepsilon_1^{(n)} = \frac{1}{0}$ is undefined; in practice the algorithm stops when two neighbouring entries agree.