Question

Evaluate: $\underset{n\to \infty }{lim}{\left[\frac{n!}{n^n}\right]}^{1/n}$

Answer 1

Consider the given limit.

$K=\underset{n\to \infty }{lim}{\left(\frac{n!}{n^n}\right)}^{1/n}$

Take log on both sides.

$\ln K=\underset{n\to \infty }{lim}\frac{1}{n}\ln \left(\frac{n!}{n^n}\right)$

$\ln K=\underset{n\to \infty }{lim}\frac{1}{n}\left[\ln (\frac{1}{n}.\frac{2}{n}.\frac{3}{n}........\frac{n}{n})\right]$

$\ln K=\underset{n\to \infty }{lim}\frac{1}{n}\sum _{r=1}^n\ln \left(\frac{r}{n}\right)$

$\ln K=\underset{0}{\overset{1}{\int }}\ln xdx$

$\ln K={\left[x\ln x\right]}_0^1-{\left[x\right]}_0^1$

$\ln K=0-0-1$

$\ln K=-1$

$K=e^{-1}$

$K=\frac{1}{e}$

Hence, this is the required value of the limit.