if we know $\displaystyle n! ~ \sqrt{2 \pi n}(\frac{n}{e})^n $

as $\displaystyle n \rightarrow \infty $

(i) define $\displaystyle r_{n} = \frac{\sqrt{n}}{n!}}(\frac{n}{e})^n$

express $\displaystyle log(\frac{r_{n+1}}{r_{n}})$

(ii)prove that the following limit exists and calculate it

$\displaystyle lim_{x \rightarrow 0} \frac{(1+x/2)log(1+x) -x}{x^3}$