The sum of the shaded areas is [area of biggest square]-[area of biggest circle]+[area of next biggest square]-[area of next biggest circle]+...
Split this into two series
[area of biggest square]+[area of next biggest square]+...
And
-[area of biggest circle]-[area of next biggest circle]-...
Examining the two largest squares you can see using pythagoras' theorem you can see that the smaller square's side length is $\displaystyle \sqrt{2}$ times smaller than the larger square's side. This ratio is constant for each set of squares so the first series mentioned above is an infinite geometric series.
The diameter of the circle is equal to the length of the side of the square it is inside, the square's side length is decreasing in a geometric series so the diameter of each circle will also decrease in a geometric series. Use this information to determine the sum of the second series mentioned above.