Rewriting summations

dwsmith

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Mar 2010
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Can all double summations be rewritten as a single sum?

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

The answer isn't important this is just an example.
 

undefined

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Mar 2010
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Can all double summations be rewritten as a single sum?

For example, can this, \(\displaystyle \sum_{i=-1}^{1}\sum_{j=0}^{2}(2i+3j)\), 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:

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

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

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

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

Depending on the sums you're dealing with, similar strategies could be possible.
 

dwsmith

MHF Hall of Honor
Mar 2010
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Well, I don't know of any general method, but I would attack the above double sum as follows:

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

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

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

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

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.
 

undefined

MHF Hall of Honor
Mar 2010
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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.
 
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dwsmith

MHF Hall of Honor
Mar 2010
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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?
 
Nov 2009
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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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dwsmith

MHF Hall of Honor
Mar 2010
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582
Florida
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?
 
Nov 2009
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184
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.