1/1×2 + 1/2×3 + 1/3×4 + ...+ 1/2006×2007
= 1 - 1/2 + 1/2 - 1/3 + 1/3 - 1/4 + ...+ 1/2006 - 1/2007
= 1 - 1/2007
= 2006/2007