[tex]\it lg\dfrac{1}{2}+lg\dfrac{2}{3}+lg\dfrac{3}{4}+\ ...\ +lg\dfrac{999}{1000}=lg\dfrac{1}{\not2}\cdot\dfrac{\not2}{\not3}\cdot\dfrac{\not3}{\not4}\cdot\ ...\ \cdot\dfrac{\not999}{1000}=lg\dfrac{1}{1000}=\\ \\ \\ =lg10^{-3}=-3lg10=-3\cdot1=-3[/tex]
