Home
Class 12
MATHS
The value of underset(nrarrinfty)(lim)(1...

The value of `underset(nrarrinfty)(lim)(1)/(n).sum_(r=1)^(2n)(r)/(sqrt(n^(2)+r^(2)))` is equal to

A

`1+sqrt(5)`

B

`-1+sqrt(5)`

C

`-1+sqrt(2)`

D

`1sqrt(2)`

Text Solution

Verified by Experts

The correct Answer is:
B
Let
`S=underset(n to oo)lim(1)/(2)" " underset(r=+1)overset(2n)sum (r)/(sqrt(n^(2)+r^(2)))`
`S=underset(n to oo)lim" " underset(r=+1)overset(2n)sum (r//n)/(sqrt(1+(r^(2))/(n^(2))))xx(1)/(n)`
`rArr S=underset(0) overset(2)int (x)/(sqrt(1+x^(2)))dx=[sqrt(x^(2)+1)]_(0)^(2)=sqrt(5)-1`
Promotional Banner

Topper's Solved these Questions

  • DEFINITE INTEGRALS

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

  • DEFINITE INTEGRALS

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

  • CONTINUITY AND DIFFERENTIABILITY

    OBJECTIVE RD SHARMA ENGLISH|Exercise Exercise|87 Videos

  • DERIVATIVE AS A RATE MEASURER

    OBJECTIVE RD SHARMA ENGLISH|Exercise Exercise|26 Videos

Similar Questions

Explore conceptually related problems

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

lim_(n->oo)1/nsum_(r=1)^(2n)r/(sqrt(n^2+r^2)) equals

lim_(nrarroo) sum_(r=0)^(n-1) (1)/(sqrt(n^(2)-r^(2)))

sum_(r=0)^(n)((r+2)/(r+1))*""^(n)C_(r) is equal to :

sum_(r=0)^(n)((r+2)/(r+1))*""^(n)C_(r) is equal to :

(sum_(r=1)^n r^4)/(sum_(r=1)^n r^2) is equal to

lim_(nrarr0) sum_(r=1)^(n) ((r^(3))/(r^(4)+n^(4))) equals to :

l isum_(n-gtoo)sum_(r=1)^n1/(sqrt(4n^2-r^2))

The value of ("lim")_(n rarr oo)sum_(r=1)^(4n)(sqrt(n))/(sqrt(r)(3sqrt(r)+sqrt(n))^2) is equal to

The value of ("lim")_(nvecoo)sum_(r=1)^(4n)(sqrt(n))/(sqrt(r)(3sqrt(r)+sqrt(n))^2) is equal to 1/(35) (b) 1/4 (c) 1/(10) (d) 1/5

OBJECTIVE RD SHARMA ENGLISH-DEFINITE INTEGRALS-Chapter Test 2
  1. The value of underset(nrarrinfty)(lim)(1)/(n).sum(r=1)^(2n)(r)/(sqrt(n...

    Text Solution

    |

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

    Text Solution

    |

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

    Text Solution

    |

  4. Let f (x) be a function satisfying f(x)=f(x) with f(0) = 1 and g be th...

    Text Solution

    |

  5. If I=int(0)^(1)cos(2 cot^(-1)sqrt(((1-x)/(1+x))))dx then :

    Text Solution

    |

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

    Text Solution

    |

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

    Text Solution

    |

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

    Text Solution

    |

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

    Text Solution

    |

  10. Evaluate int(0)^(1)log(sin((pix)/(2)))dx

    Text Solution

    |

  11. Evaluate int(0)^(pi) xlog sinx dx

    Text Solution

    |

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

    Text Solution

    |

  13. If f(x)={{:(x,xlt1),(x-1,xge1):}, then underset(0)overset(2)intx^(2)f(...

    Text Solution

    |

  14. The value of the integral overset(1)underset(0)int (1)/((1+x^(2))^(3//...

    Text Solution

    |

  15. Prove that: int0^(2a)f(x)dx=int0^(2a)f(2a-x)dxdot

    Text Solution

    |

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

    Text Solution

    |

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

    Text Solution

    |

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

    Text Solution

    |

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

    Text Solution

    |

  20. The value of alpha in [0,2pi] which does not satify the equation int(p...

    Text Solution

    |

  21. lim(x to 0)(int(0)^(x^(2))sinsqrt(t) dt)/(x^(3)) is equl to

    Text Solution

    |