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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?
     
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