Dougall's Hypergeometric Theorem/Corollary 3/Lemma

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Lemma for Dougall's Hypergeometric Theorem/Corollary 3

$\ds \lim_{z \mathop \to \infty} \dfrac {\paren {x + z + n + 1}^{\overline y} } {\paren {z+ n + 1}^{\overline y} } = 1$

where $x^{\overline y}$ denotes the $y$th rising factorial of $x$.


Proof

From Properties of Limit at Infinity of Real Function: Product Rule, we have:

\(\ds \lim_{z \mathop \to \infty} \dfrac {\paren {x + z + n + 1}^{\overline y} } {\paren {z+ n + 1}^{\overline y} }\) \(=\) \(\ds \lim_{z \mathop \to \infty} \paren {\paren {\dfrac {\paren {x + z + n + 1} } {\paren {z + n + 1} } } \paren {\dfrac {\paren {x + z + n + 2} } {\paren {z + n + 2} } } \cdots \paren {\dfrac {\paren {x + z + n + y} } {\paren {z + n + y} } } }\) Definition of Rising Factorial
\(\ds \) \(=\) \(\ds \lim_{z \mathop \to \infty} \paren {\dfrac {\paren {x + z + n + 1} } {\paren {z + n + 1} } } \lim_{z \mathop \to \infty} \paren {\dfrac {\paren {x + z + n + 2} } {\paren {z + n + 2} } } \cdots \lim_{z \mathop \to \infty} \paren {\dfrac {\paren {x + z + n + y} } {\paren {z + n + y} } }\) Properties of Limit at Infinity of Real Function: Product Rule


From L'Hôpital's Rule:Corollary 2, we have:

$\ds \lim_{z \mathop \to a^+} \frac {\map f z} {\map g z} = \lim_{z \mathop \to a^+} \frac {\map {f'} z} {\map {g'} z}$

In the present example, for the kth limit, we have $\ds \map {f_k} z = \paren {x + z + n + y}$ and $\ds \map {g_k} z = \paren {z + n + y}$

Therefore taking the derivative of the numerator $\map {f_k} z$ and denominator $\map {g_k} z$ with respect to $z$, we obtain:

$\ds \lim_{z \mathop \to \infty}\paren {\dfrac {\paren {x + z + n + y} } {\paren {z + n + y} } } = \lim_{z \mathop \to \infty} \dfrac 1 1 = 1$

Therefore:

\(\ds \lim_{z \mathop \to \infty} \dfrac {\paren {x + z + n + 1}^{\overline y} } {\paren {z+ n + 1}^{\overline y} }\) \(=\) \(\ds \lim_{z \mathop \to \infty} \paren {\paren {\dfrac {\paren {x + z + n + 1} } {\paren {z + n + 1} } } \paren {\dfrac {\paren {x + z + n + 2} } {\paren {z + n + 2} } } \cdots \paren {\dfrac {\paren {x + z + n + y} } {\paren {z + n + y} } } }\) Definition of Rising Factorial
\(\ds \) \(=\) \(\ds \lim_{z \mathop \to \infty} \paren {\dfrac {\paren {x + z + n + 1} } {\paren {z + n + 1} } } \lim_{z \mathop \to \infty} \paren {\dfrac {\paren {x + z + n + 2} } {\paren {z + n + 2} } } \cdots \lim_{z \mathop \to \infty} \paren {\dfrac {\paren {x + z + n + y} } {\paren {z + n + y} } }\) Properties of Limit at Infinity of Real Function: Product Rule
\(\ds \) \(=\) \(\ds \lim_{z \mathop \to \infty} \dfrac 1 1 \lim_{z \mathop \to \infty} \dfrac 1 1 \cdots \lim_{z \mathop \to \infty} \dfrac 1 1\) L'Hôpital's Rule:Corollary 2
\(\ds \) \(=\) \(\ds 1^y\)
\(\ds \) \(=\) \(\ds 1\)

$\blacksquare$