[tex]\displaystyle \frac{1}{1 \cdot 2} + \frac{1}{2 \cdot 3} + \frac{1}{3 \cdot 4} +...+ \frac{1}{2012\cdot 2013} = \\ \\ = \frac{1}{1} - \frac{1}{2} + \frac{1}{2} - \frac{1}{3} + \frac{1}{3} - \frac{1}{4} +...+ \frac{1}{2012} - \frac{1}{2013}= \\ \\ =1- \frac{1}{2013}= \frac{2013-1}{2013}= \frac{2012}{2013} [/tex]
