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  "body": "\n## Statement\n\nAs $n \\to \\infty$,\n$$n! \\sim \\sqrt{2 \\pi n} \\left( \\frac{n}{e} \\right)^n.$$\nThat is, the ratio $\\frac{n!}{\\sqrt{2 \\pi n} \\, (n/e)^n}$ tends to $1$.\n\n## Remarks\n\nThe relation $\\sim$ is [[asymptotic equivalence|00SB5EA3VW]]. The absolute error grows, but the relative error tends to $0$.\nComparing $\\ln n! = \\sum_{k \\le n} \\ln k$ with $\\int_1^n \\ln x \\, dx$ gives the form $n! \\sim C \\sqrt{n} (n/e)^n$. The [[Wallis product|0XCYA4YERS]] gives $C = \\sqrt{2\\pi}$.\nA sharper form is $n! = \\sqrt{2 \\pi n} (n/e)^n \\big(1 + \\frac{1}{12 n} + O(n^{-2})\\big)$. Mathlib states the limit as `Stirling.tendsto_stirlingSeq_sqrt_pi`.\n\n## History\n\nAbraham de Moivre first stated a less precise form of the result.\nIn 1730, James Stirling showed that the constant is $\\sqrt{2\\pi}$.\n\n## Explanations\n\n### L1 human\n\nStirling's formula gives the size of $n!$ for large $n$.\nFor example, it gives $10! \\approx 3.599 \\times 10^6$, while $10! = 3628800$.\n",
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