Squared Difference of Two Series: Why?

[Dustinsfl]

$$
\left(\sum_{i = 1}^n(x_i - y_i)\right)^2 = \sum_{i = 1}^n(x_i - y_i)^2 + \sum_{1\leq i\leq j\leq n}|x_i - y_i||x_j-y_j|
$$
Why is this true?

 

[Sudharaka]

$$
\left(\sum_{i = 1}^n(x_i - y_i)\right)^2 = \sum_{i = 1}^n(x_i - y_i)^2 + \sum_{1\leq i{\color{red}<}j\leq n}|x_i - y_i||x_j-y_j|
$$
Why is this true?

Hi dwsmith, :)

I think you will be able prove this by mathematical induction. I further think that in the second summation the inequality should be changed as highlighted (you can verify this by taking \(n=1\)).

Kind Regards,
Sudharaka.

 

[Opalg]

$$
\left(\sum_{i = 1}^n(x_i - y_i)\right)^2 = \sum_{i = 1}^n(x_i - y_i)^2 + \sum_{1\leq i\leq j\leq n}|x_i - y_i||x_j-y_j|
$$
Why is this true?

Even with the correction suggested by Sudharaka, this formula is not true. To start with, simplify it by writing $z_i = x_i-y_i$. The formula becomes $$ \Bigl(\sum_{i = 1}^nz_i\Bigr)^2 = \sum_{i = 1}^nz_i^2 + \sum_{1\leqslant i < j\leqslant n}|z_i||z_j|.$$

Test that formula by putting n=2. It becomes $(z_1+z_{\,2})^2 = z_1^2 + z_{\,2}^2 + |z_1||z_{\,2}|$. Compare that with the correct formula $(z_1+z_{\,2})^2 = z_1^2 + z_{\,2}^2 + 2z_1z_{\,2}$ and you see that two things are wrong: there is a missing 2, and the absolute value signs should not be there.

 

[Sudharaka]

Even with the correction suggested by Sudharaka, this formula is not true. To start with, simplify it by writing $z_i = x_i-y_i$. The formula becomes $$ \Bigl(\sum_{i = 1}^nz_i\Bigr)^2 = \sum_{i = 1}^nz_i^2 + \sum_{1\leqslant i < j\leqslant n}|z_i||z_j|.$$

Test that formula by putting n=2. It becomes $(z_1+z_{\,2})^2 = z_1^2 + z_{\,2}^2 + |z_1||z_{\,2}|$. Compare that with the correct formula $(z_1+z_{\,2})^2 = z_1^2 + z_{\,2}^2 + 2z_1z_{\,2}$ and you see that two things are wrong: there is a missing 2, and the absolute value signs should not be there.

Thank you for pointing that out. Thinking about this further I came up with the following. :)

\[\left(\sum_{i = 1}^nz_i\right)^2=\sum_{i=1}^{n}z_i^2 + \sum_{i\neq j}z_{i}z_{j}\]

Hence,

\[\left(\sum_{i = 1}^n(x_{i}-y_{i})\right)^2=\sum_{i=1}^{n}(x_{i}-y_{i})^2 + \sum_{i\neq j}(x_{i}-y_{i})(x_{j}-y_{j})\]

 

[blue_raver22]

The squared difference of two series is a useful mathematical concept in various fields of science. It allows us to quantify the difference between two sets of data and understand the relationship between them. The equation provided shows that the squared difference of two series can be expressed as the sum of two terms, one of which is the squared difference of each individual data point, and the other is the product of the absolute differences between all possible pairs of data points.

This is true because when we square a quantity, we are essentially multiplying it by itself. In this case, we are squaring the difference between two data points, which gives us a positive value regardless of the sign of the difference. This allows us to eliminate any negative values and focus on the magnitude of the difference.

The first term in the equation, $\sum_{i = 1}^n(x_i - y_i)^2$, represents the sum of the squared differences between each individual data point. This term gives us an overall measure of the difference between the two series and is useful in determining the overall trend or pattern between them.

The second term, $\sum_{1\leq i\leq j\leq n}|x_i - y_i||x_j-y_j|$, takes into account the absolute differences between all possible pairs of data points. This term captures the interaction between the individual data points and provides a more comprehensive understanding of the relationship between the two series.

In conclusion, the squared difference of two series is a valuable tool for analyzing and understanding data. It allows us to consider both the individual differences and the interactions between data points, providing a more complete picture of the relationship between two sets of data.