Ta có: \(\frac14=\frac14;\frac15<\frac14;\frac17<\frac14;\frac18<\frac14\)
Do đó: \(\frac14+\frac15+\frac17+\frac18<\frac14+\frac14+\frac14+\frac14=1\) (1)
Ta có: \(\frac14>\frac18;\frac15>\frac18;\frac17>\frac18;\frac18=\frac18\)
Do đó: \(\frac14+\frac15+\frac17+\frac18>\frac18+\frac18+\frac18+\frac18=\frac48=\frac12\left(2\right)\)
Từ (1),(2) suy ra \(\frac12<\frac14+\frac15+\frac17+\frac18<1\) (3)
Ta có: \(\frac19>\frac{1}{18};\frac{1}{10}>\frac{1}{18};\ldots;\frac{1}{17}>\frac{1}{18}\)
Do đó: \(\frac19+\frac{1}{10}+\cdots+\frac{1}{17}>\frac{1}{18}+\frac{1}{18}+\cdots+\frac{1}{18}=\frac{9}{18}=\frac12\) (4)
Ta có: \(\frac19=\frac19;\frac{1}{10}<\frac19;\ldots;\frac{1}{18}<\frac19\)
Do đó: \(\frac19+\frac{1}{10}+\cdots+\frac{1}{17}<\frac19+\frac19+\cdots+\frac19=\frac99=1\) (5)
Từ (4),(5) suy ra \(\frac12<\frac19+\frac{1}{10}+\cdots+\frac{1}{17}<1\) (6)
Ta có: \(\frac12+\frac13+\frac16\)
\(=\frac36+\frac26+\frac16=\frac66=1\) (7)
Từ (3),(6),(7) suy ra \(\frac12+\frac12+1<\frac12+\frac13+\cdots+\frac{1}{17}<1+1+1\)
=>2<T<3
=>T không là số tự nhiên
