Question

If [.] denote the greatest integer function then $\lim _ { n \rightarrow \infty }$ $\frac{\lbrack x\rbrack + \lbrack 2x\rbrack + .... + \lbrack nx\rbrack}{n^{2}}$is –

• A. 0
• B. x
• C. x/2
• D. x²/2

Answer 1

Answer: (C)

x/2

Answer 2

nx – 1 < [nx] ≤ nx. Putting n = 1, 2, 3,…. , n and adding them,

x Σ n – n < Σ [nx] ≤ x Σ n

∴ x . $\frac { \Sigma \mathrm { n } } { \mathrm { n } ^ { 2 } }$ – $\frac { 1 } { \mathrm { n } }$ < $\frac { \Sigma [ \mathrm { nx } ] } { \mathrm { n } ^ { 2 } }$ ≤ x . … (1)

Now,

= x . – =

As the two limits are equal, by (1)

$\frac { \Sigma [ \mathrm { nx } ] } { \mathrm { n } ^ { 2 } }$ = .

Hence (3) is correct answer.