Logarithm of Factorial/Proof

Theorem

\(\ds \ln n!\)

\(=\)

\(\ds \sum_{k \mathop \ge 0} a_{k + 1} n^{\underline k}\)

\(\ds \)

\(=\)

\(\ds a_1 n + a_2 n \paren {n - 1} + a_3 n \paren {n - 1} \paren {n - 2} + \cdots\)

where:

$\ds a_m = \dfrac {\paren {-1}^m} {m!} \sum_{k \mathop \ge 1} \paren {-1}^k \binom {m - 1} {k - 1} \ln k$

Proof

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Sources

• 1997: Donald E. Knuth: The Art of Computer Programming: Volume 1: Fundamental Algorithms (3rd ed.) ... (previous) ... (next): $\S 1.2.6$: Binomial Coefficients: Exercise $57$