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  "body": "\n## Definition\n\nLet $\\sum_{k \\ge 0} (-1)^k a_k$ be an [[alternating series|017BR1M98J]] with [[partial sums|0NB643K60Q]] $s_k = \\sum_{i=0}^{k} (-1)^i a_i$.\nThe *van Wijngaarden transformation* builds the table $(s_{j,k})_{j, k \\ge 0}$ by\n$$s_{0,k} = s_k, \\qquad s_{j+1,k} = \\frac{s_{j,k} + s_{j,k+1}}{2}.$$\nThe entries $s_{j,k}$ with $j \\ge 1$ serve as approximations of the sum of the series.\n\n## Remarks\n\nBy induction, $s_{j,k} = 2^{-j} \\sum_{i=0}^{j} \\binom{j}{i} s_{k+i}$, a binomially weighted mean of $j + 1$ consecutive partial sums.\nExample: for the Leibniz series, $s_0, \\dots, s_3$ are $1, 0.6667, 0.8667, 0.7238$.\nThe next rows are $0.8333, 0.7667, 0.7952$, then $0.8000, 0.7810$, then $0.7905$, which is within $0.006$ of $\\frac{\\pi}{4} \\approx 0.7854$.\nNon-example: the series $\\sum 2^{-k}$ is not alternating, and its partial sums $1, 1.5, 1.75$ give the averages $1.25, 1.625$.\nThese are worse than the last partial sum; the method relies on the alternation of signs.\n\n## Explanations\n\n### L1 human\n\nWrite the partial sums in a row, then write the averages of neighbours below them, and repeat.\n",
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