For all integers $k \geqslant 2$, we denote:
$$w_{k} = \frac{1}{2} \int_{k-1}^{k} \frac{(t-k+1)(k-t)}{t^{2}} dt$$
Justify the convergence of the series $\sum_{k \geqslant 2} w_{k}$.
Deduce that there exists a real number $a$ such that:
$$\ln n! = n \ln n - n + \frac{1}{2} \ln n + a + v_{n}$$
where $v_{n} = \sum_{k=n+1}^{+\infty} w_{k}$.