\(\text{Ta có:}\frac{9}{9.19}+\frac{9}{19.29}+\frac{9}{29.39}+....+\frac{9}{2009.2019}\)
\(=\frac{9}{10}.\left(\frac{10}{9.19}+\frac{10}{19.29}+\frac{10}{29.39}+.....+\frac{10}{2009.2019}\right)\)
\(=\frac{9}{10}\left(\frac{1}{9}-\frac{1}{2019}\right)\)
\(=\frac{9}{10}.\frac{670}{6057}\)