Answer: The semiperimeters of regular polygons with k sides that inscribe and

Chapter 4, Problem 15

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The semiperimeters of regular polygons with k sides that inscribe and circumscribe the unit circle were used by Archimedes before 200 b.c.e. to approximate , 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 k and Pk = k tan k , with pk << 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 to within 104 by computing pk and Pk until Pk pk < 104. d. Use Taylor Series to show that = pk + 3 3! 1 k 2 5 5! 1 k 4 + and = Pk 3 3 1 k 2 + 25 15 1 k 4 . e. Use extrapolation with h = 1/k to better approximate .