\(\dfrac{1+\dfrac{1}{3}+\dfrac{1}{5}+...+\dfrac{1}{999}}{\dfrac{1}{1.999}+\dfrac{1}{3.997}+...+\dfrac{1}{997.3}+\dfrac{1}{999.1}}\)
\(=\dfrac{\left(1+\dfrac{1}{999}\right)+\left(\dfrac{1}{3}+\dfrac{1}{997}\right)+...+\left(\dfrac{1}{499}+\dfrac{1}{501}\right)}{2\left(\dfrac{1}{1.999}+\dfrac{1}{3.997}+...+\dfrac{1}{499.501}\right)}\)
\(=\dfrac{\dfrac{1000}{1.999}+\dfrac{1000}{3.997}+...+\dfrac{1000}{499.501}}{2\left(\dfrac{1}{1.999}+\dfrac{1}{3.997}+...+\dfrac{1}{499.501}\right)}\)
\(=\dfrac{1000\left(\dfrac{1}{1.999}+\dfrac{1}{3.997}+...+\dfrac{1}{499.501}\right)}{2\left(\dfrac{1}{1.999}+\dfrac{1}{3.997}+...+\dfrac{1}{499.501}\right)}\)
\(=\dfrac{1000}{2}=500\)
