[tex]S_n = \sum\limits_{k=1}^{8} \dfrac{1}{k!\cdot (k+2)} = \sum\limits_{k=1}^{8} \dfrac{k+1}{(k+1)!\cdot (k+2)} = \\ \\ = \sum\limits_{k=1}^8 \dfrac{k+2-1}{(k+1)!\cdot (k+2)} = \\ \\ = \sum\limits_{k=1}^8 \dfrac{k+2}{(k+1)!\cdot (k+2)} - \sum\limits_{k=1}^8 \dfrac{1}{(k+1)!\cdot (k+2)} = \\ \\ = \sum\limits_{k=1}^8 \dfrac{1}{(k+1)!} - \sum\limits_{k=1}^8 \dfrac{1}{(k+2)!}= [/tex]
[tex]= \dfrac{1}{2!} + \dfrac{1}{3!} +...+\dfrac{1}{9!} - \Big(\dfrac{1}{3!} + \dfrac{1}{4!} +... +\dfrac{1}{10!}\Big) = \\ \\ = \dfrac{1}{2!} - \dfrac{1}{10!}[/tex]
