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Finite Series Expansion

  1. Jul 11, 2013 #1
    Hello :blushing:

    How to do expand this: [itex](\sum_{j=1}^{n}(X(t_j)-X(t_{j-1}))^2 - t)^2[/itex] where [itex]X(t_j)-X(t_{j-1}) = \Delta X_j[/itex]

    to this: [itex](\sum_{j=1}^{n}(\Delta X_j)^4 + 2*\sum_{i=1}^{n}\sum_{j<i}^{ }(\Delta X_i)^2(\Delta X_j)^2[/itex] [itex]-2*t*\sum_{j=1}^{n}(\Delta X_j)^2+t^2[/itex]


    I get near the North Pole.... but it seems that I've forgotten some fundamental rules of finite series to do the last part of the manipulation :smile:.I assume that the blue part has to be expaned further. This is what I've done:

    [itex]\sum_{j=1}^{n}(\Delta X_i)^2 = a,t=b[/itex]

    [itex]E[(a-b)^2]=E[a^2-2ab+b^2]=E[\color{blue} {(\sum_{j=1}^{n}(\Delta X)^2)^2} \color{black} -2*t*\sum_{j=1}^{n}(\Delta X)^2+t^2][/itex]

    Any help would be greatly appreciated.
     
    Last edited: Jul 11, 2013
  2. jcsd
  3. Jul 11, 2013 #2
    I may be misunderstanding something, but I think the second term (with the double sum in [itex]i[/itex] and [itex]j[/itex]) should be multiplied by [itex]2[/itex]. Either that or the sum in [itex]j[/itex] should be over [itex]j \neq i[/itex] rather than [itex]j<i[/itex].

    Anyway, you are correct to say that the blue term needs to be expanded further. Just try writing out one explicit example, say for [itex]n=2[/itex]. Often, the compact summation notation obscures otherwise obvious patterns.

    Also, you can use [itex]( \sum_j f_j )^2 = ( \sum_j f_j ) ( \sum_i f_i )[/itex].
     
  4. Jul 11, 2013 #3
    krome you're correct, its supposed to be multiplied by 2
     
  5. Jul 11, 2013 #4

    Ray Vickson

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    No, it should not be multiplied by 2. If ##a_i = (\Delta X_i)^2##, you have to expand the sum
    [tex]\left(\sum_i (a_i-t)\right)^2 = \left( \sum_i a_i - nt \right)^2
    = \sum_i a_i^2 + 2\sum_{i<j} a_i a_j - 2nt \sum_i a_i + n^2 t^2.[/tex]
     
  6. Jul 11, 2013 #5
    Good lord! I must be going mad or selectively blind. I swear when I read this last night the second term did not have a factor of 2! :confused:
     
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