: Matemáticas: Stirling's approximation

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mathematical analysis > combinatorics > factorial > Stirling's approximation

algebra > combinatorics > factorial > Stirling's approximation

number > number theory > arithmetic function > factorial > Stirling's approximation

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Stirling's approximation

In mathematics, Stirling's approximation (or Stirling's formula) is an asymptotic approximation for factorials. It is a good approximation, leading to accurate results even for small values of ${\\displaystyle n}$ . It is named after James Stirling, though a related but less precise result was first stated by Abraham de Moivre. One way of stating the approximation involves the logarithm of the factorial: ${\\displaystyle \\ln(n!)=n\\ln n-n+O(\\ln n),}$ where the big O notation means that, for all sufficiently large values of ${\\displaystyle n}$ , the difference between ${\\displaystyle \\ln(n!)}$ and ${\\displaystyle n\\ln n-n}$ will be at most proportional to the logarithm. In computer science applications such as the worst-case lower bound for comparison sorting, it is convenient to instead use the binary logarithm, giving the equivalent form ${\\displaystyle \\log _{2}(n!)=n\\log _{2}n-n\\log _{2}e+O(\\log _{2}n).}$ The error term in either base can be expressed more precisely as ${\\displaystyle {\ frac {1}{2}}\\log(2\\pi n)+O({\ frac {1}{n}})}$ , corresponding to an approximate formula for the factorial itself, ${\\displaystyle n!\\sim {\\sqrt {2\\pi n}}\\left({\ rac {n}{e}}\ ight)^{n}.}$ Here the sign ${\\displaystyle \\sim }$ means that the two quantities are asymptotic, that is, that their ratio tends to 1 as ${\\displaystyle n}$ tends to infinity. The following version of the bound holds for all ${\\displaystyle n\\geq 1}$ , rather than only asymptotically: ${\\displaystyle {\\sqrt {2\\pi n}}\\ \\left({\ rac {n}{e}}\ ight)^{n}e^{\ rac {1}{12n+1}}<n!<{\\sqrt {2\\pi n}}\\ \\left({\ rac {n}{e}}\ ight)^{n}e^{\ rac {1}{12n}}.}$
(Wikipedia, The Free Encyclopedia, https://en.wikipedia.org/wiki/Stirling%27s_approximation)