[tex] \frac{1}{k(k+1)}= \frac{(k+1)-k}{k(k+1)}= \frac{k+1}{k(k+1)}- \frac{k}{k(k+1)}= \frac{1}{k}- \frac{1}{k+1} . \\ \\ S= \frac{1}{1 \cdot 2} + \frac{1}{2 \cdot 3}+ \frac{1}{3 \cdot 4} +...+ \frac{1}{2013 \cdot 2014} = \\ \\ ~~~= \frac{1}{1}- \frac{1}{2}+ \frac{1}{2}- \frac{1}{3}+ \frac{1}{3}- \frac{1}{4}+...+ \frac{1}{2013}- \frac{1}{2014}= \\ \\ ~~~= \frac{1}{1}- \frac{1}{2014}= \\ \\ ~~~= \frac{2013}{2014}. [/tex]
