The value of _ n [ 1 1 - n^2 + 2 1 - n^2 + + n 1 - n^2 ]^3 is

Step-by-step solution

Let L = _ n [ 1 1 - n^2 + 2 1 - n^2 + + n 1 - n^2 ]^3 The expression inside the bracket can be simplified by taking the common denominator: S_n = 1 + 2 + + n 1 - n^2 Using the sum of the first n natural numbers, 1 + 2 + + n = n(n+1) 2 : S_n = n(n+1) 2 1 - n^2 S_n = n(n+1) 2(1 - n)(1 + n) S_n = n 2(1 - n) Taking the limit as n : _ n S_n = _ n n 2(1 - n) = _ n 1 2 ( 1 n - 1 ) = - 1 2 Therefore, the required limit is: L = (- 1 2 )^3 = - 1 8 Answer: -1 8