Circumscribed polygons and circles

Starting with a one unit circle, circumscribed by a regular triangle, circumscribed by a circle outside, circumscribed by a square, circumscribed by a circle, circumscribed by a regular pentagon and so on by circles and regular polygon. Will the radius of the circle diverge or converge when we continue to add more polygon and circles? This presentation will use Taylor series and comparison test to answer whether the radius will approach infinity or converge at some point.