设函数 $f(x)=\left\{\begin{array}{l}\frac{1}{(x-1)^{\alpha-1}}, 1 < x < e \\ \frac{1}{x \ln ^{\alpha+1} x}, x \geq e\end{array}\right.$, 若反常积分 $\int_1^{+\infty} f(x) \mathrm{d} x$ 收敛，则
$\text{A.}$ $\alpha < -2$
$\text{B.}$ $\alpha>2$
$\text{C.}$ $-2 < \alpha < 0$
$\text{D.}$ $0 < \alpha < 2$