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  "body": "\n## Definition\n\nLet $(s_n)_{n \\ge 0}$ be a [[sequence|0QE8STGJH2]] of real or complex numbers.\nWrite $\\Delta s_n = s_{n+1} - s_n$ and $\\Delta^2 s_n = s_{n+2} - 2 s_{n+1} + s_n$.\nFor each $n$ with $\\Delta^2 s_n \\ne 0$, *Aitken's delta-squared process* gives\n$$A_n = s_n - \\frac{(\\Delta s_n)^2}{\\Delta^2 s_n}.$$\n\n## Remarks\n\nEquivalently, $A_n = \\dfrac{s_n s_{n+2} - s_{n+1}^2}{s_{n+2} - 2 s_{n+1} + s_n}$.\nExample: 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 $n$.\nNon-example: for an arithmetic sequence such as $s_n = n$ one has $\\Delta^2 s_n = 0$, so $A_n$ is not defined.\nA 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|90ANMX55DP]] the convergence of $(s_n)$.\n\n## Explanations\n\n### L1 human\n\nThe formula subtracts from $s_n$ the correction that a perfectly geometric error would require.\nIt uses only the first and second differences of the sequence.\n",
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