Home
Class 12
MATHS
underset(n to oo)lim(1)/(2)" " under...

`underset(n to oo)lim(1)/(2)" " underset(r=+1)overset(2n)sum (r)/(sqrt(n^(2)+r^(2)))` equals

A

`1+sqrt5`

B

`-1+sqrt5`

C

`-1+sqrt2`

D

`1+sqrt2`

Text Solution

Verified by Experts

The correct Answer is:
B
Let `l=underset(nrarroo)(lim)(1)/(n)sum_(r=1)^(2n)(r)/(sqrt(n^(2)-r^(2)))`
`=underset(nrarroo)(lim)(1)/(n)sum_(r=1)^(2n)(r)/(sqrt(1+((r)/(n))^(2)))`
`=underset(nrarroo)(lim)(1)/(n)sum_(r=1)^(2n)((r)/(n))/(sqrt(1+((r)/(n))^(2)))`
Put `(r)/(n)=x,(1)/(n)=dx, underset(nrarroo)(lim)sum_(r=1)^(2n)=int_(0)^(2)`
`therefore" "l=int_(0)^(2)(x)/(sqrt(1+x^(2)))dx=[sqrt(1+x^(2))]_(0)^(2)`
`=sqrt5-1`
Promotional Banner

Topper's Solved these Questions

  • DEFINITE INTEGRALS

    MHTCET PREVIOUS YEAR PAPERS AND PRACTICE PAPERS|Exercise PRACTICE EXERCISE (Exercise 2) (MISCELLANEOUS PROBLEMS)|76 Videos

  • DEFINITE INTEGRALS

    MHTCET PREVIOUS YEAR PAPERS AND PRACTICE PAPERS|Exercise MHT CET Corner|22 Videos

  • CONTINUITY

    MHTCET PREVIOUS YEAR PAPERS AND PRACTICE PAPERS|Exercise MHT CET Corner|16 Videos

  • DIFFERENTIAL EQUATION

    MHTCET PREVIOUS YEAR PAPERS AND PRACTICE PAPERS|Exercise MHT CET Corner|27 Videos

Similar Questions

Explore conceptually related problems

underset(r=1)overset(n)sum ( r )/(r^(4)+r^(2)+1) is equal to

underset(r=1)overset(n)(sum)r(.^(n)C_(r)-.^(n)C_(r-1)) is equal to

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

The value of underset( n rarroo)(lim) underset(r=1)overset(r=4n)(sum)(sqrt(n))/(sqrt(r ) (3 sqrt(r)+4)sqrt(n)^(2)) is equal to

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

If k=underset(r=0)overset(n)(sum)(1)/(.^(n)C_(r)) , then underset(r=0)overset(n)(sum)(r)/(.^(n)C_(r)) is equal to

Use Sandwich theorem to evaluate : underset( n rarr oo) ( "Lim") underset(k=n^(2))overset((n+1)^(2))sum (1)/( sqrt( k ))

MHTCET PREVIOUS YEAR PAPERS AND PRACTICE PAPERS-DEFINITE INTEGRALS-MHT CET Corner
  1. underset(n to oo)lim(1)/(2)" " underset(r=+1)overset(2n)sum (r)/(s...

    Text Solution

    |

  2. int (-pi/2)^(pi/2)log((2-sin x)/(2+sinx))dx is equal to

    Text Solution

    |

  3. int (0)^(pi //2)((root(n)(secx))/(root(n)(secx)+root(n)("cosec"x)))dx=

    Text Solution

    |

  4. The value of int 0 ^ 1 x ^ 2 ( 1 - x ^ 2 ) ^ (3//2 ) dx ...

    Text Solution

    |

  5. The value of int0^oox/((1+x)(x^2+1))dx is

    Text Solution

    |

  6. Evaluate int(0)^(pi)(x dx)/(1+cos alpha sin x),where 0lt alpha lt pi.

    Text Solution

    |

  7. int(pi//2)^(pi//2)(cosx)/(1+e^(x))dx is equal to

    Text Solution

    |

  8. int(0)^(pi//2)(1)/((1+tanx))dx=?

    Text Solution

    |

  9. If int(0)^(1) tan^(-1) x dx = p , then the value of int(0)^(1) tan^(-1...

    Text Solution

    |

  10. The value of int (0)^(pi//2) log ("cosec "x) dx is

    Text Solution

    |

  11. Which of the following is true ?

    Text Solution

    |

  12. int(0)^(5) 1/((x-1)(x-2))dx is equal to

    Text Solution

    |

  13. int(pi/4)^(pi/2) e^x(logsinx+cotx)dx

    Text Solution

    |

  14. The value of int(0)^(pi) x sin^(3) x dx is

    Text Solution

    |

  15. The value of int0 ^(pi/2) (cos3x+1)/(cosx - 1) dx is equal to

    Text Solution

    |

  16. The value of underset(0)overset(1)int tan^(-1) ((2x-1)/(1+x-x^(2)))dx ...

    Text Solution

    |

  17. If f is a continous function, then

    Text Solution

    |

  18. The value of int(-pi)^(pi) sin^(3) x cos^(2) x dx is equal to

    Text Solution

    |

  19. The value of int(-1)^(1) log ((x-1)/(x+1))dx is

    Text Solution

    |

  20. int(pi//6)^(pi//3)(1)/((1+sqrt(tanx)))dx=(pi)/(12)

    Text Solution

    |

  21. int (1)^(2)e^(x) (1/x - 1/(x^(2)))dx is qual to

    Text Solution

    |