Question

Calculati suma:

S=(1-\frac{1}{2})+(1-\frac{1}{3})+(1-\frac{1}{4})+...+(1-\frac{1}{2007})-(\frac{1}{1}+\frac{1}{2}+\frac{2}{3}+\frac{3}{4}+...+\frac{2006}{2007})

Answer 1

   
[tex]\displaystyle S = (1-\frac{1}{2})+(1-\frac{1}{3})+(1-\frac{1}{4})+...+(1-\frac{1}{2007})- \\ \\ -(\frac{1}{1}+\frac{1}{2}+\frac{2}{3}+\frac{3}{4}+...+\frac{2006}{2007})= \\ \\ \\ =\frac{1}{2}+\frac{2}{3}+\frac{3}{4}+ \cdots +\frac{2006}{2007}- \frac{1}{1}-\frac{1}{2}-\frac{2}{3}-\frac{3}{4}- \cdots -\frac{2006}{2007}= \\ \\ \\ =\frac{1}{2}-\frac{1}{2}+\frac{2}{3}-\frac{2}{3}+\frac{3}{4}-\frac{3}{4} + \cdots + \frac{2006}{2007}-\frac{2006}{2007} - \frac{1}{1} =-\frac{1}{1} =\boxed{-1}[/tex]