Results 1 to 6 of 6

Math Help - Help with proving that Improper Integral is Divergent

  1. #1
    Newbie
    Joined
    Mar 2014
    From
    United States
    Posts
    2

    Help with proving that Improper Integral is Divergent

    The problem is attached in this post.


    Lim t -> ∞ ∫ dx/xlnx from 1 to t

    u-substitution:

    u=lnx
    du=1/x dx

    Lim t -> ∞ ∫ 1/u du

    Lim t -> ∞ ln u

    Lim t -> ∞ ln(lnx) from 1 to t

    Lim t -> ∞ ln(lnt) - ln(0)

    = ∞ - ∞ = 0 (This is incorrect since the answer is that the integral is divergent).

    Attached Thumbnails Attached Thumbnails Help with proving that Improper Integral is Divergent-screen-shot-2014-03-06-3.14.52-am.png  
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor
    Joined
    Nov 2013
    From
    California
    Posts
    2,207
    Thanks
    849

    Re: Help with proving that Improper Integral is Divergent

    Quote Originally Posted by student93 View Post
    The problem is attached in this post.


    Lim t -> ∞ ∫ dx/xlnx from 1 to t

    u-substitution:

    u=lnx
    du=1/x dx

    Lim t -> ∞ ∫ 1/u du

    Lim t -> ∞ ln u

    Lim t -> ∞ ln(lnx) from 1 to t

    Lim t -> ∞ ln(lnt) - ln(0)

    = ∞ - ∞ = 0 (This is incorrect since the answer is that the integral is divergent).

    you're fine up until trying to evaluate the definite integral. You can't do algebra with infinities and get usable answers. What you have to do is use the integral test with the harmonic series. Since you know the harmonic series diverges it must be that the integral of 1/x diverges as well.
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Newbie
    Joined
    Mar 2014
    From
    United States
    Posts
    2

    Re: Help with proving that Improper Integral is Divergent

    Do you mean the Direct Comparison Test? Also could you show how would do an integral test for this question?
    Follow Math Help Forum on Facebook and Google+

  4. #4
    MHF Contributor
    Joined
    Nov 2013
    From
    California
    Posts
    2,207
    Thanks
    849

    Re: Help with proving that Improper Integral is Divergent

    No I mean the Integral test.

    Just read that link on what the test is. Given that the harmonic series $\displaystyle{\sum_{k=1}^{\infty}} \dfrac{1}{k}$ diverges it should be obvious how to apply the integral test to this problem.
    Follow Math Help Forum on Facebook and Google+

  5. #5
    Super Member
    Joined
    Mar 2010
    Posts
    980
    Thanks
    236

    Re: Help with proving that Improper Integral is Divergent

    You don't need to know about the integral test for convergence of a series for this problem. Since the integral is improper at both endpoints, the convergence needs to be analyzed separately (you should have a special definition for this case), and it needs to converge on both sides to be called convergent. So just take some a, 1<a<\infty, and

    \int_1^\infty \frac{1}{x\ln{x}}\,dx = \int_1^a \frac{1}{x\ln{x}}\,dx + \int_a^\infty \frac{1}{x\ln{x}}\,dx

    and you correctly calculated that both integrals diverge. So the original integral diverges.

    - Hollywood
    Follow Math Help Forum on Facebook and Google+

  6. #6
    Senior Member
    Joined
    Sep 2013
    From
    Portland
    Posts
    410
    Thanks
    49

    Re: Help with proving that Improper Integral is Divergent

    you can take the lim as b approaches infinity with b being your upper bound of integration then solve the indefinite integral then take the limit of F(b) - F(1)

    or you can use the limit of the integral a smaller function and if it diverges then the larger function will also diverge, this is the comparison test.

    it does diverge
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Improper integral (convergent/divergent)
    Posted in the Calculus Forum
    Replies: 2
    Last Post: August 25th 2010, 03:47 PM
  2. Improper integral convergent or divergent?
    Posted in the Calculus Forum
    Replies: 1
    Last Post: July 5th 2010, 12:47 PM
  3. Improper integral. Convergent or divergent?
    Posted in the Calculus Forum
    Replies: 2
    Last Post: May 2nd 2010, 06:20 AM
  4. Divergent Improper Integral
    Posted in the Calculus Forum
    Replies: 3
    Last Post: April 18th 2010, 06:21 PM
  5. Proving an improper integral divergent.
    Posted in the Calculus Forum
    Replies: 7
    Last Post: April 7th 2010, 12:11 AM

Search Tags


/mathhelpforum @mathhelpforum