*Draw a unit circle.*

Around it, circumscribe an equilateral triangle.

*Circumscribe another circle around that.*

Circumscribe a square around this circle.

*Circumscribe another circle around that.*

Circumscribe a regular pentagon around this circle.

*Circumscribe another circle around that.*

Continue, ad infinitum, with the next regular polygon.

Do the radii of these circles converge? If so, what is the limiting radius?