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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))`

Text Solution

Verified by Experts

The correct Answer is:
A, D
`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)`
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