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  "body": "\n## Summary\n\nTruncation error is the error made by replacing an infinite or exact mathematical process by a finite approximation, even when all arithmetic is exact.\nExamples are cutting a [[Taylor series|0Z49YH6ZWY]] after finitely many terms, replacing a derivative by a difference quotient, or an integral by a finite sum.\nIt usually shrinks as the number of terms grows or the step size $h$ goes to $0$, often like a power $h^p$.\nFor ODE methods, the error made in one step is the [[local truncation error|T220HF0562]].\nIt is the counterpart of [[round-off error|CPN6SD3BTV]], which comes from finite-precision arithmetic.\n\n## Explanations\n\n### L1 human\n\nMany quantities are defined by a process that never ends, such as an infinite sum or a limit.\nA computer must stop after finitely many steps, and what is left out is the truncation error.\nFor example, $e^x = 1 + x + x^2/2 + \\cdots$, and stopping after $1 + x$ gives an error of about $x^2/2$ for small $x$.\nTaking more terms or a smaller step makes the truncation error smaller, but it may make the round-off error larger, so one has to balance the two.\n",
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