1. ## Rewriting summations

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.

2. Originally Posted by dwsmith
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:

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

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

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

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

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

3. Originally Posted by undefined
Well, I don't know of any general method, but I would attack the above double sum as follows:

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

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

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

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

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.

4. 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.

5. 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?

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

7. 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?

8. 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.