Answer
$\displaystyle \frac{1}{3}$ of the area of the initial square.
Work Step by Step
Say the square has area 1.
The largest shaded square has area $\displaystyle \frac{1}{4}$,
the next is a quarter of the previous, $\displaystyle \frac{1}{4^{2}}$, and so on.
$\displaystyle \frac{1}{4}+\frac{1}{4^{2}}+\frac{1}{4^{3}}+\frac{1}{4^{4}}+...$ is an infinite geometric series with
$a_{1}=\displaystyle \frac{1}{4}, r=\frac{1}{4}$
Since $|r| \lt 1,$ the infinite geometric series converges. Its sum is
$\displaystyle \sum_{k=1}^{\infty}a_{1}r^{k-1}=\frac{a_{1}}{1-r}=\frac{\frac{1}{4}}{\frac{3}{4}}=\frac{1}{3}$
