设函数 $f(x) = \frac{x}{x + 2}$（$x > 0$），观察：$f_{1}(x) = f(x) = \frac{x}{x + 2}$，$f_{2}(x) = f\!\left(f_{1}(x)\right) = \frac{x}{3x + 4}$，$f_{3}(x) = f\!\left(f_{2}(x)\right) = \frac{x}{7x + 8}$，$f_{4}(x) = f\!\left(f_{3}(x)\right) = \frac{x}{15x + 16}$. 根据以上事实，由归纳推理可得：当 $n \in \mathbf{N}^{*}$ 且 $n \geqslant 2$ 时，$f_n(x) = f(f_{n-1}(x)) =$______.