_ n [ 1 1- n ^4 + 8 1- n ^4 + . .+ n ^3 1- n ^4 ]=

Step-by-step solution

Limit evaluation using cube sum formula. Combining the terms over the common denominator 1 - n^4 yields the numerator as the sum of cubes from 1^3 to n^3 : L = _ n 1^3 + 2^3 + + n^3 1 - n^4 Applying the known identity _ k=1 ^ n k^3 = n^2(n+1)^2 4 : L = _ n n^2(n+1)^2 4(1 - n^4) = _ n n^4 + 2n^3 + n^2 4 - 4n^4 Dividing numerator and denominator by n^4 : L = _ n 1 + 2 n + 1 n^2 4 n^4 - 4 As n , terms with n in the denominator vanish, resulting in: L = 1 -4 = - 1 4 This matches option D.