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Math Help - Rewriting summations

  1. #1
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    Rewriting summations

    Can all double summations be rewritten as a single sum?

    For example, can this, \sum_{i=-1}^{1}\sum_{j=0}^{2}(2i+3j), be rewritten as a single sum, and if so, how?

    The answer isn't important this is just an example.
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  2. #2
    MHF Contributor undefined's Avatar
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    Quote Originally Posted by dwsmith View Post
    Can all double summations be rewritten as a single sum?

    For example, can this, \sum_{i=-1}^{1}\sum_{j=0}^{2}(2i+3j), be rewritten as a single sum, and if so, how?

    The answer isn't important this is just an example.
    Well, I don't know of any general method, but I would attack the above double sum as follows:

    \sum_{i=-1}^{1}\sum_{j=0}^{2}(2i+3j)

    =\sum_{i=-1}^{1}\left(2i\sum_{j=0}^{2}1+3\sum_{j=0}^{2}j\rig  ht)

    =\sum_{i=-1}^{1}\left((2i)(3)+3\frac{(2)(2+1)}{2}\right)

    =\sum_{i=-1}^{1}6i+9

    Depending on the sums you're dealing with, similar strategies could be possible.
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    Quote Originally Posted by undefined View Post
    Well, I don't know of any general method, but I would attack the above double sum as follows:

    \sum_{i=-1}^{1}\sum_{j=0}^{2}(2i+3j)

    =\sum_{i=-1}^{1}\left(2i\sum_{j=0}^{2}1+3\sum_{j=0}^{2}j\rig  ht)

    =\sum_{i=-1}^{1}\left((2i)(3)+3\frac{(2)(2+1)}{2}\right)

    =\sum_{i=-1}^{1}6i+9

    Depending on the sums you're dealing with, similar strategies could be possible.
    I was wondering if there was a way of combining the summations without prior to summing the inside summation.
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  4. #4
    MHF Contributor undefined's Avatar
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    Quote Originally Posted by dwsmith View Post
    I was wondering if there was a way of combining the summations without prior to summing the inside summation.
    Well it's not possible in general to rewrite a double integral as a single integral, without trying to compute the inner one (before or after a change in order of integration), right? So I expect you may be looking for a method that does not exist.

    Edit: I stand corrected. But I probably won't be of much further help in this discussion.. I am familiar with Kronecker Delta but don't see how it relates to the question, also it's been years since I used Green's Theorem and I don't really remember it.
    Last edited by undefined; May 20th 2010 at 05:00 PM.
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    Quote Originally Posted by undefined View Post
    Well it's not possible in general to rewrite a double integral as a single integral, without trying to compute the inner one (before or after a change in order of integration), right? So I expect you may be looking for a method that does not exist.
    So the Kronecker Delta is a special case then?
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  6. #6
    Senior Member roninpro's Avatar
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    There are some ways to write double integrals as single integrals; I should point out that Green's Theorem does this for us.
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    Quote Originally Posted by roninpro View Post
    There are some ways to write double integrals as single integrals; I should point out that Green's Theorem does this for us.

    How can Green's Theorem be applied to the summations?
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  8. #8
    Senior Member roninpro's Avatar
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    This mostly just has application to rewriting an infinite double sum as an infinite single sum. I'm not sure if it can be used in the finite case.
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