Kummer's Hypergeometric Theorem/Lemma 1

Lemma for Kummer's Hypergeometric Theorem

 * $\ds \lim_{y \mathop \to \infty} \dfrac {y^{\underline k} } {\paren {y + n + 1}^{\overline k} } = 1$

where $y^{\underline k}$ denotes the $k$th falling factorial of $y$.

Proof
From L'Hôpital's Rule:Corollary 2, we have:
 * $\ds \lim_{x \mathop \to a^+} \frac {\map f x} {\map g x} = \lim_{x \mathop \to a^+} \frac {\map {f'} x} {\map {g'} x}$

Therefore taking the the derivative of the numerator and denominator $y$, we obtain:
 * $\ds \lim_{y \mathop \to \infty} \paren {\dfrac {\paren {y + 1 - k} } {\paren {y + n + k} } } = \dfrac 1 1 = 1$

Therefore: