lim-x-n-n-n-

Question Number 168256 by mathlove last updated on 07/Apr/22

\lim_{x \to \infty} \frac{n!}{n^{n}} = ?

Answered by MJS_new last updated on 07/Apr/22

first thought \frac{n!}{n^{n}} means \frac{1 \times 2 \times \dots \times n}{n \times n \times \dots \times n} means

\frac{n - 1 n u m b e r s s m a l l e r t h a n n}{n - 1 t i m e s n} which looks

a lot like 0

prove :

n! \sim {(\frac{n}{e})}^{n} \sqrt{2 π n}

\frac{n!}{n^{n}} \sim \frac{\sqrt{2 π n}}{e^{n}}

obviously the answer is 0

Facebook Tweet Pin