Solve to find $y(x)$ of the $\frac{1}{\sum_{n=0}^{\infty }y^n}-\sum_{n=0}^{\infty }x^n=0$ Solve the equation to find the $y$ as a function to respect $x$ without $n$ $$\frac{1}{\sum_{n=0}^{\infty }y^n}-\sum_{n=0}^{\infty }x^n=0$$

Assuming $|x| < 1$ and $|y| < 1$, the two geometric series simplify to:
$$\frac{1}{\frac{1}{1 - y}} = \frac{1}{1 - x}.$$
Consequently:
$$(1 - x)(1 - y) = 1.$$
Solving for $y$ in terms of $x$, you get:
$$1 - y = \frac{1}{1 - x}$$
which implies that
$$y(x) = 1 - \frac{1}{1 - x} = \frac{x}{x - 1}.$$
Note that $|y(x)| < 1$, so that $|x/(x - 1)| < 1$.  This implies that:
$$|x| < |x - 1|,$$
which further implies that
$$x^2 < (x - 1)^2 = x^2 - 2x + 1.$$
Consequently,
$$x < \frac{1}{2}.$$
