Fungrim entry: fc8d5d

n ! \lt \sqrt{2 \pi} {n}^{n + 1 / 2} {e}^{-n} \exp\!\left(\frac{1}{12 n}\right)

Assumptions:

n \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 ! \lt \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
`Factorial`	$n !$	Factorial
`Sqrt`	$\sqrt{z}$	Principal square root
`ConstPi`	$\pi$	The constant pi (3.14...)
`Pow`	${a}^{b}$	Power
`Exp`	${e}^{z}$	Exponential function
`ZZGreaterEqual`	$\mathbb{Z}_{\ge n}$	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."))

Topics using this entry

Factorials and binomial coefficients