Đặt
\(A=\frac{1}{1.2.3}+\frac{1}{2.3.4}+...+\frac{1}{37.38.39}\Rightarrow2A=\frac{2}{1.2.3}+\frac{2}{2.3.4}+...+\frac{2}{37.38.39}=\frac{1}{1.2}-\frac{1}{2.3}+\frac{1}{2.3}-\frac{1}{3.4}+...+\frac{1}{37.38}-\frac{1}{38.39}=\frac{1}{1.2}-\frac{1}{38.39}=\frac{1}{2}-\frac{1}{1428}\Rightarrow A=\left(\frac{1}{2}-\frac{1}{1428}\right):2=\frac{713}{1428}.\frac{1}{2}\)
=>S=\(\frac{713}{1428}.\frac{1}{2}.1428+185.8=\frac{713}{2}+185.8=\frac{713}{2}+1480=356+\frac{1}{2}+1480=1836\frac{1}{2}\)