Với `n in NN^**`, ta có:

`1/((k+1).sqrt(k))`

`= (sqrt(k))/(k (k+1)) `

`= sqrt(k) . 1/(k (k+1)) `

`= sqrt(k) . (1/k - 1/(k+1))`

`= sqrt(k) . (1/(sqrt(k)) - 1/(sqrt(k+1)))(1/(sqrt(k)) +1/(sqrt(k+1)))`

`< sqrt(k) . (1/(sqrt(k)) - 1/(sqrt(k + 1)))(1/(sqrt(k)) + 1/(sqrt(k)))`

`= sqrt(k) . 2/(sqrt(k)) . (1/(sqrt(k)) - 1/(sqrt(k+1)))`

`= 2 .  (1/(sqrt(k)) - 1/(sqrt(k+1)))`

Khi đó:

`VT = 1/(2sqrt(1)) + 1/(3sqrt(2)) + ... + 1/((n+1)sqrt(n))`

`< 2 . (1/sqrt(1) - 1/(sqrt(2) + 1/sqrt(2) - 1/sqrt(3) + ... - 1/sqrt(n) + 1/sqrt(n) - 1/sqrt(n+1))`

`= 2 . (1 - 1/(sqrt(n+1)))`

Vì `1 - 1/sqrt(n+1) < 1` nên `2.(1-1/sqrt(n+1)) < 2`

Khi đó: `VT < 2` (đpcm)