Home
Class 12
MATHS
lim(n to oo) " " sum(r=2n+1)^(3n) (...

`lim_(n to oo) " " sum_(r=2n+1)^(3n) (n)/(r^(2)-n^(2))` is equal to

A

`ln sqrt((2)/(3))`

B

ln `sqrt((3)/(2))`

C

`"ln" (2)/(3)`

D

`"ln" (3)/(2)`

Text Solution

Verified by Experts

The correct Answer is:
B
We have,
`S=underset(n to oo)lim" " underset(r=2n+1)overset(3n)sum (n)/(r^(2)-n^(2))`
`rArr S=underset(n to oo)lim" " underset(r=2n+1)overset(3n)sum (1)/(((r)/(n))^(2)-1)*(1)/(n)`
`rArr S=underset(2)overset(3)int(1)/(x^(2)-1)dx=(1)/(2)[ log_(e)|(x-1)/(x+1)|]_(2)^(3)`
`rArr S=(1)/(2)[ "ln" (1)/(2)-"ln"(1)/(3)]=(1)/(2)ln ((3)/(2))= lnsqrt((3)/(2))`
Promotional Banner

Topper's Solved these Questions

  • DEFINITE INTEGRALS

    OBJECTIVE RD SHARMA|Exercise Section I - Solved Mcqs|145 Videos

  • DEFINITE INTEGRALS

    OBJECTIVE RD SHARMA|Exercise Section II - Assertion Reason Type|12 Videos

  • CONTINUITY AND DIFFERENTIABILITY

    OBJECTIVE RD SHARMA|Exercise Exercise|86 Videos

  • DERIVATIVE AS A RATE MEASURER

    OBJECTIVE RD SHARMA|Exercise Exercise|26 Videos

Similar Questions

Explore conceptually related problems

lim_(n rarr oo)sum_(r=2n+1)^(3n)(n)/(r^(2)-n^(2)) is equal to

lim_(n to oo)(1)/(2)" " sum_(r=+1)^(2n) (r)/(sqrt(n^(2)+r^(2))) equals

The value of lim_(n to oo)(1)/(n).sum_(r=1)^(2n)(r)/(sqrt(n^(2)+r^(2))) is equal to

lim_(n to oo) sum_(r=1)^(n) (1)/(n)e^(r//n) is

lim_(n to oo) sum_(r=1)^(n) ((r^(3))/(r^(4) + n^(4))) equals to-

lim_ (n rarr oo) sum_ (k = 1) ^ (n) (k) / (n ^ (2) + k ^ (2)) is equals to

The value of lim_(n rarr oo)sum_(i=1)^(n)(i)/(n^2)sin((pi i^(2))/(n^(2))) is equal to

lim_(x to oo ) tan { sum_(r=1)^(n) tan^(-1) ((1)/( 1 +r +r^2))} is equal to ________.

OBJECTIVE RD SHARMA-DEFINITE INTEGRALS-Chapter Test 2
  1. lim(n to oo) " " sum(r=2n+1)^(3n) (n)/(r^(2)-n^(2)) is equal to

    Text Solution

    |

  2. The integral int(0)^(r pi) sin^(2x)x dx is equal to

    Text Solution

    |

  3. The value of the integral int(0)^(2)x[x]dx

    Text Solution

    |

  4. The value of integral sum (k=1)^(n) int (0)^(1) f(k - 1+x) dx is

    Text Solution

    |

  5. Let f(x) be a funntion satifying f'(x)=f(x) with f(0)=1 and g(x) be th...

    Text Solution

    |

  6. If I=int(0)^(1) cos{ 2 "cot"^(-1)sqrt((1-x)/(1+x))}dx then

    Text Solution

    |

  7. The value of int(a)^(a+(pi//2))(sin^(4)x+cos^(4)x)dx is

    Text Solution

    |

  8. The vaue of int(-1)^(2) (|x|)/(x)dx is

    Text Solution

    |

  9. The value of int0^1 (x^(3))/(1+x^(8))dx is

    Text Solution

    |

  10. The value of int(0)^(3) xsqrt(1+x)dx, is

    Text Solution

    |

  11. The value of the integral int(0)^(1) log sin ((pix)/(2))dx is

    Text Solution

    |

  12. The value of the integral int(0)^(pi)x log sin x dx is

    Text Solution

    |

  13. If I(1)=int(0)^(oo) (dx)/(1+x^(4))dx and I(2)underset(0)overset(oo)in...

    Text Solution

    |

  14. If f(x)={{:(x,"for " x lt 1),(x-1,"for " x ge1):},"then" int(0)^(2) x...

    Text Solution

    |

  15. The value of the integral int(0)^(2) (1)/((x^(2)+1)^(3//2))dx is

    Text Solution

    |

  16. If int(0)^(2a) f(x)dx=int(0)^(2a) f(x)dx, then

    Text Solution

    |

  17. If int(0)^(36) (1)/(2x+9)dx =log k, is equal to

    Text Solution

    |

  18. The value of the integral int(0)^(pi//2) sin^(6) x dx, is

    Text Solution

    |

  19. If int(0)^(oo) e^(-x^(2))dx=sqrt((pi)/(2))"then"int(0)^(oo) e^(-ax^(2)...

    Text Solution

    |

  20. The value of the integral int 0^oo 1/(1+x^4)dx is

    Text Solution

    |

  21. If int(pi//2)^(x) sqrt(3-2sin^(2)u) dx+int(dx)^(dy) equal pi//2

    Text Solution

    |