+ \(\frac{1}{1\cdot2}+\frac{1}{3\cdot4}+...+\frac{1}{399\cdot400}\)
\(=1-\frac{1}{2}+\frac{1}{3}-\frac{1}{4}+...+\frac{1}{399}-\frac{1}{400}\)
\(=\left(1+\frac{1}{2}+\frac{1}{3}+...+\frac{1}{400}\right)-2\left(\frac{1}{2}+\frac{1}{4}+...+\frac{1}{400}\right)\)
\(=\left(1+\frac{1}{2}+\frac{1}{3}+...+\frac{1}{400}\right)-\left(1+\frac{1}{2}+\frac{1}{3}+...+\frac{1}{200}\right)\)
\(=\frac{1}{201}+\frac{1}{202}+\frac{1}{203}+...+\frac{1}{400}\)
+ \(\frac{1}{201\cdot400}+\frac{1}{202\cdot399}+...+\frac{1}{300\cdot301}\)
\(=\frac{1}{601}\cdot\left(\frac{201+400}{201\cdot400}+\frac{202+399}{202\cdot399}+...+\frac{300+301}{300\cdot301}\right)\)
\(=\frac{1}{601}\cdot\left(\frac{1}{201}+\frac{1}{400}+\frac{1}{202}+\frac{1}{399}+...+\frac{1}{300}+\frac{1}{301}\right)\)
\(=\frac{1}{601}\left(\frac{1}{201}+\frac{1}{202}+\frac{1}{203}+...+\frac{1}{400}\right)\)
Do đó : \(A=\frac{1}{\frac{1}{601}}=601\)
