If `S_1=Sigman, S_2=Sigman^2,S_3=Sigma n^3, "then the value of " lim_(n to oo) (S_1(1+(S_3)/(8)))/(S_2^2)` is equal to

Options

A. `(3)/(32)`
B. ` (3)/(64)`
C. `(9)/(32)`
D. `(9)/(64)`

Answer

D

Explanation

We have,
`S_1=Sigma (n =(n+1))/(2) ,S_2 = Sigman^2=(n(n+1) (2n+1))/(6)`
and m,`S_3=Sigma n^3={(n(n+1))/(2)}^2`
`therefore lim_(xto oo) (S_1((1+S_3)/(8)))/S_(2)^(2)`
` = lim_(xtooo) ((n(n+1))/(2){1_+(n^2(n+1)^2)/(32)})/((n^2(n+1)^2(2n+1)^2)/(36))`
`=lim_(xtooo) (18)/(n(n+1)(2n+1)^2){1+(n^2(n+1)^2)/(32)}`
` =lim_(xto oo) (18)/(32)xx ({32+n^2(n+1)^2})/(n(n+1)(2n+1)^2)`
` =(9)/(16)lim_(xtooo) ((32)/(n^4)+(1+(1)/(n))^2)/((1+(1)/(n))(2+(1)/(n))^2)=(9)/(16)xx(1)/(1xx4)=(9)/(64)`.