Name ____________________________________
From Euclid to Spigotry
Assignment 14 (outline 13)
Last modified: September 14, 2004
1. If you drop all the minus signs in Gregory's series, then the sum just keeps getting bigger and bigger ("the series diverges.") Here is the trick for proving this.
After the initial 1, chose groups of 1, 2, 4, 8, 16, 32, .... terms, so that the series is

1 + 
a. Show that each group has a sum that is greater than 1/4.

b. Suppose that you add up the first n terms, where n is a power of 2.
Show that whenever you double n, the sum increases by at least 1/4.
Explain how, if someone asked you to make the sum greater than 5, you could choose a value
of n that meets the request. 

2.  Consider the two functions f(y) =    g(x) = .  You know that 
a. Show that f(1-x) = g(x).  Draw graphs of f(y) and g(x) and consider areas to show that 
 and that p = 
b. Show that g(x) =  so that 4 g(x) = 
c. By doing a binomial expansion of  , convert   to a formula for p as a sum of three infinite series in which each term is smaller than its predecessor by at least a factor of 4.  Work out the first three terms of each series. A variation of this technique was discovered by accident in the 1990s by Canadian mathematicians Bailey, Borwein, and Plouffe, who then used it to calculate the ten-trillionth digit of p in the hexadecimal number system.