# Rewriting summations

#### dwsmith

MHF Hall of Honor
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

MHF Hall of Honor
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
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
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:

#### dwsmith

MHF Hall of Honor
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

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

• undefined

#### dwsmith

MHF Hall of Honor
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

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.