Homework Help: Summation notation

1. Apr 15, 2006

Benny

Hi, can someone please tell me whether or not I can switch the 'order' of the indices over which a double sum is taken? To clarify, my question is whether or not the following is true.

$$\sum\limits_{j = 1}^n {\sum\limits_{i = 1}^n {\left( {a_i b_j } \right)} } \mathop = \limits^? \sum\limits_{i = 1}^n {\sum\limits_{j = 1}^n {\left( {a_i b_j } \right)} }$$

Any help would be appreciated.

2. Apr 15, 2006

Hurkyl

Staff Emeritus
Yes, by Fubini's theorem.

Er... I mean by associativity and commutativity! (That's just a finite sum of numbers!)

Or, you could always try proving it by induction!

3. Apr 15, 2006

Astronuc

Staff Emeritus
Well, one can simply show

a1(b1+b2) + a2(b1+b2) = (a1+a2) b1 + (a1+a2) b2

and the show it for 1,2,3 or 1,2, . . . n

4. Apr 15, 2006

Benny

Oh ok thanks for the help guys. I was just after a 'yes' or 'no' answer seeing as I assumed this property for one of the questions I was doing.

Seeing as Fubini's theorem was mentioned, is it related to my question in some way that I'm not seeing? I know that Fubini's theorem has something to do with multiple integrals which in turn has something to do with multiple sums but I don't see the connection with my question. It doesn't really matter though because I was working on a question which wasn't related to integrals, well not exactly anyway - it was a vector identity.

Thanks for the help.

5. Apr 16, 2006

Hurkyl

Staff Emeritus
There is a perspective in which a summation really is just an integral, and this interchange property could be proved with Fubini's theorem in this context.

But there's no reason you would go through such great lengths to prove such an elementary result -- I just like being silly. (And I had a teacher who liked to justify interchanging finite sums with Fubini's theorem)

6. Apr 16, 2006

HallsofIvy

Notice, by the way, that this is true for finite sums. It is not necessarily true that we can interchange infinite sums.

7. Apr 17, 2006

Benny

Oh ok, I remember googling something about double sums and there were different representations given for finite and infinite double sums.