Problem involving Stirling's formula

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March 14th 2011, 09:42 AM
maximus101

Problem involving Stirling's formula

if we know $n! ~ \sqrt{2 \pi n}(\frac{n}{e})^n$
as $n \rightarrow \infty$
(i) define $r_{n} = \frac{\sqrt{n}}{n!}}(\frac{n}{e})^n$
express $log(\frac{r_{n+1}}{r_{n}})$
(ii)prove that the following limit exists and calculate it
$lim_{x \rightarrow 0} \frac{(1+x/2)log(1+x) -x}{x^3}$

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