There is a figure of an equilateral triangle whose side is 1. There is a big circle in the middle touching all sides. Then more circles are added that touch each circle and approaching all 3 vertices. What is the total area occupied by the circles?
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The radius of the center circle is R = (1/2)(1/√3). Thereafter, the successive smaller radii are (1/3^n)R, so when all worked out, the total area of the circles is (11/96)π = 0.35997..., as compared to the area of the equilateral triangle, which is (1/4)√3 = 0.4331... I don't know how this is a calculus problem, through.