# Archimedes approximated the value of p by computing the perimeter of a regular polygon inscribing…

Archimedes approximated the value of π by computing the perimeter of a regular polygon inscribing or circumscribing a circle of diameter 1. The perimeter of an inscribed polygon with n sides is given by

and these values provide lower and upper bounds, respectively, on the value of π.

(a) Using the power series expansions for the sine and tangent functions, show that pn and qn can be expressed in the form

where h = 1/n. What are the true values of a0 and b0?

(b) Given the values p6 = 3.0000 and p12 = 3.1058, use Richardson extrapolation to produce a better estimate for π. Similarly, given the values q6 = 3.4641 and q12 = 3.2154, use Richardson extrapolation to produce a better estimate for π.

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