If a sum is 0, is the summand 0?

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In summary, the conversation discusses solving an equation of the form \sum_{i=1}^{n}\frac{1}{x_{i}}(a-by_{i})=0 and determining the variables to solve for. It is mentioned that simply setting the summand to 0 may not yield a solution and another equation involving the variables may be needed. The conversation also clarifies the summation and variables involved.
  • #1
Dixanadu
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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?
 
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  • #2
Dixanadu said:
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?
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.
 
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  • #3
Dixanadu said:
how would I solve this?
What variables are you solving for? What symbols represent known values ?
 
  • #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 I am trying to solve for alpha and beta but one at a time...
 
  • #5
Dixanadu said:
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]

So it's "[itex] x [/itex]" instead of "[itex] x_i [/itex]"?`
 
  • #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].
 
  • #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
 
  • #8
Dixanadu said:
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

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.
 
  • #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
 

1. What does it mean for a sum to be 0?

When a sum is 0, it means that the result of adding all the numbers in the sum is equal to 0. In other words, the sum has a total value of nothing.

2. Can a sum have multiple summands that are 0?

Yes, a sum can have multiple summands that are 0. For example, 0 + 0 + 0 = 0. This is because the sum of any number and 0 is equal to the original number.

3. If a sum is 0, does it mean all the summands are 0?

No, a sum can be 0 even if not all the summands are 0. For instance, 5 + 0 + (-5) = 0. In this case, only one summand is 0, but the overall sum is still 0.

4. What happens if a summand is equal to -0?

If a summand is equal to -0, it is essentially equivalent to 0. This is because the concept of negative 0 does not exist in mathematics.

5. Can a sum be 0 if there are no summands?

No, a sum cannot be 0 if there are no summands. In order for a sum to have a value of 0, there must be at least one summand present. Otherwise, the sum does not exist.

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