Analyzing the circles of harmonic radii. They almost do not overlap when centered at (2ln(n),1/n).

So when they are spread around the circle of radius 1, they don't overlap and it can be shown that for each time we go around the circle we can place so many circles that the radius of the first circle in the next round is exp(-pi) times the radius of the circle in the preceeding round. Hence we have a geometric series of radii as an upper bound for the thickness of the layer of circles.