1. Not finding help here? Sign up for a free 30min tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Test for integrability

  1. Dec 30, 2006 #1
    It started out as an attempt to solve a HW question (which I also posted in the appropriate forum), but now I'm just curious as to the general case;

    Assume f>0 is a measurable function from [0,infinity) to itself. Then if xf(x) tends to zero as x tends to zero, there is a positive [tex]\epsilon[/tex] for which the integral of f over [tex][0,\epsilon ][/tex] is finite.

    This is following the intuition that while 1/x isn't integrable, multypling it by anything that tends to zero is.

    What do you say? True, not true?
    Last edited: Dec 30, 2006
  2. jcsd
  3. Dec 30, 2006 #2


    User Avatar
    Homework Helper

    No, take f(x)=1/(x ln(x)).
  4. Dec 30, 2006 #3

    matt grime

    User Avatar
    Science Advisor
    Homework Helper

    1/x isn't even a function from [0,infinity) to itself, never mind a measurable one.
  5. Dec 30, 2006 #4
    Ok, one by one:

    StatusX, 1/ln(x) doesn't tend to zero when x tends to zero, it tends to infinity... Edit: wait, it does, I'm an idiot.

    matt - It's almost everywhere defined, what's the problem? Define 1/0=81...
    Last edited: Dec 30, 2006
  6. Dec 30, 2006 #5
    Ok, why is 1/(xln(x)) not integrable?

    Another stupid question, I answered myself... Thanks!
    Last edited: Dec 30, 2006
  7. Dec 30, 2006 #6

    Gib Z

    User Avatar
    Homework Helper

    What baffles me is that you define 1/0 to be 81...
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?

Similar Discussions: Test for integrability
  1. The integral test (Replies: 8)

  2. Integral test HELP! (Replies: 4)