The semi perimeters of regular polygons with k sides that inscribe and circumscribe the unit circle were used by Archimedes before 200 b.c.e. to approximate π, the circumference of a semicircle. Geometry can be used to show that the sequence of inscribed and circumscribed semiperimeters {pk}and {Pk}, respectively, satisfy pk = k sin(π/k) and Pk = k tan(π/k), With pk < π < Pk , whenever k ≥ 4.

a. Show that p4 = 2√2 and P4 = 4.

b. Show that for k ≥ 4, the sequences satisfy the recurrence relations P2k = (2pkPk)/(pk + Pk) and p2k = √(pkP2k) .

c. Approximate π to within 10−4 by computing pk and Pk until Pk − pk < 10−4.

d. Use Taylor Series to show that π = pk + π3/3!(1/k)2 − π5/5!(1/k)4 +· · · and π = Pk − π3/3(1/k)2 + 2π5/15(1/k)4 −· · · .