Answer
$\frac{1}{2}$
Work Step by Step
$\frac{1}{n(n-1)}$ = $\frac{1}{n-1}-\frac{1}{n}$
$\Sigma_{n=3}^{\infty}\frac{1}{n(n-1)}$ = $\Sigma_{n=3}^{\infty}(\frac{1}{n-1}-\frac{1}{n})$
$S_{N}$ = $(\frac{1}{2}-\frac{1}{3})+(\frac{1}{3}-\frac{1}{4})+(\frac{1}{4}-\frac{1}{5})+...+(\frac{1}{N-1}-\frac{1}{N})$ = $\frac{1}{2}-\frac{1}{N}$
$S$ = $\lim\limits_{N \to \infty}S_{N}$ = $\lim\limits_{N \to \infty}(\frac{1}{2}-\frac{1}{N})$ = $\frac{1}{2}$
