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Let Sn=sum(k=0)^n n/(n^2+k n+k^2) and ...

Let `S_n=sum_(k=0)^n n/(n^2+k n+k^2) and T_n=sum_(k=0)^(n-1)n/(n^2+k n+k^2)`,for `n=1,2,3,.......,` then

A

`S_(n)lt(pi)/(3sqrt(3))`

B

`T_(n)lt(pi)/(3sqrt(3))`

C

`T_(n)lt(pi)/(2sqrt(2))`

D

`T_(n)gt(pi)/(3sqrt(3))`

Text Solution

Verified by Experts

The correct Answer is:
A, D
Given , `S_(n)=sum_(k=n)^(n)(n)/(n^(2)+kn+k^(2))`
`=sum_(k=0)^(1)(1)/(n) *((1)/(1+(k)/(n)+(k^(2))/(n^(2))))ltunderset(n tooo)(lim)sum_(k=0)^(n)(1)/(n)((1)/(1+(k)/(n)+((k)/(n))^(2)))`
`=int_(0)^(1)(1)/(1+x+x^(2))dx`
`= [(2)/(sqrt(3))tan^(-1)((2)/(sqrt(3))(x+(1)/(2)))]_(0)^(1)`
` = (2)/(sqrt(3))*((pi)/(3)-(pi)/(6))=(pi)/(3sqrt(3))i.e S_(n)lt(pi)/(3sqrt(3))`
Similarly , `T_(n)gt(pi)/(3sqrt(3))`
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