\(E=\frac{1}{2\times9}+\frac{1}{9\times7}+\frac{1}{7\times19}+...+\frac{1}{252\times509}\)
\(E=\frac{2}{4\times9}+\frac{2}{9\times14}+\frac{2}{14\times19}+...+\frac{2}{504\times509}\)
\(E=\frac{2}{5}\times\left(\frac{5}{4\times9}+\frac{5}{9\times14}+\frac{5}{14\times19}+...+\frac{5}{504\times509}\right)\)
\(E=\frac{2}{5}\times\left(\frac{9-4}{4\times9}+\frac{14-9}{9\times14}+\frac{19-14}{14\times19}+...+\frac{509-504}{504\times509}\right)\)
\(E=\frac{2}{5}\times\left(\frac{1}{4}-\frac{1}{9}+\frac{1}{9}-\frac{1}{14}+...+\frac{1}{504}-\frac{1}{509}\right)\)
\(E=\frac{2}{5}\times\left(\frac{1}{4}-\frac{1}{509}\right)\)
\(E=\frac{101}{1018}\)
