\(\dfrac{1}{1+2}+\dfrac{1}{1+2+3}+\dfrac{1}{1+2+3+4}+...+\dfrac{1}{1+2+3+...+2007}\)
\(=\dfrac{1}{\dfrac{2\times3}{2}}+\dfrac{1}{\dfrac{3\times4}{2}}+\dfrac{1}{\dfrac{4\times5}{2}}+...+\dfrac{1}{\dfrac{2007\times2008}{2}}\)
\(=\dfrac{2}{2\times3}+\dfrac{2}{3\times4}+\dfrac{2}{4\times5}+...+\dfrac{2}{2007\times2008}\)
\(=2\times(\dfrac{1}{2\times3}+\dfrac{1}{3\times4}+\dfrac{1}{4\times5}+...+\dfrac{1}{2007\times2008})\)
\(=2\left(\dfrac{1}{2}-\dfrac{1}{3}+\dfrac{1}{3}-\dfrac{1}{4}+\dfrac{1}{4}-\dfrac{1}{5}+...+\dfrac{1}{2006}-\dfrac{1}{2007}\right)\)
\(=2\left(\dfrac{1}{2}-\dfrac{1}{2007}\right)\)
\(=1-\dfrac{2}{2007}\)
\(=\dfrac{2005}{2007}\)
