$\begin{array}{l}\text { Example } 3 . u_{k}=\frac{1}{k(k+1)(k+2)(k+3)} \\ \text { Find } \sum_{k=1}^{n} u_{k} .\end{array}$
\begin{array}{l} \text { Sol } u_{k}=\frac{1}{3}\left[\frac{1}{k(k+1)(k+2)}\right. \\ -\left.\frac{1}{(k+1)(k+2)(k+3)}\right], \\ \therefore \sum_{k=1}^{n} u_{k} \\ = \frac{1}{3}\left[\frac{1}{6}-\frac{1}{(n+1)(n+2)(n+3)}\right]\end{array}