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If a sum is 0, is the summand 0?

  1. Nov 30, 2014 #1
    Hey guys,

    Was just wondering something. Suppose I have an equation of the form

    [itex]\sum_{i=0}^{n}\frac{1}{x_{i}}(a-by_{i})=0[/itex],

    how would I solve this? do I just set the summand = 0?
     
  2. jcsd
  3. Nov 30, 2014 #2

    Mark44

    Staff: Mentor

    No. If the terms in the sum can be positive or negative, their sum can be zero without any of them being zero. As a simple example, 2 + (-1) + (-1) = 0, but no single term equals zero.
     
    Last edited: Nov 30, 2014
  4. Nov 30, 2014 #3

    Stephen Tashi

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    Science Advisor

    What variables are you solving for? What symbols represent known values ?
     
  5. Nov 30, 2014 #4
    The real equation I'm dealing with is
    [itex]\sum_{i=1}^{n}\frac{1}{\sigma_{i}^{2}}2x(y_{i}-\alpha x -\beta x^{2})=0[/itex]

    and im trying to solve for alpha and beta but one at a time...
     
  6. Nov 30, 2014 #5

    Stephen Tashi

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    Science Advisor

    So it's "[itex] x [/itex]" instead of "[itex] x_i [/itex]"?`
     
  7. Nov 30, 2014 #6
    Yes I wrote it completely differently in the original post as I didnt think I would need to put the actual equation here but I changed my mind, sorry.

    There is no summation over [itex]x[/itex], only over the [itex]\sigma_{i}[/itex] and [itex]y_{i}[/itex].
     
  8. Nov 30, 2014 #7
    The closest I can get is

    [itex](\alpha x +\beta x^{2})\sum_{i=1}^{n}\frac{1}{\sigma_{i}^{2}}=\sum_{i=1}^{n}\frac{y_{i}}{\sigma_{i}^{2}}[/itex]

    No idea what to do from here
     
  9. Dec 1, 2014 #8

    Stephen Tashi

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    Science Advisor

    That equation doesn't have a unique solution. To get a unique solution, you need to know another equation involving [itex] \alpha [/itex] and [itex] \beta [/itex].

    Does this equation come from setting a partial derivative equal to zero? If so, perhaps there is another partial derivative that's supposed to be set equal to zero.
     
  10. Dec 1, 2014 #9
    Yes you are right - it turns out that the [itex]x[/itex] are also summed in addition to [itex]y_{i},\sigma_{i}[/itex], i had interpreted the situation inocorrectly.

    Everything is fine now thank you for your help! :D
     
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