Euler's Number as Limit of n over nth Root of n Factorial
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Theorem
- $\ds e = \lim_{n \mathop \to \infty} \dfrac n {\sqrt [n] {n!} }$
where:
- $e$ denotes Euler's number
- $n!$ denotes $n$ factorial.
Proof
\(\ds \lim_{n \mathop \to \infty} {n!} {n^n \sqrt n e^{-n} }\) | \(=\) | \(\ds \sqrt {2 \pi}\) | Lemma for Stirling's Formula | |||||||||||
\(\ds \leadsto \ \ \) | \(\ds e\) | \(=\) | \(\ds \lim_{n \mathop \to \infty} \dfrac {n \paren {2 \pi n}^{1 / 2 n} } {\sqrt [n] {n!} }\) | |||||||||||
\(\ds \leadsto \ \ \) | \(\ds e\) | \(=\) | \(\ds \lim_{n \mathop \to \infty} \dfrac {n \sqrt {\paren {2 \pi n}^{1 / n} } } {\sqrt [n] {n!} }\) |
From Limit of Root of Positive Real Number:
- $\ds \lim_{n \mathop \to \infty} \paren {2 \pi}^{1 / 2 n} = 1$
and from Limit of Integer to Reciprocal Power:
- $n^{1 / n} = 1$
Hence the result.
$\blacksquare$
Sources
- 1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $2 \cdotp 71828 \, 18284 \, 59045 \, 23536 \, 02874 \, 71352 \, 66249 \, 77572 \, 47093 \, 69995 \ldots$