Ta có: \(\frac{1}{41}<\frac{1}{40};\frac{1}{42}<\frac{1}{40};\ldots;\frac{1}{60}<\frac{1}{40}\)
Do đó: \(\frac{1}{41}+\frac{1}{42}+\cdots+\frac{1}{60}<\frac{1}{40}+\frac{1}{40}+\cdots+\frac{1}{40}=\frac{20}{40}=\frac12\) (1)
ta có: \(\frac{1}{61}<\frac{1}{60};\frac{1}{62}<\frac{1}{60};\ldots;\frac{1}{80}<\frac{1}{60}\)
Do đó: \(\frac{1}{61}+\frac{1}{62}+\cdots+\frac{1}{80}<\frac{1}{60}+\frac{1}{60}+\cdots+\frac{1}{60}=\frac{20}{60}=\frac13\) (2)
Từ (1),(2) suy ra \(\left(\frac{1}{41}+\frac{1}{42}+\cdots+\frac{1}{60}\right)+\left(\frac{1}{61}+\frac{1}{62}+\cdots+\frac{1}{80}\right)<\frac12+\frac13\)
=>\(\left(\frac{1}{41}+\frac{1}{42}+\cdots+\frac{1}{60}\right)+\left(\frac{1}{61}+\frac{1}{62}+\cdots+\frac{1}{80}\right)<\frac56\) (3)
ta có: \(\frac{1}{41}>\frac{1}{60};\frac{1}{42}>\frac{1}{60};\ldots;\frac{1}{59}>\frac{1}{60};\frac{1}{60}=\frac{1}{60}\)
Do đó: \(\frac{1}{41}+\frac{1}{42}+\cdots+\frac{1}{60}>\frac{1}{60}+\frac{1}{60}+\cdots+\frac{1}{60}=\frac{20}{60}=\frac13\) (4)
ta có: \(\frac{1}{61}>\frac{1}{80};\frac{1}{62}>\frac{1}{80};\ldots;\frac{1}{80}=\frac{1}{80}\)
Do đó: \(\frac{1}{61}+\frac{1}{62}+\cdots+\frac{1}{80}>\frac{1}{80}+\frac{1}{80}+\cdots+\frac{1}{80}=\frac{20}{80}=\frac14\) (5)
Từ (4),(5) suy ra \(\left(\frac{1}{41}+\frac{1}{42}+\cdots+\frac{1}{60}\right)+\left(\frac{1}{61}+\frac{1}{62}+\cdots+\frac{1}{80}\right)>\frac13+\frac14=\frac{7}{12}\) (6)
Từ (3),(6) suy ra \(\frac{7}{12}<\frac{1}{41}+\frac{1}{42}+\cdots+\frac{1}{80}<\frac56\)
