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Fungrim entry: fc8d5d

n!<2πnn+1/2enexp ⁣(112n)n ! < \sqrt{2 \pi} {n}^{n + 1 / 2} {e}^{-n} \exp\!\left(\frac{1}{12 n}\right)
Assumptions:nZ1n \in \mathbb{Z}_{\ge 1}
References:
  • H. Robbins (1955), A remark on Stirling's formula, Am. Math. Monthly 62(1), pp. 26-29.
TeX:
n ! < \sqrt{2 \pi} {n}^{n + 1 / 2} {e}^{-n} \exp\!\left(\frac{1}{12 n}\right)

n \in \mathbb{Z}_{\ge 1}
Definitions:
Fungrim symbol Notation Short description
Factorialn!n ! Factorial
Sqrtz\sqrt{z} Principal square root
ConstPiπ\pi The constant pi (3.14...)
Powab{a}^{b} Power
Expez{e}^{z} Exponential function
ZZGreaterEqualZn\mathbb{Z}_{\ge n} Integers greater than or equal to n
Source code for this entry:
Entry(ID("fc8d5d"),
    Formula(Less(Factorial(n), Mul(Mul(Mul(Sqrt(Mul(2, ConstPi)), Pow(n, Add(n, Div(1, 2)))), Exp(Neg(n))), Exp(Div(1, Mul(12, n)))))),
    Variables(n),
    Assumptions(Element(n, ZZGreaterEqual(1))),
    References("H. Robbins (1955), A remark on Stirling's formula, Am. Math. Monthly 62(1), pp. 26-29."))

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2019-10-05 13:11:19.856591 UTC