Factorial of Integer plus Reciprocal of Integer

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Theorem

Let $x \in \Z$ be a positive integer.

Then:

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


Proof

We have that:

\(\ds \dfrac {\paren {n + x}!} {n! n^x}\) \(=\) \(\ds \dfrac {\paren {n + 1} \paren {n + 2} \cdots \paren {n + x} } {n^x}\)
\(\ds \) \(=\) \(\ds \paren {1 + \frac 1 n} \paren {1 + \frac 2 n} \cdots \paren {1 + \frac x n}\)

As $n \to \infty$, the quantity on the right hand side indeed tends to $1$.

$\blacksquare$


Sources