# find the limit

• Feb 14th 2010, 11:03 PM
chialin4
find the limit
$\lim_{n->oo}\frac{(n!)^{1/n}}{n}$
• Feb 15th 2010, 03:16 AM
Krizalid
it's a Riemann sum.

first write $a=e^{\ln a}.$
• Feb 15th 2010, 10:11 AM
chialin4
Quote:

Originally Posted by Krizalid
it's a Riemann sum.

first write $a=e^{\ln a}.$

sorry,i can not see it
can u give me more idea??pls thx
• Feb 23rd 2010, 07:11 PM
Drexel28
Quote:

Originally Posted by chialin4
$\lim_{n->oo}\frac{(n!)^{1/n}}{n}$

It is commonly known that for convergent sequences the ration and root test gives the same result, in particular

$\lim_{n\to\infty}\frac{(n!)^{\frac{1}{n}}}{n}=\lim _{n\to\infty}\frac{(n+1)!}{(n+1)^{n+1}}\cdot\frac{ n^n}{n!}=\lim_{n\to\infty}\frac{n^n}{(n+1)^n}=\lim _{n\to\infty}\left(1+\frac{1}{n}\right)^n=e$