Limit to Infinity of Binomial Coefficient over Power

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Theorem

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

Then:

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


Proof 1

\(\ds \lim_{r \mathop \to \infty} \frac {\dbinom r k} {r^k}\) \(=\) \(\ds \lim_{r \mathop \to \infty} \frac {\map \Gamma {r + 1} } {\map \Gamma {k + 1} \map \Gamma {r - k + 1} r^k}\) 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
\(\ds \) \(=\) \(\ds \lim_{r \mathop \to \infty} \frac 1 {\map \Gamma {k + 1} } \sqrt {\frac r {r - k} } \paren {\frac r e}^r \paren {\frac e {r - k} }^{r - k} \frac 1 {r^k}\)
\(\ds \) \(=\) \(\ds \lim_{r \mathop \to \infty} \frac 1 {\map \Gamma {k + 1} } \sqrt {\frac r {r - k} } \frac 1 {e^k} \frac {r^{r - k} } {\paren {r - k}^{r - k} }\)
\(\ds \) \(=\) \(\ds \lim_{r \mathop \to \infty} \frac 1 {\map \Gamma {k + 1} } \sqrt {\frac r {r - k} } \frac 1 {e^k} \frac {\paren {1 - k / r}^k} {\paren {1 - k / r}^r}\)
\(\ds \) \(=\) \(\ds \frac 1 {\map \Gamma {k + 1} }\)


The last equality is justified, since as $r \to \infty$:

\(\ds \sqrt {\frac r {r - k} }\) \(\to\) \(\ds 1\)
\(\ds \paren {1 - k / r}^k\) \(\to\) \(\ds 1\)
\(\ds \paren {1 - k / r}^r\) \(\to\) \(\ds e^{-k}\) Definition of Euler's Number $e$

$\blacksquare$


Proof 2

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

Then, by hypothesis, we need only consider:

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

By Gamma Function Extends Factorial, it suffices to show:

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


We have:

\(\ds \forall r \in \R: \, \) \(\ds \frac {\dbinom r k} {r^k}\) \(=\) \(\ds \frac 1 {k !} \cdot \frac {r^{\underline k} } {r^k}\) Definition of Binomial Coefficient
\(\ds \) \(=\) \(\ds \frac 1 {k !} \prod_{j \mathop = 0}^{k - 1} \frac {r - j} r\) Definition of Falling Factorial, Definition of Integer Power
\(\ds \) \(=\) \(\ds \frac 1 {k !} \prod_{j \mathop = 0}^{k - 1} \paren {1 - \frac j r}\)
\(\ds \leadsto \ \ \) \(\ds \lim_{r \mathop \to \infty} \frac {\dbinom r k} {r^k}\) \(=\) \(\ds \lim_{r \mathop \to \infty} \frac 1 {k !} \prod_{j \mathop = 0}^{k - 1} \paren {1 - \frac j r}\)
\(\ds \) \(=\) \(\ds \frac 1 {k !} \prod_{j \mathop = 0}^{k - 1} \lim_{r \mathop \to \infty} \paren {1 - \frac j r}\) Product Rule for Real Sequences
\(\ds \) \(=\) \(\ds \frac 1 {k !} \prod_{j \mathop = 0}^{k - 1} \paren {1 - j \lim_{r \mathop \to \infty} \frac 1 r}\) Combined Sum Rule for Real Sequences
\(\ds \) \(=\) \(\ds \frac 1 {k !} \prod_{j \mathop = 0}^{k - 1} 1\) Sequence of Reciprocals is Null Sequence
\(\ds \) \(=\) \(\ds \frac 1 {k !}\)

$\blacksquare$

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