Due:  Wednesday, April 2

I    Machin's formula for computing p*

(* This problem, in part, is based upon project 1 of Hughes-Hallett et al, Calculus, 3rd edition, page 474.)

In 1706, John Machin - a professor of Astronomy in London - discovered the formula

1.  Use the 5^th degree Maclaurin polynomial for arctan(x) to approximate the value of p.   (Historical note:  In 1873, William Shanks used this approach to calculate p to 707 decimal places.  In 1949, John von Neumann used the ENIAC to compute p up to 2000 digits using Machin's formula.  Currently, several billion decimal places are known!)

2.   In addition to Machin's formula, a group of related identities have been discovered, some by famous mathematicians.  These are known as Machin-like formulas.
Using appropriate Maclaurin polynomials, check the validity of each of the following:

(A)   p/4 = arctan(1/2) + arctan(1/3)   (Hutton, 1776)

(B)  p/4 = 2 arctan(1/3) + arctan(1/7)     (Hutton, 1776)

(C)   p/4  = 5 arctan(1/7) + 2 arctan(3/79)    (Euler, 1755)

(D)   p/4  = 4 arctan(1/5) -  arctan(1/70) + arctan (1/99)   (Euler, 1764)

(E)  p/4  = 12 arctan(1/18) + 8 arctan(1/57) - 5 arctan (1/239)   (Gauss, 1863)

(F)    p/2  = 2 arctan(1/sqrt(2)) + arctan(1/sqrt(8))    (Wetherfield, 1996)
 
 

II        Wallis' formula for computing p.

In 1655, John Wallis discovered a beautiful formula for p.

1.  Plot the sequence, a(n), and show that the limit as n increases without bound is p.  How does the graph indicate the slowness of this convergence?

2.  Describe briefly the historical development of the calculation of p.