\(\frac{1}{3}+\frac{1}{6}+\frac{1}{10}+...+\frac{2}{x\left[x+1\right]}=\frac{2017}{2019}\)
\(\Leftrightarrow\frac{2}{2.3}+\frac{2}{3.4}+\frac{2}{4.5}+...+\frac{2}{x\left[x+1\right]}=\frac{2017}{2019}\)
\(\Rightarrow2\left[\frac{1}{2\cdot3}+\frac{1}{3\cdot4}+\frac{1}{4\cdot5}+...+\frac{1}{x\left[x+1\right]}\right]=\frac{2017}{2019}\)
\(\Rightarrow2\left[\frac{1}{2}-\frac{1}{3}+\frac{1}{3}-\frac{1}{4}+\frac{1}{4}-\frac{1}{5}+...+\frac{1}{x}-\frac{1}{x+1}\right]=\frac{2017}{2019}\)
\(\Rightarrow2\left[\frac{1}{2}-\frac{1}{x+1}\right]=\frac{2017}{2019}\)
\(\Rightarrow\frac{1}{2}-\frac{1}{x+1}=\frac{\frac{2017}{2019}}{2}=\frac{2017}{4038}\)
\(\Rightarrow\frac{1}{x+1}=\frac{1}{2}-\frac{2017}{4038}=\frac{1}{2019}\)
=> x + 1 = 2019 <=> x = 2018
