Proving Sigma Notation Equals Zero

In summary, using sigma notation, we can show that the sum of (x_i - \overline{x}) from i=1 to n equals 0. This can be proven by expanding the expression and showing that both sums equal 0.
  • #1
bob1182006
492
1

Homework Statement


Show:
[tex]\sum_{i=1}^n (x_i - \overline{x}) = 0[/tex]


Homework Equations


Sigma notation


The Attempt at a Solution



[tex]\sum_{i=1}^n x_i - \sum_{i=1}^n \overline{x} = \sum_{i=1}^n x_i - \frac{1}{n}\sum_{i=1}^n \sum_{i=1}^n x_i = 0[/tex]
[tex]\sum_{i=1}^n x_i = \frac{1}{n}\sum_{i=1}^n \sum_{i=1}^n x_i [/tex]

By Inspection I know i need to show that:
[tex]\sum_{i=1}^n \frac{1}{n}=1[/tex]

Since the LHS has no [itex]x_i[/itex] how can i show that the sum will result in n/n =1?

Is it just:
[tex]\sum_{i=1}^n 1 = n?[/tex]
 
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  • #2
consider that [tex]\overline{x}[/tex] is idependent of [tex]i[/tex] therefore [tex]\sum_{i=1}^n\overline{x}[/tex] simply equals [tex]n \overline{x}[/tex]. Also IMPORTANT NOTE: In the second line of your attempted solution you use the fact that [tex]\sum_{i=1}^n (x_i - \overline{x}) = 0[/tex] how ever that's what you're trying to SHOW so you can't assume it's true. You have to work entirely on the LHS and get it equal to 0.
 
  • #3
Ah, I get it now thanks alot!

So on the first step it'd just be the n*xbar/n leaving only sum of x - sum of x which =0.

Thank You!
 
  • #4
Err no it'd be
[tex]\sum_{i=1}^n (x_i - \overline{x}) = 0[/tex]
[tex]\sum_{i=1}^n x_i - n\overline{x} = 0[/tex]
[tex]n \sum_{i=1}^n \frac{x_i}{n} - n\overline{x} = 0[/tex]

[tex]n\overline{x} - n\overline{x} = 0[/tex]

which gives you zero... hope he comes back to read that.
 
  • #5
ah okay, I was expanding the xbar so there would be 2 equal sums being subrated which would just =0.

[tex]\sum_{i=1}^n x_i - n \frac{1}{n} \sum_{j=1}^n x_j = 0[/tex]

since n/n =1, and both sums start and end at the same place they are both equal and thus will give 0 as the answer.
 

Q: What is sigma notation?

Sigma notation is a mathematical notation used to represent the summation of a series of numbers. It is denoted by the Greek letter sigma (Σ) and a variable, an expression, and the range of values over which the variable is summed.

Q: How do you prove that sigma notation equals zero?

To prove that sigma notation equals zero, you need to show that the sum of all the terms in the series is zero. This can be done by evaluating each term and adding them together. If the sum is equal to zero, then the sigma notation equals zero.

Q: What are the steps for proving sigma notation equals zero?

The steps for proving sigma notation equals zero are as follows:

  • 1. Write the given sigma notation in expanded form.
  • 2. Evaluate each term in the series.
  • 3. Add all the terms together.
  • 4. If the sum is equal to zero, then the sigma notation equals zero.

Q: Can sigma notation equal zero for any series of numbers?

Yes, it is possible for sigma notation to equal zero for certain series of numbers. However, it is not always the case and it depends on the values and the range of the series.

Q: How is proving sigma notation equals zero useful in science?

Proving sigma notation equals zero can be useful in scientific research and data analysis. It allows scientists to accurately sum up large amounts of data and determine if the overall result is equal to zero. This can help in identifying patterns, trends, and relationships in the data.

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