examday

/TMUA

A sequence of squares $S_1,S_2,S_3,\ldots$ is arranged as shown. Their horizontal diagonals lie consecutively on the positive $x$-axis. The horizontal diagonal of $S_1$ has length $4$, and each subsequent diagonal has half the length of the preceding one.

For odd $n$, let $P_n$ be the upper vertex of $S_n$. For even $n$, let $P_n$ be the lower vertex of $S_n$.

Starting at the origin, a broken line is drawn successively through

The points $P_n$ approach a point on the $x$-axis.

What is the sum of the areas of all the bounded regions between the broken line and the $x$-axis?