Đặt
\(S=\frac{1}{1.3}+\frac{1}{3.5}+...+\frac{1}{2003+2005}\)
\(2S=\frac{2}{1.3}+\frac{2}{3.5}+...+\frac{2}{2003+2005}\)
\(2S=\frac{1}{1}-\frac{1}{3}+\frac{1}{3}-\frac{1}{5}+...+\frac{1}{2003}-\frac{1}{2005}\)
\(2S=\frac{1}{1}-\frac{1}{2005}=\frac{2004}{2005}\)
\(S=\frac{2004}{2005}.\frac{1}{2}=\frac{1002}{2005}\)