Question

Calculati: 

$\frac{1}{1*2} + \frac{1}{2*3} + \frac{1}{3*4} +...+ \frac{1}{2013*2014}$

Answer 1

$\frac{1}{1*2}= \frac{1}{1}-\frac{1}{2} \\ \\ \frac{1}{2*3}= \frac{1}{2} -\frac{1}{3} \\ \\ \frac{1}{3*4}= \frac{1}{3} -\frac{1}{4} \\ .............................................. \\ \frac{1}{2013*2014}= \frac{1}{2013}-\frac{1}{2014} \\ \\ \\ S= \frac{1}{1}^{(2014} -\frac{1}{2014}= \frac{2014-1}{2014} \\ \\ \\\boxed{S=\frac{2013}{2014}}$