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How to Integrate a Sigma Sum?

  1. Oct 21, 2013 #1
    How would you Integrate a Sigma Sum?
     
  2. jcsd
  3. Oct 21, 2013 #2

    arildno

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    If you with a sigma sum means a sum written with the big sigma symbol, you integrate it just like any other sum.
     
  4. Oct 21, 2013 #3

    Mark44

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    I.e., term by term.
     
  5. Oct 21, 2013 #4
  6. Oct 21, 2013 #5

    arildno

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    Not necessarily, but usually beneficially! :smile:
     
  7. Oct 21, 2013 #6

    FeDeX_LaTeX

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    On a related note, an interesting question to consider would be when the orders of a sum and an integral can be interchanged.
     
  8. Oct 21, 2013 #7
  9. Oct 21, 2013 #8
    Exactly. If it's an infinite sum and it doesn't converge uniformly, then we cannot do so term by term. The usual example is from Kresyzig:

    Let: [itex]u_m(x)=mxe^{-mx^2}[/itex]

    and consider:

    [tex]\sum_{n=1}^{\infty} f_n(x)[/tex]

    where [itex]f_n(x)=u_m(x)-u_{m-1}(x)[/itex]

    then:

    [tex]\int_0^1 \sum_{n=1}^{\infty}f_n(x) dx\neq \sum_{n=1}^{\infty} \int_0^1 f_n(x)dx[/tex]
     
  10. Oct 21, 2013 #9

    WannabeNewton

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    Of course you have to make sure the sum and integral are actually interchangeable because this is not always the case. The sufficient condition is a special case of Fubini's theorem.
     
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