$\begin{array}{l}B=\dfrac{1+2+2^2+2^3+\ldots+2^{2008}}{1-2^{2009}}\\\to 2B=\dfrac{2\left(1+2+2^2+2^3+\ldots+2^{2008}\right)}{1-2^{2009}}\\\to 2B=\dfrac{2+2^2+2^3+\ldots+2^{2009}}{1-2^{2009}}\\\to 2B-B=\dfrac{\left(2+2^2+2^3+\ldots+2^{2009}\right)-\left(1+2+2^2+2^3+\ldots+2^{2008}\right)}{1-2^{2009}}\\\to B=\dfrac{2^{2009}-1}{1-2^{2009}}\\\to B=\dfrac{-1.\left(1-2^{2009}\right)}{1-2^{2009}}\\\to B=-1 \end{array}$
