Consider an infinite amount of squares stacked on top of each other where the top left corners are touching the edge of a circle:
Call the largest blue square x. How would I find the ratio of square n's side lengths to x's side lengths?
For example: $\frac{x}{square_n}$
Let $R$ be the radius of the circle and $a_n$ be the sidelength of the $n$-th square.
Let the bottom left corner of the image be the origin, and let the bottom and left edges be the $x$ and $y$ axes respectively. Then, the equation of the circle is $x^2+(y-R)^2 = R^2$, and the top left corner of the $n$-th square is at $(x,y) = \left(R-a_n,\displaystyle\sum_{k = 1}^{n}a_k\right)$. But this corner must lie on the circle.
So, $(R-a_n)^2 + \left(\displaystyle\sum_{k = 1}^{n}a_k - R\right)^2 = R^2$. Solve for $a_n$ and discard the extraneous solution to get
$a_n = R - \dfrac{1}{2}\displaystyle\sum_{k = 1}^{n-1}a_k - \dfrac{1}{2}\sqrt{2R^2 - \left(\sum_{k = 1}^{n-1}a_k\right)^2}$ for all integers $n \ge 1$.
Note: For $n = 1$, we have $\displaystyle\sum_{k = 1}^{n-1}a_k = 0$, and so, $a_1 = \left(1-\dfrac{1}{\sqrt{2}}\right)R$