Let `S_n=sum_(k=0)^n n/(n^2+kn+k^2)` and `T_n=sum_(k=0)^(n-1) n/(n^2+kn+k^2)` for n=1,2,3... then(correct options may be more than one) (a)`S_n lt pi/(3sqrt(3))` b) `S_n gt pi/(3sqrt(3))`(c) `T_n lt pi/(3sqrt(3))` (d) `T_n gt pi/(3sqrt(3))`

Options

A. `S_(n)lt(pi)/(3sqrt(3))`
B. `S_(n)gt(pi)/(3sqrt(3))`
C. `T_(n) lt (pi)/(3sqrt(3))`
D. `T_(n)gt(pi)/(3sqrt(3))`

Answer

A, D

Explanation

`S_(n)lt lim_(nto oo) S_(n)=lim_(n to oo) sum_(k=1)^(n)1/n 1/(1+k//n+(k//n)^(2))`
`=int_(0)^(1)(dx)/(1+x+x^(2))`
`=int_(0)^(1)(dx)/((x+1/2)^(2)+3/4)`
`=[2/(sqrt(3))"tan"^(-1)((x+1/2)/((sqrt(3))/2))]_(0)^(1)=(pi)/(3sqrt(3))`
Now `T_(n)gt(pi)/(3sqrt(3))`as
`h sum_(k=0)^(n-1)f(k//n)gt int_(0)^(1)f(x)dxgt h sum_(k=1)^(n)f(k//n)`