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  "body": "\n## Definition\n\nLet $(A_n)_{n \\ge 0}$ be a [[sequence|0QE8STGJH2]] of real or complex numbers, and write $\\Delta A_m = A_{m+1} - A_m$.\nFor $n \\ge 1$, the *Shanks transformation* of $(A_n)$ is\n$$S(A_n) = \\frac{A_{n+1} A_{n-1} - A_n^2}{A_{n+1} - 2A_n + A_{n-1}},$$\ndefined when the denominator is not $0$.\nMore generally, let $k \\ge 1$ and $n \\ge k$.\nThe transformation of order $k$ is the quotient of two [[determinants|0C3CFHDN6R]]:\n$$e_k(A_n) = \\frac{\\begin{vmatrix} A_{n-k} \u0026 \\cdots \u0026 A_n \\\\ \\Delta A_{n-k} \u0026 \\cdots \u0026 \\Delta A_n \\\\ \\vdots \u0026 \u0026 \\vdots \\\\ \\Delta A_{n-1} \u0026 \\cdots \u0026 \\Delta A_{n+k-1} \\end{vmatrix}}{\\begin{vmatrix} 1 \u0026 \\cdots \u0026 1 \\\\ \\Delta A_{n-k} \u0026 \\cdots \u0026 \\Delta A_n \\\\ \\vdots \u0026 \u0026 \\vdots \\\\ \\Delta A_{n-1} \u0026 \\cdots \u0026 \\Delta A_{n+k-1} \\end{vmatrix}}.$$\nFor $1 \\le j \\le k$, row $j$ of each determinant is $\\Delta A_{n-k+j-1}, \\dots, \\Delta A_{n+j-1}$.\nThe denominator must not be $0$.\nThus $e_1(A_n) = S(A_n)$.\n\n## Remarks\n\nCheck of $e_1 = S$: the numerator is $A_{n-1}\\Delta A_n - A_n \\Delta A_{n-1} = A_{n+1}A_{n-1} - A_n^2$ and the denominator is $\\Delta A_n - \\Delta A_{n-1}$.\nExample: for the partial sums $A_0 = 1$, $A_1 = \\frac23$, $A_2 = \\frac{13}{15}$ of the Leibniz series,\n$S(A_1) = \\dfrac{\\frac{13}{15} - \\frac49}{\\frac{13}{15} - \\frac43 + 1} = \\dfrac{19/45}{8/15} = \\dfrac{19}{24}$.\nNon-example: for an arithmetic sequence such as $A_n = n$ the denominator $A_{n+1} - 2A_n + A_{n-1}$ is $0$, so $S(A_n)$ is not defined.\nUp to a shift of index, $S$ is [[Aitken's delta-squared process|R40BXAKSV4]]: $S(A_{n+1}) = A_n - \\frac{(\\Delta A_n)^2}{\\Delta^2 A_n}$.\nThe transformation can be iterated, as in $S(S(A_n))$; this is different from $e_2$ in general.\n\n## Explanations\n\n### L1 human\n\nThe first-order formula uses three consecutive terms; the order-$k$ formula uses $2k + 1$ terms and fits a more flexible model of the error.\n",
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