\(B=\dfrac{2008}{1}+\dfrac{2007}{2}+\dfrac{2006}{3}+...+\dfrac{2}{2007}+\dfrac{1}{2008}\)
\(=\left(1+\dfrac{2007}{2}\right)+\left(1+\dfrac{2006}{3}\right)+...+\left(\dfrac{2}{1007}+1\right)+\left(\dfrac{1}{2008}+1\right)+1\)
\(=\dfrac{2009}{1}+\dfrac{2009}{2}+...+\dfrac{2009}{2008}\)
\(=2009\)