Name __________________________________
   From Euclid to Spigotry
Assignment 16 (outline 14)
Last modified: September 14, 2004
1. Charles Hutton, in 1776, proved that p = 4 arctan  + 4 arctan  
In 1877, it is reported, "Tseng Chi-hung used this formula to compute p to 100 decimals in less than a month."
a. Suppose Tseng evaluated arctan  as
 with x = 1/2.  Roughly how many terms would he have needed in order to be sure that the error in ignoring the rest of the series was less than ? A useful fact, that will let you avoid logarithms, is that  and  are both very nearly equal to .

b. More likely, Tseng knew Euler's speeded-up version of the series,
arctan  =  with y = 
Roughly how many terms would he have needed in order to be sure
that the error in ignoring the rest of the series was less than ?
Hint: ,  so  is very nearly equal to 

2. Another series for which Euler's trick works is 
a. This is a "geometric series" of the form 1+x-x2+.... Show that its sum is .

b. Carry out Euler's trick: split each term in two, keep the first term intact, and combine the
others in pairs, combining each negative term with the preceding positive term.  What series do you obtain?

c. Do the same thing again. This time you leave the first term alone.

d. Do the same thing yet again. This time you leave the first two terms alone.

e. At this point you can guess the general rule, which gives a series that has all positive terms, with an extra factor of 4 in the denominator for each successive term.  Show that this series also has a sum of 2/3.