Brouncker's Fraction

Last revised: September 8, 2003
Inspired by the infinite product of his friend Wallis, Lord Brouncker devised a continued 
fraction that converges to pi.  A century later, Euler showed that this fraction is equivalent 
to a well-known infinite series for pi.  Continued fractions, once a hot topic in mathematics 
but now almost forgotten, provide a way of getting rational approximations to pi that are 
better than 22/7. 
Requires
Wallis's Product
Newton's Arc Sine
References:
Beckmann, pages 129-131
Minilecture 1: Brouncker and Gregory
Show that Brouncker's continued fraction is equivalent to Gregory's series.
The proof is due to Euler, who lived a century after Brouncker.  How Brouncker came up with this continued fraction appears to be a total mystery. 
Basically, the proof is in Beckmann, but he is a bit cavalier in saying "is easily seen to be equivalent" in the middle of page 131. The attached hint from Pi and the AGM might be more useful.

To see what is going on, flip over Brouncker's fraction to get
 
For comparison purposes, define
 
This gives the partial sums of Gregory's series.
Now let's calculate some convergents of Brouncker's fraction:
while 

while 
 while 
So there's nothing new here.  The convergence of the continued fraction is as painfully slow as the convergence of Gregory's series

Minilecture 2: Simple continued fractions
Given a real number that you can calculate, show how to develop an expansion of it as a simple continued fraction.
Notes on this minilecture:
This is a good way to get excellent rational approximations to irrational numbers like p, e, or   All you have to do is to keep subtracting off the integer part of the number and inverting the fraction that remains.  Here's how to do it for e (which you can change to p or anything else in the interactive version of Mathcad that will be available in class)

And in fact, is a pretty good rational approximation to e.
If you change x0 to  above you will get and you will also notice that the values quickly start to repeat.  This happens for any square root.
And if you change x0 to p you will rediscover and some other rational approximations to p 
Minilecture 3: Biography of Lord Brouncker:
You can learn about Lord Brouncker in Cabot Library in the book
Mathematics of Great Amateurs by Julian Coolidge
It is claimed that Wallis showed that Brouncker's fraction was equivalent to his product, but no one seems to have reproduced the proof in English.  Some authors are skeptical. Harvard appears to own Wallis's original work, but it's in Latin and only in Houghton.  If you can read scientific Latin, you might look it up.  There must be more than just the little bit that is on page 80 of the Pi Source Book. 
Minilecture 4 (computational):
Use Mathematica to get a rational approximation to p that is accurate to 100 decimal places.  Just carry out the simple continued fraction calculation of minilecture 2.