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Let S(n)=sum(k=1)^(n)n/(n^(2)+kn+k^(2)) ...

Let `S_(n)=sum_(k=1)^(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

A

`S_(n) lt(pi)/(3sqrt3)`

B

`S_(n) gt (pi)/(3sqrt3)`

C

`T_(n) lt (pi)/(3sqrt3)`

D

`T_(n) gt (pi)/(3sqrt3)`

Text Solution

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The correct Answer is:
D
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ARIHANT MATHS-DEFINITE INTEGRAL-Exercise (Questions Asked In Previous 13 Years Exam)
  1. For a epsilonR (the set of all real numbers) a!=-1, lim(n to oo) ((1^...

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  2. Let f:[0,1]toR (the set of all real numbers ) be a function. Suppose t...

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  3. Which of the following is true?

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  4. f(x)={(x-[x]",",if[x] "is odd"),(1+[x]-x",",if[x]" is even"):} Then,...

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  5. Let f be a non-negative function defined on the interval [0,1]. If int...

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  6. Let I(n)=int(-pi)^(pi)(sin x)/((1+pi^(x))sin x )dx, n=0,1,2,... The va...

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  7. Let S(n)=sum(k=1)^(n)n/(n^(2)+kn+k^(2)) and T(n)=sum(k=0)^(n-1)n/(n^(2...

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  8. Then integral int(pi//4)^((3pi)/4) (dx)/(1+cosx) is equal to

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  9. "Let " I(n)=int tan^(n)x dx,(n gt 1). I(4)+I(6)=a tan^(5)x+bx^(5)+C, "...

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  10. lim(nto oo)[((n+1)(n+2)…3n)/(n^(2n))]^(1//n) is equal to

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  11. The integral int(2)^(4)(logx^(2))/(logx^(2)+log(36-12x+x^(2))) dx is e...

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  12. The integral int(0)^(pi)sqrt(1+4"sin"^(2)x/2-4"sin"x/2)dx is equals to...

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  13. Prove that the value of the integral int(pi//6)^(pi//3) (dx)/(1+sqrt(c...

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  14. The intercepts on x- axis made by tangents to the curve, y=int(0)^(x)|...

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  15. If g(x)=int(0)^(x)cos^(4)t dt, then g(x+pi) equals to (a)(g(x))/(g(pi)...

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  16. The value of int(0)^(1)(8log(1+x))/(1+x^(2))dx is

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  17. For x epsilon(0,(5pi)/2), definite f(x)=int(0)^(x)sqrt(t) sin t dt. T...

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  18. Let p(x) be a function defined on R such that p'(x)=p'(1-x),"for all"...

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  19. int(0)^(pi)[cos x] dx, [ ] denotes the greatest integer function , is ...

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  20. Let I=int(0)^(1)(sinx)/(sqrtx) dx and f= int(0)^(1)( cos x)/(sqrtx) dx...

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