`S=5/{2^2}+5/{3^2}+5/{4^2}+...+5/{100^2}`
Ta có:
`S<5/{1.2}+5/{2.3}+5/{3.4}+...+5/{99.100}`
`S<5.(1/{1.2}+1/{2.3}+1/{3.4}+...+1/{99.100})`
`S<5.(1-1/2+1/2-1/3+1/3-1/4+...+1/{99}-1/{100})`
`S<5.(1-1/{100})`
`S<5.1` (vì `1-1/{100}<1`)
`=>S<5` $(1)$
Ta lại có:
`S>5/{2.3}+5/{3.4}+5/{4…5}+...+5/{100.101}`
`S>5.(1/{2.3}+1/{3.4}+1/{4.5}+...+1/{100.101})`
`S>5.(1/2-1/3+1/3-1/4+1/4-1/5+...+1/{100}-1/{101})`
`S>5.(1/2-1/{101})`
`S>5. ({101}/{202}-2/{202})=5. {99}/{202}`
`=>S>5. {99}/{205}>5.{82}/{205}`
`=>S>5. 2/ 5`
`=>S>2` $(2)$
Từ `(1);(2)=>2<S<5`
