4. If let $a_{n}=\frac{1}{3}+\frac{1}{12}+\frac{1}{30}+\frac{1}{60}+\cdots+\frac{2}{n(n-1)(n-2)}+\frac{2}{(n+1) n(n-1)}$, then $\lim _{n \rightarrow \infty} a_{n}=$
$\begin{array}{l}\text { 4. } \frac{1}{2} \quad \text { Detailed Explanation: } a_{n}=\frac{1}{3 C_{2}^{0}}+\frac{1}{4 C_{3}^{1}}+\frac{1}{5 C_{4}^{2}}+\cdots+\frac{1}{n C_{n-1}^{n-3}}+\frac{1}{(n+1) C_{n}^{n-2}}+\frac{1}{(n+1) C_{n}^{n-1}}-\frac{1}{(n+1) C_{n}^{n-1}} \\ =\frac{1}{3 C_{2}^{0}}+\frac{1}{4 C_{3}^{1}}+\frac{1}{5 C_{4}^{2}}+\cdots+\frac{1}{n C_{n-1}^{n-3}}+\frac{1}{n C_{n-1}^{n-2}}-\frac{1}{(n+1) C_{n}^{n-1}}=\cdots=\frac{1}{2}-\frac{1}{(n+1) C_{n}^{n-1}}, \therefore \lim _{n \rightarrow \infty} a_{n}=\frac{1}{2} .\end{array}$