Đặt A = \(\frac{1}{1.3}+\frac{1}{3.5}+\frac{1}{5.7}+.....+\frac{1}{17.19}\)
\(\Rightarrow2A=\frac{2}{1.3}+\frac{2}{3.5}+\frac{2}{5.7}+.....+\frac{2}{17.19}\)
\(\Rightarrow2A=1-\frac{1}{3}+\frac{1}{3}-\frac{1}{5}+.....+\frac{1}{17}-\frac{1}{19}\)
\(\Rightarrow2A=1-\frac{1}{19}\)
\(\Rightarrow2A=\frac{18}{19}\)
\(\Rightarrow A=\frac{18}{19}.\frac{1}{2}=\frac{9}{19}\)
\(\frac{1}{1.3}+\frac{1}{3.5}+\frac{1}{5.7}+...+\frac{1}{17.19}\)
\(=\frac{1}{2}.\left(1-\frac{1}{3}+\frac{1}{3}-\frac{1}{5}+\frac{1}{5}-\frac{1}{7}+...+\frac{1}{17}-\frac{1}{19}\right)\)
\(=\frac{1}{2}.\left(1-\frac{1}{19}\right)\)
\(=\frac{1}{2}.\frac{18}{19}=\frac{9}{19}\)
