Home
Class 12
MATHS
lim(n to oo) {(1)/(n)+(1)/(n+1)+(1)/(n+2...

`lim_(n to oo) {(1)/(n)+(1)/(n+1)+(1)/(n+2)+...+(1)/(3n)}=`

A

log2

B

log3

C

log5

D

0

Text Solution

Verified by Experts

The correct Answer is:
B
`S=underset(n to oo)lim{(1)/(n)+(1)/(n+1)+(1)/(n+2)+...+(1)/(3n)}`
`S=underset(n to oo)lim{(1)/(n)underset(r=0)overset(2n)sum(1)/(n+r)}`
`S=underset(n to oo)lim{(1)/(n)underset(r=0)overset(2n)sum(n)/(n+r)}=underset(n to oo)lim{(1)/(n)underset(r=0)overset(2n)sum(1)/(1+(r//n))}`
Now, Lower limit `underset(nto oo)lim(r)/(n)`
`underset(nto oo)lim(r)/(n)=0" " [ :. r=0 "for the first term"]`
Upper Limit `underset(nto 0)lim(r)/(n)`
`underset(nto oo)lim(r)/(n)" " [ :. r=2n "for the last term"]`
`:. S=underset(n to oo)lim{(1)/(n)underset(r=0)overset(2n)sum(1)/(1+r(//n))}`
`rArr S=underset(0)overset(2)int(1)/(1+x)dx=[ log(1+x)]_(0)^(2)=log3-log=log3`
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_(nto oo) {(1)/(n+1)+(1)/(n+2)+(1)/(n+3)+...+(1)/(n+n)} is, equal to

lim_(n rarr oo)[(1)/(n)+(1)/(n+1)+(1)/(n+2)+.....+(1)/(2n)]

lim_(n rarr oo)((-1)^(n)n)/(n+1)

lim_(n to oo) ((1)/(n)+(1)/(n+1)+...+(1)/(3n)) is equal to

lim_ (n rarr oo) ((1) / (n + 1) + (1) / (n + 2) + (1) / (n + 3) + ...... + (1) / (6n ))

The sum S = lim_ (n rarr oo) [(1) / (n + 1) + (1) / (n + 2) + (1) / (n + 3) + ----- + (1) / (2n)]

lim_ (n rarr oo) (1) / (n) [(1) / (n + 1) + (2) / (n + 2) + ... + (3n) / (4n)]

OBJECTIVE RD SHARMA-DEFINITE INTEGRALS-Chapter Test 2
  1. lim(n to oo) {(1)/(n)+(1)/(n+1)+(1)/(n+2)+...+(1)/(3n)}=

    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

    |