Question

Evaluate: lim_(n→∞) [x]+[2x]+[3x]+....+[nx]/n² Where [ ] denotes the greatest integer function.

Answer 1

We know that, x – 1 < |x| ≤ x 

⇒ 2x – 1 < [2x] ≤ 2x 

⇒ 3x – 1 < [3x] ≤ 3x

.................... 

.................... 

⇒ nx – 1 < [nx] ≤ nx 

∴ (x + 2x + 3x + .... + nx) – n < [x] + [2x] + ..... +[nx] ≤ (x + 2x + .... + nx)

Aliter 

We know that [x] = x – {x}