\(A=\frac{1}{1.2.3}+\frac{1}{2.3.4}+...+\frac{1}{37.38.39}\)
\(\Rightarrow2A=\frac{1}{1.2}-\frac{1}{2.3}+\frac{1}{2.3}-\frac{1}{3.4}+...+\frac{1}{37.38}-\frac{1}{38.39}\)
\(\Rightarrow2A=\frac{1}{1.2}-\frac{1}{38.39}=\frac{1}{2}-\frac{1}{1482}=\frac{740}{1482}\)
\(\Rightarrow A=\frac{740}{1482}\div2\)