Rewriting summations

**dwsmith** · May 20th 2010, 07:47 AM

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.

**undefined** · May 20th 2010, 08:06 AM

Originally Posted by dwsmith

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.

**dwsmith** · May 20th 2010, 11:00 AM

Originally Posted by undefined

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.

**undefined** · May 20th 2010, 11:32 AM

Originally Posted by dwsmith

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.

**dwsmith** · May 20th 2010, 11:35 AM

Originally Posted by undefined

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?

**roninpro** · May 20th 2010, 04:24 PM

There are some ways to write double integrals as single integrals; I should point out that Green's Theorem does this for us.

**dwsmith** · May 20th 2010, 04:25 PM

Originally Posted by roninpro

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?

**roninpro** · May 20th 2010, 07:57 PM

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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