Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Homework Help: Finding Where an Improper Integral Converges

  1. Oct 17, 2012 #1
    Hi all,

    This is a case of a book answer going against Wolfram's and my answer.

    The problem is ∫e(ln(x)/x)dx

    The book claims the answer is ∞.

    I would think it is a case of ∞/∞ and use L'Hospital's Rule. Wolfram has the same solution.

    *= lima->∞(1/x)/1
    = 0

    Which would be correct?
     
  2. jcsd
  3. Oct 17, 2012 #2

    pwsnafu

    User Avatar
    Science Advisor

  4. Oct 17, 2012 #3
    Good call. I must have typed it in incorrectly.

    Is it correct that the integral diverges because lim as x approaches infinity of lnx/x is an indeterminate?

    Thanks!
     
  5. Oct 17, 2012 #4

    SammyS

    User Avatar
    Staff Emeritus
    Science Advisor
    Homework Helper
    Gold Member

    No. Being indeterminate has nothing to do with it.

    It's true that [itex]\displaystyle \lim_{x\,\to\,\infty} \frac{\ln(x)}{x}=0\,,\ [/itex] as well as [itex]\displaystyle \lim_{x\,\to\,\infty} \frac{1}{x}=0\ .[/itex]

    Those results are necessary conditions (but not sufficient) for [itex]\displaystyle \int_{e}^{\infty} \frac{\ln(x)}{x}\,dx\ [/itex] to converge.

    Does [itex]\displaystyle \int_{e}^{\infty} \frac{1}{x}\,dx\ [/itex] converge?
     
  6. Oct 19, 2012 #5
    No, but I'm not sure why.


    EDIT:

    It's because ∫1/x dx = ln(x)

    and ln(∞) = ∞

    So since

    ∫ln(x)/x = ln(x)2/2, and ln(∞)2/2= ∞,

    the integral diverges.

    Correct?
     
    Last edited: Oct 19, 2012
  7. Oct 20, 2012 #6

    SammyS

    User Avatar
    Staff Emeritus
    Science Advisor
    Homework Helper
    Gold Member

    We know that [itex]\displaystyle \int_{e}^{\infty} \frac{1}{x}\,dx[/itex] diverges, because we know that [itex]\displaystyle \sum_{n=3}^{\infty} \frac{1}{n}[/itex] diverges.
     
  8. Oct 21, 2012 #7
    But 1/infinity goes to zero, correct?
     
  9. Oct 21, 2012 #8

    SammyS

    User Avatar
    Staff Emeritus
    Science Advisor
    Homework Helper
    Gold Member

    Yes 1/n goes to zero as n goes to ∞ . That's basically looking at the sequence [1/n] .

    However, the infinite series, [itex]\displaystyle \sum \frac{1}{n}\ ,[/itex] diverges.
     
  10. Oct 21, 2012 #9
    Thanks for clarifying that; I need to practice these.
     
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook