There are 86 pages on this site alone under a search for: area, square, circle, convergent, approximation. I found one that arguably asks the same question here, but I am not sure.
The question is, if we are trying to approximate the area of a square with identical circles, does the sequence of approximations converge (and does it converge to the area of the square)?
Because for 1,4,9,16, and 25 circles, the ratio r of the area of the approximating circles to that of the square is the same ($\frac{\pi}{4}$), I thought that this pattern surely continued. So my original question was whether, if we remove the square-numbered approximations, we get a convergent sequence.
I see that this is related to the problem of packing circles into a square, and having looked at the sequence of ratios (given here) I see that after 25 the "best" packing is not a nice military arrangement but something else, something that allows the circles to swell a bit. So the "best" packing improves for square numbers as well.
And so it does appear that we have a non-monotone sequence of approximations that might or might not converge to a ratio r = 1. But appearances can be deceiving.
I did a quick review of the literature (at the link above) but it is addressed primarily at the question of optimal packing for given numbers of circles, and not convergence, which I hope is much simpler.
Thanks for any insight.
If there were no walls and you were just trying to squeze circles together as tightly as possible, then the best way to do it is the hexagonal packing. Even in this packing the circles only cover 90.69% of the area, the other 9.31% lies in the gaps between the circles. So the approximation is always going to be less than 90.69% of the total area.
Now consider putting really small circles into your square. You can use a hexagonal packing in the middle, and continue it out toward the edges. Since the circles are very small, you'll be able to get very close to the edges without them messing up your pattern. So almost all the square will be covered with the hexagonal packing. So the limit does in fact converge to the ratio 0.9069...