Factorial as Sum of Series of Subfactorial by Falling Factorial over Factorial/Historical Note

Historical Note on Factorial as Sum of Series of Subfactorial by Falling Factorial over Factorial

This result was established by James Stirling during the course of his attempt extend the factorial to the real numbers.

While it provides an exact value for $n!$ when $n$ is a positive integer, the infinite sum exists only under this condition.

Sources

• 1997: Donald E. Knuth: The Art of Computer Programming: Volume 1: Fundamental Algorithms (3rd ed.) ... (previous) ... (next): $\S 1.2.5$: Permutations and Factorials