1. Limited time only! Sign up for a free 30min personal 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!

Problem with divergent integral

  1. Oct 29, 2009 #1
    I'm confused with the following integral.

    Let a > 1.

    [tex] \int_{a}^{\infty} \left( \dfrac{1}{t} - \dfrac{1}{t-1} \right) ~ dt = \left[ \log \left| \dfrac{t}{t-1} \right| \right]_{a}^{\infty} = - \log \left| \dfrac{a}{a-1} \right| [/tex]

    This should be the correct result. But I could also decompose the integral into two parts (because the integrand is a sum) and compute:

    [tex] \int_{a}^{\infty} \left( \dfrac{1}{t} - \dfrac{1}{t-1} \right) ~ dt = \left| \log \vert t \vert \right|_{a}^{\infty} - \left[ \log \vert t - 1 \vert \right]_{a}^{\infty} = - \log \left| \dfrac{a}{a-1} \right| + \infty - \infty [/tex]

    But [tex] \infty - \infty [/tex] is of course not defined!

    Where did I make a mistake? I don't find it.
  2. jcsd
  3. Oct 29, 2009 #2
    Decomposing the integral into two parts is justified only if both integrals are finite.
  4. Oct 29, 2009 #3
    Thanks :smile:
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Similar Discussions: Problem with divergent integral
  1. Divergence Problem (Replies: 3)

  2. Divergence Problem (Replies: 1)

  3. Divergent Integral (Replies: 3)