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  "body": "\n## Summary\n\nThe Shanks transformation is a nonlinear [[series acceleration|B1S2XJPGYD]] method.\nIts first-order form takes three consecutive partial sums and returns the limit they would have if the error were exactly geometric.\nThis form coincides with [[Aitken's delta-squared process|WF08BMN6Q6]].\nHigher orders $e_k$ use $2k + 1$ terms and model the error as a sum of $k$ geometric sequences.\nIn practice it is computed with [[Wynn's epsilon algorithm|6X0C4HNMAZ]], and applied to the partial sums of a power series it produces [[Padé approximants|1F16G4V64X]].\n\n## Explanations\n\n### L1 human\n\nMany slowly converging series have partial sums that overshoot and undershoot their limit, like a damped oscillation.\nThe Shanks transformation fits a simple model to a few consecutive partial sums and reads off the limit of the model.\nFor the Leibniz series $1 - \\frac13 + \\frac15 - \\cdots = \\frac{\\pi}{4}$, the first three partial sums $1, \\frac23, \\frac{13}{15}$ already give $\\frac{19}{24} \\approx 0.792$, close to $\\frac{\\pi}{4} \\approx 0.785$.\nApplying the transformation again to the new sequence improves the values further.\n",
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