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

Interchange of integrals and limits

  1. Aug 22, 2009 #1
    Can anybody please show me an example of a convergent sequence of real valued functions of a real variable where the integral of the limit of the sequence is different than the limit of the integrals of functions in the sequence ?

    Thank you
  2. jcsd
  3. Aug 22, 2009 #2
    Try a sequence where convergence is not uniform
  4. Aug 22, 2009 #3
    [tex]\lim _{n\rightarrow \infty } \left( \int _{0}^{1}\!{x}^{n} \left( 1-x
    \right) \left( n+1 \right) \left( n+2 \right) {dx} \right) [/tex]
  5. Aug 22, 2009 #4
    So the integral of the limit is zero and the limit of the integrals is 1.
    Thank you g_edgar
  6. Oct 12, 2009 #5
    If we were taking the Lebesgue integral on this sequence, we would also have the same result, right?
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook