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  "body": "\n## Definition\n\nA *sequence transformation* $T$ assigns to a sequence $(s_n)$ of real or complex numbers a new sequence $(t_n) = T(s_n)$.\nEach $t_n$ depends on finitely many terms of $(s_n)$.\nLet $(s_n)$ [[converge|0XKWSSY8P2]] to $s$, with $s_n \\ne s$ for all large $n$.\nThe transformation $T$ *accelerates the convergence* of $(s_n)$ if $(t_n)$ converges to $s$ and\n$$\\lim_{n \\to \\infty} \\frac{t_n - s}{s_n - s} = 0.$$\n*Series acceleration* means applying such a transformation to the sequence of [[partial sums|0NB643K60Q]] of a convergent [[series|0JH4MN4K1C]].\n\n## Remarks\n\nExample: for $s_n = 2^{-n}$ with limit $0$, the transformation $t_n = s_n s_{n+1} = 2^{-2n-1}$ gives $t_n / s_n = 2^{-n-1} \\to 0$, so it accelerates this sequence.\nNon-example: the shift $t_n = s_{n+1}$ gives $t_n / s_n = \\frac12$ for the same sequence.\nThe error is halved, but the ratio does not tend to $0$, so this is not acceleration in the sense above.\nA given transformation accelerates some sequences and not others.\n\n## Explanations\n\n### L1 human\n\nThe new sequence must have the same limit and its error must become negligible compared with the old error.\nGaining a fixed factor is not enough.\n",
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