Ta có: \(\frac{1}{21}>\frac{1}{30};\frac{1}{22}>\frac{1}{30};\ldots;\frac{1}{30}=\frac{1}{30}\)
Do đó: \(\frac{1}{21}+\frac{1}{22}+\cdots+\frac{1}{30}>\frac{1}{30}+\frac{1}{30}+\cdots+\frac{1}{30}=\frac{10}{30}=\frac13\) (1)
Ta có: \(\frac{1}{31}>\frac{1}{40};\frac{1}{32}>\frac{1}{40};\ldots;\frac{1}{40}=\frac{1}{40}\)
Do đó: \(\frac{1}{31}+\frac{1}{32}+\cdots+\frac{1}{40}>\frac{1}{40}+\frac{1}{40}+\cdots+\frac{1}{40}=\frac{10}{40}=\frac14\) (2)
Ta có: \(\frac{1}{41}>\frac{1}{50};\frac{1}{42}>\frac{1}{50};\ldots;\frac{1}{50}=\frac{1}{50}\)
Do đó: \(\frac{1}{41}+\frac{1}{42}+\cdots+\frac{1}{50}>\frac{1}{50}+\frac{1}{50}+\cdots+\frac{1}{50}=\frac{10}{50}=\frac15\) (3)
Từ (1),(2),(3) suy ra \(\left(\frac{1}{21}+\frac{1}{22}+\cdots+\frac{1}{30}\right)+\left(\frac{1}{31}+\frac{1}{32}+\cdots+\frac{1}{40}\right)+\left(\frac{1}{41}+\frac{1}{42}+\cdots+\frac{1}{50}\right)>\frac13+\frac14+\frac15=\frac{47}{60}\)
