\(\dfrac{1}{1.4}+\dfrac{1}{4.7}+\dfrac{1}{7.10}+...+\dfrac{1}{397.400}\)
\(=\dfrac{1}{3}.\left(\dfrac{3}{1.4}+\dfrac{3}{4.7}+\dfrac{3}{7.10}+...+\dfrac{3}{397.400}\right)\)
\(=\dfrac{1}{3}.\left(\dfrac{4-1}{1.4}+\dfrac{7-4}{4.7}+\dfrac{10-7}{7.10}+...+\dfrac{400-397}{397.400}\right)\)
\(=\dfrac{1}{3}.\left(1-\dfrac{1}{4}+\dfrac{1}{4}-\dfrac{1}{7}+\dfrac{1}{7}-\dfrac{1}{10}+...+\dfrac{1}{397}-\dfrac{1}{400}\right)\)
\(=\dfrac{1}{3}.\left(1-\dfrac{1}{400}\right)\)
\(=\dfrac{1}{3}.\dfrac{399}{400}\)
\(=\dfrac{133}{400}\)