Given the infinite series y = racx2 + racx24 + racx38 + racx416 + racx532 + racx664 + racx7128 + racx8256 + racx9512 + racx101024 + racx112048 + racx124096 + racx138192 + racx1416384 + racx1532768 + r

The first term is $a = \frac{x}{2}$.
The common ratio is $r = \frac{x^2/4}{x/2} = \frac{x^2}{4} \cdot \frac{2}{x} = \frac{x}{2}$.
The condition $0 < x < 2$ ensures that $|r| = \left|\frac{x}{2}\right| < 1$, so the series converges.

Multiply both sides by $(2-x)$:

Distribute $y$:

Move all terms containing $x$ to one side:

Factor out $x$:

Divide by $(1+y)$ to solve for $x$:

This proves the required relationship.

The first term is $a = \frac{2}{3}x$.
The common ratio is $r = \frac{\frac{4}{9}x^2}{\frac{2}{3}x} = \frac{4x^2}{9} \cdot \frac{3}{2x} = \frac{2x}{3}$.
The condition $0 < x < \frac{3}{2}$ ensures that $|r| = \left|\frac{2x}{3}\right| < 1$, so the series converges.

Multiply both sides by $(3-2x)$:

Distribute $y$:

Move all terms containing $x$ to one side:

Factor out $2x$:

Divide by $2(1+y)$ to solve for $x$:

This shows the required relationship.