Find The Limiting Ratio

Relevant wiki: Limits of Sequences - Medium

Let $x_n$ be the total length of the rectangle when $n-1$ rectangles has contributed to the length.

Let $y_n$ be the total height of the rectangle when $n$ rectangles has contributed to the height.

Through considering the area of the rectangle,

$x_n y_n = 2n$

$x_{n-1}y_n=2n-1$

With initial conditions: $x_1=2, y_1=1$

Solving these recurrence relations gives

$x_n=2^{2n}\frac{\left(n!\right)^2}{\left(2n\right)!}$

$y_n=\frac{2^{1-2n}n\cdot \left(2n\right)!}{\left(n!\right)^2}$

$\frac{x_n}{y_n}=\frac{2^{4n-1}\left(n!\right)^4}{n\cdot \left(\left(2n\right)!\right)^2}$

Using these and Stirling's approximation gives

$\lim_{n \to \infty} \frac{x_n}{y_n}=\frac{\pi}{2}$