Note: for the purposes of this lab, and until otherwise instructed, the
definition of m significant digits (or m digits of accuracy) is when
|x_T - x_A|/x_T <= .5E-m.

1.  It may _not_ be more efficient to use the built-in taylor command. 
In particular, for the second series, there are no even terms in the
expansion.  Using the index k as the series is written in the problem,
is a better count of how many terms are used in the estimate, than is
the order of the taylor polynomial.  Also note that the entire summation
need not be computed at each iteration, just the kth term, which is
added to the previous estimate.  (If you didn't know the exact solution,
you could stop iterating once your kth term was small enough, since it
is in essence the difference between your kth and (k-1)th estimates.)

It is interesting to also look at the ratio of the current error to the 
previous error in order to compare rates of convergence for the two 
methods.

2.  Brute force is a good place to start, but you should not have to use 
a range of values for both the numerator and denominator.  In 
particular, if you have narrowed down your numerators to a set of 
integers stored in the vector x, then for each x_i there should be a 
best choice of the denominator (think x_i/y_i ~= pi, solve for the 
integer y_i).  Now you can define a vector y of denominator candidates, 
and therefore a vector of errors which can be minimized.  Be sure to use 
"./" for component-wise division.  Just for fun, find a rational 
approximation for e with a 2 digit numerator (never know when it might 
come in handy!)

3. Might be more clear to say "decreasing by 10", but you know what I 
mean:  x=10^(-i), i=1:20.  For (6b), see the expansion on page 5 (or 13, 
or 16...), but replace x with -x.  Use a "sufficient" number of terms in 
your expansion.

Wed Oct 11 16:19:18 PDT 2006