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By Pugh G.R.

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Should indeed equal 1), it is generally applied to counting arguments in combinatorics and probability. This is the end of the line for some students. Others continue on to study calculus, where n! is again encountered in the freshman year, typically in the result dn n x = n! dxn for n a non-negative integer. Closely related to this result is the appearance of n! in Taylor’s formula. Some are asked to prove ∞ n! 1) “makes sense” for non-integer n. Some even go so far as to prove √ n! ∼ 2πn nn e−n as n → ∞ .

3 × 10−34 . This bound is impressive, but bear in mind that the corresponding Bernoulli number required to achieve this is |B76 | ≈ 8 × 1050 . This is the best possible accuracy for z = 6 + 13i with N = 0; greater accuracy is possible as N is increased from zero. 7 × 10−42 . Uniform Bound of |UN,n(z)| As demonstrated, for fixed z and N there is a limit to the accuracy which can be achieved by taking more and more terms of the Stirling series. It is worthwhile to determine this limiting accuracy in the form of the best possible uniform error bound as a function of N ≥ 0.

3) converges for real x > −1. 3) into the more familiar form ∞ x! = ux e−u du . 0 In 1808 Legendre introduced the notation which would give the gamma function its name1 . 4) 0 so that Γ(n + 1) = n!. It is unclear why Legendre chose to shift the argument in his notation. Lanczos remarks in the opening paragraph of his paper [14]: The normalization of the gamma function Γ(n + 1) instead of Γ(n) is due to Legendre and void of any rationality. This unfortunate circumstance compels us to utilize the notation z!

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