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A question on a summation

  1. Nov 28, 2011 #1
    1. The problem statement, all variables and given/known data
    The Spivak's Calculus Answer Book (3ed) states that, on page 17,

    [itex]\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})[/itex]

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

    [itex]\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})[/itex]

    Could you check which is right?

    Thanks.


    2. Relevant equations



    3. The attempt at a solution

    [itex]\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})[/itex]
     
  2. jcsd
  3. Nov 28, 2011 #2

    CompuChip

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    Homework Helper

    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
    [tex]\sum_{i \neq j} a_{ij} = \sum_{i < j} a_{ij} + \sum_{i > j} a_{ij}[/tex]
    where
    [tex]\sum_{i > j} a_{ij} = \sum_{j > i} a_{ji} = \sum_{i < j} a_{ji}[/tex]
     
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