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