Not that large

Algebra Level 5

\[ \begin{eqnarray} A&=&\frac{1}{1\times2}+\frac{1}{3\times4}+\cdots+\frac{1}{2003\times2004}+\frac{1}{2005\times2006} \\ B&=&\frac{1}{1004\times2006}+\frac{1}{1005\times2005}+\cdots+\frac{1}{2006\times1004} \end{eqnarray} \]

Find the value of \( {\frac{A}{B}} \).

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