What does this summation do?

**fortenbt** · February 20th 2006, 11:47 AM

I was given this summation today and asked to find out what it did. So far my friend and I have run into a brick wall at every turn.

Here's the summation:

$<br /> \sum_{i=1}^N\sum_{j=i+1}^N \frac{a_i a_j}{N(N-1)/2}<br />$

This is the only information we were given, and we interpreted $a_i$ and $a_j$ as being integers in some set of N integers, where the first integer in the set is $a_1$

The problem we've run into is that when i gets to be N, j = N+1 and $a_j$ is therefore undefined. So we've been interpreting it as i goes from 1 to N-1. Does anyone recognize this as a famous series of some sort, or know what it does?

Thanks for any information anyone can provide,
-Ben

**ThePerfectHacker** · February 20th 2006, 11:58 AM

I did not spend much time on this problem thus I do not know how it simplifies but I noticed that $N(N-1)/2$ is the $n-1$ th triangular number. And the infinite sum of the reciprocals of triangular numbers is 2.

**fortenbt** · February 20th 2006, 12:29 PM

Originally Posted by ThePerfectHacker

I noticed that $N(N-1)/2$ is the $n-1$ th triangular number.

We had noticed this as well. Since a triangular number can be defined as:

$\sum_{k=1}^N k$

You can just take that outside the summations and you can rewrite everything as:

$\left[\sum_{k=1}^{N-1} k\right]\left[\sum_{i=1}^N\sum_{j=i+1}^N a_i a_j}\right]$

We still couldn't make any connection as to what this would actually do for anything, though.

**ThePerfectHacker** · February 20th 2006, 12:41 PM

The numbers $a_k$ clearly follow a certain sequence. It might help if we knew which sequence?

**fortenbt** · February 20th 2006, 12:48 PM

We don't know which sequence. As of now, we're assuming if N=10, the sequence is 1,2,3,4,....,9,10.

**topsquark** · February 20th 2006, 01:09 PM

Originally Posted by fortenbt

We had noticed this as well. Since a triangular number can be defined as:

$\sum_{k=1}^N k$

You can just take that outside the summations and you can rewrite everything as:

$\left[\sum_{k=1}^{N-1} k\right]\left[\sum_{i=1}^N\sum_{j=i+1}^N a_i a_j}\right]$

We still couldn't make any connection as to what this would actually do for anything, though.

I'm not sure exactly what it is, but I can tell you where I've seen it before:

Consider the polynomial $f(x)=ax^5+bx^4+cx^3+dx^2+ex+f$ (a,b,c,d,e, and f are rational numbers) and assume there exist 5 roots $r_1, r_2, r_3,r_4,r_5$ . We may show that we can reproduce the coefficients using these roots. What I want to point out is that, in particular:
$r_1(r_2+r_3+r_4+r_5)+r_2(r_3+...)+...r_4r_5=c/a$ .
The series in the roots is your series times N(N-1)/2. Supposedly the series mentioned is one of the "elementary symmetric functions in n-variables" according to my book.

Hope it helps!
-Dan

**fortenbt** · February 21st 2006, 04:30 AM

Well, I think we got it.

We're pretty sure that it's the expected value of the product of any sample of two elements from the set. Once it's stated and you look back at it, you wonder why you didn't see that before. I really should have gotten that.

Thanks for all your help!

-Ben

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