\(\frac{1}{1\cdot2\cdot3}+\frac{1}{2\cdot3\cdot4}+\cdots+\frac{1}{2019\cdot2020\cdot2021}\)
\(=\frac12\left(\frac{2}{1\cdot2\cdot3}+\frac{2}{2\cdot3\cdot4}+\cdots+\frac{2}{2019\cdot2020\cdot2021}\right)\)
\(=\frac12\left(\frac{1}{1\cdot2}-\frac{1}{2\cdot3}+\frac{1}{2\cdot3}-\frac{1}{3\cdot4}+\cdots+\frac{1}{2019\cdot2020}-\frac{1}{2020\cdot2021}\right)\)
\(=\frac12\left(\frac{1}{1\cdot2}-\frac{1}{2020\cdot2021}\right)\)
\(=\frac12\cdot\frac{1010\cdot2021-1}{2020\cdot2021}=\frac{20412009}{8164840}\)
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