# A question on a summation

1. Nov 28, 2011

### julypraise

1. The problem statement, all variables and given/known data
The Spivak's Calculus Answer Book (3ed) states that, on page 17,

$\sum_{i \neq j} (x_{i}^{2}y_{j}^2 - x_{i}y_{i}x_{j}y_{j}) = 2\sum_{i < j}(x_{i}^{2}y_{j}^2 + x_{j}^{2}y_{i}^2 - x_{i}y_{i}x_{j}y_{j})$

But as I speculate, I've got the following:

$\sum_{i \neq j} (x_{i}^{2}y_{j}^2 - x_{i}y_{i}x_{j}y_{j}) = \sum_{i < j}(x_{i}^{2}y_{j}^2 + x_{j}^{2}y_{i}^2 - 2x_{i}y_{i}x_{j}y_{j})$

Could you check which is right?

Thanks.

2. Relevant equations

3. The attempt at a solution

$\sum_{i \neq j} (x_{i}^{2}y_{j}^2 - x_{i}y_{i}x_{j}y_{j}) = \sum_{i < j}(x_{i}^{2}y_{j}^2 + x_{j}^{2}y_{i}^2 - 2x_{i}y_{i}x_{j}y_{j})$

2. Nov 28, 2011

### CompuChip

I suspect that the answer in the book is wrong.
You can easily write it out manually for i, j running through {1, 2}.

It's also possible to prove using
$$\sum_{i \neq j} a_{ij} = \sum_{i < j} a_{ij} + \sum_{i > j} a_{ij}$$
where
$$\sum_{i > j} a_{ij} = \sum_{j > i} a_{ji} = \sum_{i < j} a_{ji}$$

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