已知函数 $f(x)=\left\{\begin{array}{cc}x & 0 \leq x \leq 1 \\ 2-x & 1 < x \leq 2\end{array}\right.$ ，试计算下列各题:
(1) $S_0=\int_0^2 f(x) e^{-x} \mathrm{~d} x$;
(2) $S_1=\int_2^4 f(x-2) e^{-x} \mathrm{~d} x$;
(3) $S_n=\int_{2 n}^{2 n+2} f(x-2 n) e^{-x} \mathrm{~d} x(n=2,3, \cdots)$ ；
(4) $S=\sum_{n=0}^{\infty} S_n$.