Thread: How do I multiply these sums?

1. How do I multiply these sums?

I have the following expression, and I want to have a formula that gives me the coefficients of $x^{s}$ of this product.

$(x+x^2+x^3)^2$

This can be written like this:

$x^2(1-x^3)^2(1-x)^{-2}$

Now I used the binomial theorem:
$x^2(1-x^3)^2=\sum_{k=0}^{2}{2\choose k}(-1)^{k}x^{3k+2}$ and $(1-x)^{-2}=\sum_{r=0}^{\infty}{2+r-1\choose r}x^r$

So
$(x+x^2+x^3)^2=\sum_{k=0}^{2}{2\choose k}(-1)^{k}x^{3k+2}\sum_{r=0}^{\infty}{2+r-1\choose r}x^r$

But it doesn't seem right, and I don1t know how to rewrite it.

2. Re: How do I multiply these sums?

Is there a reason why you are trying to do it in that convoluted fashion rather than just multiply it?

( $(x+ x^2+x^3)^2$ is NOT equal to $x^2(1- x^3)^2(1- x)^{-2}$. One rather obvious point is that the first is defined at x= 1 and the second isn't.)

3. Re: How do I multiply these sums?

Okay, but if |x|<1 then they are equal aren't they?

Yes, there is a reason. I want to find a formula that finds these coefficients, for more general cases.

4. Re: How do I multiply these sums?

Okay, but still why are you doing that?

It's simple to see that $x(x+ x^2+ x^3)= x^2+ x^3+ x^4$,
$x^2(x+ x^2+ x^3)= x^3+ x^4+ x^5$, and
$x^3(x+ x^2+ x^3)= x^4+ x^5+ x^6$ so that

$((x+ x^2+ x^3)^2= x^2+ 2x^3+ 3x^4+ 2x^5+ x^6$

5. Re: How do I multiply these sums?

If I had a formula for this simple case than I would be able to create a similar formul for not so simple cases like.

$(x+x^2+x^3+x^4+x^5+x^6)^m$

6. Re: How do I multiply these sums?

Originally Posted by gotmejerry
If I had a formula for this simple case than I would be able to create a similar formul for not so simple cases like.

$(x+x^2+x^3+x^4+x^5+x^6)^m$
LaTeX problems: will try again

7. Re: How do I multiply these sums?

Originally Posted by gotmejerry
If I had a formula for this simple case than I would be able to create a similar formul for not so simple cases like.

$(x+x^2+x^3+x^4+x^5+x^6)^m$
Is this the problem: simplify $\displaystyle \left(\sum_{i=1}^nx^i\right)^m ,\ given\ m,\ n \in \mathbb Z\ and\ m > 0 < n\ and\ x > 0.$

$Case\ I:\ x = 1 \implies \displaystyle \left(\sum_{i=1}^nx^i\right)^m = \left(\sum_{i=1}^n1^i\right)^m = \left(\sum_{i=1}^n1\right)^m = n^m.$

$Case\ II:\ x \ne 1.$

$\dfrac{x^{(n + 1)} - 1}{x - 1} = \displaystyle \sum_{i=0}^nx^i = 1 + \sum_{i=1}^nx^i \implies$

$\displaystyle \sum_{i=1}^nx^i = \dfrac{x^{(n + 1)} - 1}{x - 1} - 1 = \dfrac{x^{(n + 1)} - 1 - x + 1}{x - 1} = \dfrac{x^{(n + 1)} - x}{x - 1} = \dfrac{x(x^n - 1)}{x - 1}\implies$

$\displaystyle \left(\sum_{i=1}^nx^i\right)^m = \left(\dfrac{x(x^n - 1)}{x - 1}\right)^m.$

8. Re: How do I multiply these sums?

Thanks, but I want to find the coefficients of the powers of x.