Ta có:
`1/2^2 = 1/4`
`1/3^2 < 1/(2.3)`
`1/4^2 < 1/(3.4)`
`1/5^2 < 1/(4.5)`
`...`
`1/100^2 < 1/(99.100)`
Công các vế lại với nhau ta có:
`1/2^2 + 1/3^2 + 1/4^2 + 1/5^2 + ... + 1/100^2 < 1/4 + 1/(2.3) + 1/(3.4) + /(4.5) + ... + 1/(99.100)`
Hay `A < 1/4 + 1/(2.3) + 1/(3.4) + /(4.5) + ... + 1/(99.100)`
Lại có:
`1/4 + 1/(2.3) 1/(3.4) + /(4.5) + ... + 1/(99.100)`
`= 1/4 + 1/2 - 1/3 + 1/3 - 1/4 + 1/4 - 1/5 + ... + 1/99 - 1/100`
`= 1/4 + 1/2 - 1/100`
`= 1/4 + 2/4 - 1/100`
`= 3/4 - 1/100 < 3/4`
Mà `A < 1/4 + 1/(2.3) + 1/(3.4) + /(4.5) + ... + 1/(99.100) < 3/4`
`=> A < 3/4`
Vậy `A < 3/4`
