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  "body": "\n## Summary\n\nThe van Wijngaarden transformation accelerates an [[alternating series|0QED0GPYKY]] by averaging its partial sums again and again.\nEach new row of the table holds the means of neighbouring entries of the row above.\nIt is a variant of the [[Euler transform|WT0HACKQV7]] that is easy to carry out by hand or by machine.\n\n## Explanations\n\n### L1 human\n\nThe partial sums of an alternating series jump above and below the sum.\nThe average of two consecutive partial sums lies between them and is usually much closer to the sum.\nAveraging the averages improves the values again.\nFor the Leibniz series $1 - \\frac13 + \\frac15 - \\cdots = \\frac{\\pi}{4}$, three rounds of averaging of the first four partial sums give about $0.7905$, against $0.7854$ for $\\frac{\\pi}{4}$ and $0.7238$ for the fourth partial sum.\n",
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