Precalculus (10th Edition)

$\frac{1}{3}$
An infinite geometric series converges if and only if $|r|\lt1$, where $r$ is the common ratio. If it converges, then it equals $\frac{a_1}{1-r}$ where $a_1$ is the first term. The common ratio is the quotient of two consecutive terms: $r=\frac{1}{4}$, because always one quarter of a larger quarter gets shaded. Hence the sum (since $a_1=\frac{1}{4}$): $\dfrac{\frac{1}{4}}{1-\frac{1}{4}}=\frac{1}{3}$