Result relating to Stirling's Formula Have a question that I'm stuck on here.
Let $$r_n= \frac{\sqrt{n}}{n!}\left(\frac{n}{e}\right)^n$$
Express $\log\left(r_{n+1}/r_n\right)$ as simply as possible. For this I got $\left(\frac{1}{2}+n\right)\log\left(1+\frac{1}{n}\right)-1$.
b) Verify that the following limit exists and calculate it:
$$\lim_{x\rightarrow 0}\frac{\left(1+\frac{x}{2}\right)\log(1+x)-x}{x^3}.$$ I got $\dfrac{1}{12}$, which I'm pretty sure is correct. I used L'Hôpital's rule. 
c) Use your answers to parts a) and b), and the comparison test to show that
$\displaystyle\sum_{n=1}^\infty \log\left(\frac{r_{n+1}}{r_n}\right)$ converges. Deduce that $\displaystyle\lim_{n\rightarrow\infty} r_n$ exists.
Struggling on part c, can't explicitly link it to parts a and b, did not find a suitable series for the comparison test.
Sorry for lack of LaTex, if someone could edit that would be great. I tried to use LaTex on the first question but it screwed up. 

I think the only thing you're missing is that if you take the expression in b) and substitute $x=1/n$ so it becomes a limit as $n\to\infty$, it becomes a limit very closely related to what you have in a). 
