Summation over k to n of Natural Logarithm of k
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Theorem
- $\ds \sum_{k \mathop = 1}^n \ln k = \map \ln {n!}$
where $n!$ denotes the $n$th factorial.
Proof
| \(\ds \sum_{k \mathop = 1}^n \ln k\) | \(=\) | \(\ds \ln \prod_{k \mathop = 1}^n k\) | Summation of General Logarithms | |||||||||||
| \(\ds \) | \(=\) | \(\ds \map \ln {n!}\) | Definition of Factorial |
$\blacksquare$
Sources
- 1997: Donald E. Knuth: The Art of Computer Programming: Volume 1: Fundamental Algorithms (3rd ed.) ... (previous) ... (next): $\S 1.2.7$: Harmonic Numbers: Exercise $8$