Way to re-express this equation by writing x and y separate?

[femiadeyemi]

Hi All,
I have an equation like this:

$\sum_{i=0}^{n} x^{i}*y^{i}$

is there a way to re-express this equation by writing x and y separate? Thank you

 

[mathman]

The expression can be looked at as the dot product of n+1 dimensional vectors. Ir can be written as:
|x||y|cosθ, where θ is the angle between the vectors and |x| and |y| are the individual vector lengths. I suspect it won't be much help, since the usual way of finding the angle is using the dot product.

 

[CompuChip]

What you wrote is not actually an equation, it is an expression. An equation should have an equals sign (=) and something on the other side of it. You may think this is a bit nitpicky but it can matter a lot whether the sum should be equal to 0, for example, or to some complex value, or to some convenient value in which stuff cancels out.

Assuming you want to rewrite your sum (and that * means scalar multiplication and not e.g. convolution) to something like
$\left( \sum f(x) \right) \left( \sum g(y) \right)$
where f(x) and g(y) are some functions of only the x's and y's, respectively, in general the answer is no - you cannot do that, except for what mathman has already pointed out.

However, I assume this question did not drop out of thin air; so maybe if you give us a bit more about the context that you asked it in we would be able to help you more.

 

[Robert1986]

Hi All,
I have an equation like this:

$\sum_{i=0}^{n} x^{i}*y^{i}$

is there a way to re-express this equation by writing x and y separate? Thank you

If you are trying to get estimates on something (like upper bounds) you can use Cauchy Schawrtz:

$\sum_{i=0}^{n} x^{i}*y^{i} \leq (\sum_{i=0}^{n} x_i^2)^{1/2}(\sum_{i=0}^{n} y_i^2)^{1/2}$

 

[blue_raver22]

.

Yes, there is a way to re-express this equation by writing x and y separately. This can be done by expanding the summation using the binomial theorem, which states that (a+b)^n = \sum_{i=0}^{n} \binom{n}{i} a^{n-i} b^i. Therefore, the given equation can be rewritten as \sum_{i=0}^{n} \binom{n}{i} x^{i} y^{i}. This allows for x and y to be written separately as x^i and y^i, respectively.