The value of `lim_(n->oo) (1^2 . n+2^2.(n-1)+......+n^2 . 1)/(1^3+2^3+......+n^3)` is equal to

Options

A. `1//3`
B. `2//3`
C. `1//2`
D. `1//6`

Answer

A

Explanation

`lim_(xto oo) (1^2.n+2^2.(n-1)+3^2.(n-2)+......+n^2.1)/(1^3+2^3......+n^3)`
`=lim_(xto oo)(sum_(r=1)^(n) r^2(n-r+1))/(sum_(r=1)^(n) r^3)`
`=lim_(xto oo)(sum_(r=1)^(n) {(n+1)r^2-r^3})/(sum_(r=1)^(n) r^3)`
`=lim_(ntooo) ((n+_1)(sum_(r=1)^(n)r^2)-(sum_(r=1)^nr^3))/((sum_(r=1)^(n)r^3))`
`=lim_(ntooo) ({((n+_1)n(n+1)(2n+1))/(6)}-{(n(n+1))/(2)}^2)/({(n(n+1))/(2)}^2) `
`lim_(xtooo) {(2)/(3) (2+(1)/(n))=(1)/(3)`