Value of `underset(ntoinfty)(lim)((n+1)^(1//3)+(n+2)^(1//3)+…+(2n)^(1//3))/(n^(4//3))` is equal to

Options

A. `4/3(2)^(3//4)`
B. `3/4(2)^(4//3)-4/3`
C. `4/3(2)^(4//3)`
D. `3/4(2)^(4//3)-3/4`

Answer

D

Explanation

`lim_(n to infty) ((n+1)^(1//3)/n^(4//3) + (n+2)^(1//3)/(n^(4//3) +……..+(2n)^(1//3)/n^(4//3)))`
`lim_(n to infty) sum_(r=1)^(n) (n+r)^(1//3)/n^(4//3), lim_(n to infty)sum_(r=1)^(n)(n^(1//3)(1+r/n)^(1//3))/(n^(4//3))`
`lim_(0)^(1)(1+x)^(1//3) dx`
`[(1+x)^(4//3)/(4//3)]_(0)^(1) rArr (2^(4//3))/(4//3) -1/(4//3) rarr 3/4 .2^(4//3) - 3/4`