Is there a reason why you are trying to do it in that convoluted fashion rather than just multiply it?
( is NOT equal to . One rather obvious point is that the first is defined at x= 1 and the second isn't.)
I have the following expression, and I want to have a formula that gives me the coefficients of of this product.
This can be written like this:
Now I used the binomial theorem:
and
So
But it doesn't seem right, and I don1t know how to rewrite it.
Is this the problem: simplify $\displaystyle \left(\sum_{i=1}^nx^i\right)^m ,\ given\ m,\ n \in \mathbb Z\ and\ m > 0 < n\ and\ x > 0.$
$Case\ I:\ x = 1 \implies \displaystyle \left(\sum_{i=1}^nx^i\right)^m = \left(\sum_{i=1}^n1^i\right)^m = \left(\sum_{i=1}^n1\right)^m = n^m.$
$Case\ II:\ x \ne 1.$
$\dfrac{x^{(n + 1)} - 1}{x - 1} = \displaystyle \sum_{i=0}^nx^i = 1 + \sum_{i=1}^nx^i \implies$
$\displaystyle \sum_{i=1}^nx^i = \dfrac{x^{(n + 1)} - 1}{x - 1} - 1 = \dfrac{x^{(n + 1)} - 1 - x + 1}{x - 1} = \dfrac{x^{(n + 1)} - x}{x - 1} = \dfrac{x(x^n - 1)}{x - 1}\implies$
$\displaystyle \left(\sum_{i=1}^nx^i\right)^m = \left(\dfrac{x(x^n - 1)}{x - 1}\right)^m.$