Limit to Infinity of Binomial Coefficient over Power

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Theorem

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

Then:

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


Integer Case

Let $k \in \N$.

Then:

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


Proof

\(\ds \lim_{r \mathop \to \infty} \frac {\dbinom r k} {r^k}\) \(=\) \(\ds \lim_{r \mathop \to \infty} \frac {\map \Gamma {r + 1} } {\map \Gamma {k + 1} \map \Gamma {r - k + 1} r^k}\) Gamma Function Extends Factorial
\(\ds \) \(=\) \(\ds \lim_{r \mathop \to \infty} \frac 1 {\map \Gamma {k + 1} } \frac {\sqrt {2 \pi r} \paren {r / e}^r} {\sqrt {2 \pi \paren {r - k} } \paren {\paren {r - k} / e}^{r - k} r^k}\) Stirling's Formula for Gamma Function; first term
\(\ds \) \(=\) \(\ds \frac 1 {\map \Gamma {k + 1} } \lim_{r \mathop \to \infty} \sqrt {\frac r {r - k} } \paren {\frac r e}^r \paren {\frac e {r - k} }^{r - k} \frac 1 {r^k}\) $\sqrt {2 \pi}$ cancels
\(\ds \) \(=\) \(\ds \frac 1 {\map \Gamma {k + 1} } \lim_{r \mathop \to \infty} \sqrt {\frac r {r - k} } \frac 1 {e^k} \frac {r^{r - k} } {\paren {r - k}^{r - k} }\) Exponent Combination Laws: Quotient of Powers
\(\ds \) \(=\) \(\ds \frac 1 {\map \Gamma {k + 1} } \frac 1 {e^k} \lim_{r \mathop \to \infty} \frac {r^{r - k + \frac 1 2} } {\paren {r - k}^{r - k + \frac 1 2} }\) Product of Powers
\(\ds \) \(=\) \(\ds \frac 1 {\map \Gamma {k + 1} } \frac 1 {e^k} \lim_{r \mathop \to \infty} \frac {r^{r - k + \frac 1 2} } {\paren {r - k}^{r - k + \frac 1 2} } \times \frac {\paren {\frac 1 r}^{r - k + \frac 1 2} } {\paren {\frac 1 r}^{r - k + \frac 1 2} }\) multiplying top and bottom by $\paren {\frac 1 r}^{r - k + \frac 1 2}$
\(\ds \) \(=\) \(\ds \frac 1 {\map \Gamma {k + 1} } \frac 1 {e^k} \lim_{r \mathop \to \infty} \frac 1 {\paren {1 - k / r}^r \times \paren {1 - k / r}^{\frac 1 2 - k} }\) Product of Powers
\(\ds \) \(=\) \(\ds \frac 1 {\map \Gamma {k + 1} } \frac 1 {e^k} \frac 1 {e^{-k} }\) Definition of Euler's Number $e^{-k} = \lim_{r \mathop \to \infty} \paren {1 - k / r}^r$
\(\ds \) \(=\) \(\ds \frac 1 {\map \Gamma {k + 1} }\)

$\blacksquare$

Sources

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