Limit to Infinity of Binomial Coefficient over Power/Proof 2

Theorem

Let $k \in \R \setminus \set {-1, -2, -3, \dotsc}$.

Then:

$\ds \lim_{r \mathop \to \infty} \dfrac {\dbinom r k} {r^k} = \frac 1 {\map \Gamma {k + 1} }$

This proof applies to the special case where $k \in \Z$.

Then, by hypothesis, we need only consider:

$k \in \set {0, 1, 2, \dotsc}$

$\ds \lim_{r \mathop \to \infty} \frac {\dbinom r k} {r^k} = \frac 1 {k !}$

We have:

$\ds \forall r \in \R: \, $

$\ds \frac {\dbinom r k} {r^k}$

$=$

$\ds \frac 1 {k !} \cdot \frac {r^{\underline k} } {r^k}$

$\ds $

$=$

$\ds \frac 1 {k !} \prod_{j \mathop = 0}^{k - 1} \frac {r - j} r$

Definition of Falling Factorial, Definition of Integer Power

$\ds $

$=$

$\ds \frac 1 {k !} \prod_{j \mathop = 0}^{k - 1} \paren {1 - \frac j r}$

$\ds \leadsto \ \ $

$\ds \lim_{r \mathop \to \infty} \frac {\dbinom r k} {r^k}$

$=$

$\ds \lim_{r \mathop \to \infty} \frac 1 {k !} \prod_{j \mathop = 0}^{k - 1} \paren {1 - \frac j r}$

$\ds $

$=$

$\ds \frac 1 {k !} \prod_{j \mathop = 0}^{k - 1} \lim_{r \mathop \to \infty} \paren {1 - \frac j r}$

$\ds $

$=$

$\ds \frac 1 {k !} \prod_{j \mathop = 0}^{k - 1} \paren {1 - j \lim_{r \mathop \to \infty} \frac 1 r}$

$\ds $

$=$

$\ds \frac 1 {k !} \prod_{j \mathop = 0}^{k - 1} 1$

$\ds $

$=$

$\ds \frac 1 {k !}$

$\blacksquare$