Euler's Number as Limit of n over nth Root of n Factorial

From ProofWiki
Jump to navigation Jump to search

Theorem

$\displaystyle e = \lim_{n \mathop \to \infty} \dfrac n {\sqrt [n] {n!} }$

where:

$e$ denotes Euler's number
$n!$ denotes $n$ factorial.


Proof

\(\displaystyle \lim_{n \mathop \to \infty} {n!} {n^n \sqrt n e^{-n} }\) \(=\) \(\displaystyle \sqrt {2 \pi}\) Lemma for Stirling's Formula
\(\displaystyle \leadsto \ \ \) \(\displaystyle e\) \(=\) \(\displaystyle \lim_{n \mathop \to \infty} \dfrac {n \paren {2 \pi n}^{1 / 2 n} } {\sqrt [n] {n!} }\)
\(\displaystyle \leadsto \ \ \) \(\displaystyle e\) \(=\) \(\displaystyle \lim_{n \mathop \to \infty} \dfrac {n \sqrt {\paren {2 \pi n}^{1 / n} } } {\sqrt [n] {n!} }\)

From Limit of Root of Positive Real Number:

$\displaystyle \lim_{n \mathop \to \infty} \paren {2 \pi}^{1 / 2 n} = 1$

and from Limit of Integer to Reciprocal Power:

$n^{1 / n} = 1$

Hence the result.

$\blacksquare$


Sources