Limit to Infinity of Binomial Coefficient over Power
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Theorem
- $\ds \lim_{r \mathop \to \infty} \dfrac {\dbinom r k} {r^k} = \frac 1 {\map \Gamma {k + 1}}$
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 | |||||||||||
\(\ds \) | \(=\) | \(\ds \lim_{r \mathop \to \infty} \frac 1 {\map \Gamma {k + 1} } \sqrt {\frac r {r - k} } \paren {\frac r e}^r \paren {\frac e {r - k} }^{r - k} \frac 1 {r^k}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \lim_{r \mathop \to \infty} \frac 1 {\map \Gamma {k + 1} } \sqrt {\frac r {r - k} } \frac 1 {e^k} \frac {r^{r - k} } {\paren {r - k}^{r - k} }\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \lim_{r \mathop \to \infty} \frac 1 {\map \Gamma {k + 1} } \sqrt {\frac r {r - k} } \frac 1 {e^k} \frac {\paren {1 - k / r}^k} {\paren {1 - k / r}^r}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \frac 1 {\map \Gamma {k + 1} }\) |
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 $e$ |
$\blacksquare$
Sources
- 1997: Donald E. Knuth: The Art of Computer Programming: Volume 1: Fundamental Algorithms (3rd ed.) ... (previous) ... (next): $\S 1.2.6$: Binomial Coefficients: Exercise $45$