A square is divided into nine smaller squares of equal area.

There's no picture?

4/9 + 4/81 + 4/729 + … = (4/9)/(1-1/9) = (4/9)/(8/9) = (4/9)(9/8) = 4/8 = 1/2

I think it is an infinite geometric sequence with   a1 = 4/9   r = 1/9

sum = a1 / (1-r) = 4/8 = 1/2

You can think of the square without the center piece. This figure would consist of $8$ squares, $4$ being shaded or $\frac{1}{2}$ shaded. We can think of the center piece of this figure as $8$ squares again with $4$ shaded. Each time you count the center piece with $8$ squares, you have $\frac{1}{2}$ shaded and $\frac{1}{2}$ unshaded. This pattern continues forever. Thus the answer is $\frac{1}{2}.$