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  "body": "\n## Summary\n\nThe Euler transform rewrites an [[alternating series|0QED0GPYKY]] $\\sum (-1)^n a_n$ as a new series built from the iterated differences of the $a_n$, with weights $2^{-n-1}$.\nWhen the $a_n$ vary smoothly, the new series converges much faster, often geometrically.\nIt is a classical tool of [[series acceleration|B1S2XJPGYD]], and the same formula gives the [[Euler summation|81Q4ERP594]] method for some divergent series.\n\n## Explanations\n\n### L1 human\n\nThe terms of an alternating series cancel each other in pairs.\nThe Euler transform groups this cancellation in a systematic way using differences of consecutive terms.\nFor the [[alternating harmonic series|0JQ5Y3WWVE]] $1 - \\frac12 + \\frac13 - \\cdots = \\ln 2$, the transformed series is $\\sum_{n \\ge 1} \\frac{1}{n 2^n}$.\nIts terms shrink by about half each time, so a few terms already give several correct digits.\n",
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