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Xét vế trái :
`3/(3.7) + 3/(7.11) + ... + 3/( (4n-1) (4n+3) ) (n \ne 1/4, n \ne (-3)/4)`
`= 3 [1/(3.7) + 1/(7.11) + ... + 1/( (4n-1) (4n+3) ) ]`
`= 3 . 1/4 [1/3 - 1/7 + 1/7 - 1/11 + ... + 1/(4n-1) - 1/(4n+3) ]`
`= 3/4 [1/3 + (-1/7 +1/7) + ... + (-1/(4n-1) + 1/(4n-1) ) - 1/(4n+3)]`
`= 3/4 [1/3 - 1/(4n+3)]`
`= 3/4 . 1/3 - 3/4 . 1/(4n + 3)`
`= 1/4 - 3/(4 (4n + 3) )`
`= (4n + 3)/(4 (4n+3) ) - 3/(4 (4n+3) )`
`= (4n+3 - 3)/(4 (4n+3) )`
`= (4n + (3-3) )/(4 (4n+3) )`
`= (4n)/(4 (4n+3) )`
`= n/(4n+3)` (Bằng vế phải)
`-> 3/(3.7) + 3/(7.11) + ... + 3/( (4n-1) (4n+3) ) = n/(4n+3) (n \ne 1/4, n \ne (-3)/4 )`
