Ta có: \(\frac{1}{201}<\frac{1}{200};\frac{1}{202}<\frac{1}{200};\ldots;\frac{1}{300}<\frac{1}{200}\)
Do đó: \(\frac{1}{201}+\frac{1}{202}+\cdots+\frac{1}{300}<\frac{1}{200}+\frac{1}{200}+\cdots+\frac{1}{200}=\frac{100}{200}=\frac12\) (1)
Ta có: \(\frac{1}{301}<\frac{1}{300};\frac{1}{302}<\frac{1}{300};\ldots;\frac{1}{400}<\frac{1}{300}\)
Do đó: \(\frac{1}{301}+\frac{1}{302}+\cdots+\frac{1}{400}<\frac{1}{300}+\frac{1}{300}+\cdots+\frac{1}{300}=\frac{100}{300}=\frac13\) (2)
Từ (1),(2) ta có: \(\left(\frac{1}{201}+\frac{1}{202}+\cdots+\frac{1}{300}\right)+\left(\frac{1}{301}+\frac{1}{302}+\cdots+\frac{1}{400}\right)<\frac12+\frac13=\frac56\)
