Gọi $A = \dfrac{1}{1.2.3}+\dfrac{1}{2.3.4}+....+\dfrac{1}{2007.2008.2009} $
$⇒ 2A = \dfrac{2}{1.2.3}+\dfrac{2}{2.3.4}+....+\dfrac{2}{2007.2008.2009} $
$=\dfrac{1}{1.2}-\dfrac{1}{2.3}+\dfrac{1}{2.3}-\dfrac{1}{3.4} +...+\dfrac{1}{2007.2008} -\dfrac{1}{2008.2009} $
$=\dfrac{1}{2} - \dfrac{1}{2008.2009} $
$⇒A = \dfrac{\dfrac{1}{2} - \dfrac{1}{2008.2009}}{2} $