For all integers $k \geqslant 2$, we denote:
$$w_{k} = \frac{1}{2} \int_{k-1}^{k} \frac{(t-k+1)(k-t)}{t^{2}} dt$$
and $v_{n} = \sum_{k=n+1}^{+\infty} w_{k}$.
Deduce that
$$\left| v_{n} - \frac{1}{12n} \right| \leqslant \frac{1}{12n^{2}}$$
then that:
$$\ln n! = n \ln n - n + \frac{1}{2} \ln n + a + \frac{1}{12n} + O\left(\frac{1}{n^{2}}\right)$$