Optimal approximation of square area with identical circles

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.

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