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
- 1997: Donald E. Knuth: The Art of Computer Programming: Volume 1: Fundamental Algorithms (3rd ed.) ... (previous) ... (next): $\S 1.2.5$: Permutations and Factorials: Exercise $22$