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  "body": "\n## Definition\n\nLet $A$ be a real number defined as the [[limit|05M5SWSZZD]] of a [[real sequence|0CTV783HYT]] $(A_n)$, or of a family $A(h)$ as $h \\to 0$.\nAssume that each $A_n$ or $A(h)$ is computable by finitely many arithmetic operations.\nThe *truncation error* of the approximation $A_n$ is $A - A_n$, and that of $A(h)$ is $A - A(h)$.\nHere $A_n$ and $A(h)$ are evaluated in exact arithmetic.\n\n## Remarks\n\nExample: let $f$ have a continuous second derivative and take the forward difference $A(h) = (f(x+h) - f(x))/h$, with limit $A = f'(x)$.\nTaylor's theorem gives $A - A(h) = -\\tfrac12 f''(\\xi) h$ for some $\\xi$ between $x$ and $x + h$, so the truncation error is of order $h$.\nExample: for $A_n = \\sum_{k=0}^{n} 1/k!$ and $A = e$, the truncation error is $\\sum_{k \u003e n} 1/k!$, which lies between $1/(n+1)!$ and $2/(n+1)!$ for $n \\ge 1$.\nNon-example: the difference between $1/3$ and its stored value $0.333$ in three-digit decimal arithmetic is a round-off error, since no limit process is cut off.\n",
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