Question

Найти сумму 1/1*2 * 1/2*3 + 1/3*4 +...+ 1/2006*2007

Answer 1

$frac{1}{1*2} + frac{1}{2*3} + frac{1}{3*4} + ...+ frac{1}{2006*2007}= \ = frac{2-1}{1*2} + frac{3-2}{2*3} + frac{4-3}{3*4} + ...+ frac{2007-2006}{2006*2007}= \ = frac{1}{1} - frac{1}{2} + frac{1}{2} - frac{1}{3}+ frac{1}{3} - frac{1}{4} + ...+ frac{1}{2006} - frac{1}{2007} = \ = 1 - frac{1}{2007} = frac{2006}{2007}$