Suppose we have two convergent geometric series ∑x^n and ∑y^n. Does it ever occur that ∑x^n∑y^n=∑[(x^n)(y^n)]?
THANKS.
I would say yes. Assume that the lead term is $\displaystyle 1$ and that $\displaystyle n$ starts at $\displaystyle n = 0$ then
$\displaystyle \sum_{n=0}^\infty x^n = \frac{1}{1-x},$ $\displaystyle \sum_{n=0}^\infty y^n = \frac{1}{1-y}$ and $\displaystyle \sum_{n=0}^\infty x^n y^n = \frac{1}{1-xy}$
so what you're asking is are there soltions to
$\displaystyle
\frac{1}{1-x} \cdot \frac{1}{1-y} = \frac{1}{1-xy} \;\; \text{on}\;\; (-1,1) \times (-1,1)
$. I believe there are an infinite number of solutions to this equation on this interval.