Answer
$\dfrac{1}{3}$ of the square will be shaded.
Work Step by Step
The largest shaded region is $\dfrac{1}{4}$ of the largest square.
We see that the next is a quarter of the previous, which is $\dfrac{1}{4^{2}}=\dfrac{1}{16}$ of the largest square, the next shaded region is $\dfrac{1}{4}$ of $\dfrac{1}{16}$, which is $\dfrac{1}{4^3}=\dfrac{1}{64}$ of the largest square, and the pattern continues.
Thus, we find $\dfrac{1}{4}+\dfrac{1}{4^{2}}+\dfrac{1}{4^{3}}+...$ as an infinite geometric series with $a_{1}= \dfrac{1}{4}$ and $r=\dfrac{1}{4}$.
Since $|r| \lt 1,$ the infinite geometric series converges and the sum for a convergent series is given by the formula $\displaystyle S_{\infty}=\sum_{k=1}^{\infty}a_{1}r^{k-1}=\frac{a_{1}}{1-r}$.
Substitute the known values to obtain:
$S_\infty=\dfrac{\frac{1}{4}}{1-\frac{1}{4}}=\dfrac{\frac{1}{4}}{\frac{3}{4}}=\dfrac{1}{4}\cdot \dfrac{4}{3}=\dfrac{1}{3}$
Therefore, $\dfrac{1}{3}$ of the largest square will be shaded.
