Answer
$\frac{1}{2}$
Work Step by Step
$S_{3}$ = $(\frac{1}{2}-\frac{1}{3})+(\frac{1}{3}-\frac{1}{4})+(\frac{1}{4}-\frac{1}{5})$ = $\frac{3}{10}$
$S_{4}$ = $(\frac{1}{2}-\frac{1}{3})+(\frac{1}{3}-\frac{1}{4})+(\frac{1}{4}-\frac{1}{5})+(\frac{1}{5}-\frac{1}{6})$ = $\frac{1}{3}$
$S_{5}$ = $(\frac{1}{2}-\frac{1}{3})+(\frac{1}{3}-\frac{1}{4})+(\frac{1}{4}-\frac{1}{5})+(\frac{1}{5}-\frac{1}{6})+(\frac{1}{6}-\frac{1}{7})$ = $\frac{5}{14}$
$S_{N}$ = $(\frac{1}{2}-\frac{1}{3})+(\frac{1}{3}-\frac{1}{4})+(\frac{1}{4}-\frac{1}{5})+...+(\frac{1}{N+1}-\frac{1}{N+2})$ = $\frac{1}{2}-\frac{1}{N+2}$
$S$ = $\lim\limits_{N \to \infty}S_{N}$ = $\lim\limits_{N \to \infty}(\frac{1}{2}-\frac{1}{N+2})$ = $\frac{1}{2}$
