Euler Form of Gamma Function at Positive Integers
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Theorem
The Euler form of the Gamma function:
- $\ds \map \Gamma z := \lim_{m \mathop \to \infty} \frac {m^z m!} {z \paren {z + 1} \paren {z + 2} \cdots \paren {z + m} }$
converges to the factorial function at positive integers:
- $\ds \lim_{m \mathop \to \infty} \frac {m^n m!} {\paren {n + 1} \paren {n + 2} \cdots \paren {n + m} } = n!$
Proof
\(\ds \lim_{m \mathop \to \infty} \frac {m^n m!} {\paren {n + 1} \paren {n + 2} \cdots \paren {n + m} }\) | \(=\) | \(\ds \lim_{m \mathop \to \infty} \frac {m^n m!} {\frac {\paren {n + m}!} {n!} }\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds n! \lim_{m \mathop \to \infty} \frac {m^n} {\paren {m + 1} \paren {m + 2} \cdots \paren {m + n} }\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds n!\) |
Now we have from Factorial of Integer plus Reciprocal of Integer that:
- $\ds \lim_{m \mathop \to \infty} \dfrac {\paren {m + n}!} {m! m^n} = 1$
Now:
\(\ds \dfrac {n! \paren {n + 1} \paren {n + 2} \cdots \paren {n + m} } {m! m^n}\) | \(=\) | \(\ds \dfrac {\paren {m + n}!} {m! m^n}\) | ||||||||||||
\(\ds \leadsto \ \ \) | \(\ds \lim_{m \mathop \to \infty} \dfrac {n! \paren {n + 1} \paren {n + 2} \cdots \paren {n + m} } {m! m^n}\) | \(=\) | \(\ds 1\) | |||||||||||
\(\ds \leadsto \ \ \) | \(\ds \lim_{m \mathop \to \infty} \dfrac {m! m^n} {\paren {n + 1} \paren {n + 2} \cdots \paren {n + m} }\) | \(=\) | \(\ds n!\) |
$\blacksquare$
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: Exercise $8$
- 1997: Donald E. Knuth: The Art of Computer Programming: Volume 1: Fundamental Algorithms (3rd ed.) ... (previous) ... (next): $\S 1.2.5$: Permutations and Factorials: Exercise $22$