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  "body": "\n## Summary\n\nLevin-type transformations are nonlinear [[series acceleration|B1S2XJPGYD]] methods that use explicit estimates of the remainder of a series.\nThey assume that the error of the $n$-th partial sum is a known estimate $\\omega_n$ times a slowly varying correction.\nThey eliminate this correction exactly when it is a polynomial in $1/(n + \\beta)$.\nLevin's $t$- and $u$-transformations take $\\omega_n$ from the terms of the series.\nThey work well both for alternating series and for many series with terms of one sign.\nThey can also assign values to some divergent series.\n\n## Explanations\n\n### L1 human\n\nFor a series, the next term is often a good guess for the size of the remaining error.\nA Levin-type transformation uses such a guess $\\omega_n$ and fits the rest of the error by a short expansion in powers of $1/n$.\nSolving for the sum then removes several orders of the error at once.\nFor the geometric series $\\sum x^n$ with $\\abs{x} \u003c 1$, the error after $n$ terms is exactly a constant times the last term, and the simplest Levin transformation returns the exact sum $\\frac{1}{1 - x}$.\n",
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