How do we know that the function ln (1 + x)
is not a polynomial? State at least one property of this function
that could not be a property of any polynomial.
The calculations in Part 3 suggest that the n-th degree Taylor polynomial
for g(x) = ln (1 + x) is
Check this by having your computer algebra system compute several Taylor polynomials
with degrees large enough to show the pattern of coefficients.
For each of the Taylor polynomials p1(x), p2(x),
p3(x), and p4(x),
plot pn(x) and g(x) together;
plot the error function, g(x) - pn(x);
describe the extent to which pn(x) does and does not
approximate g(x).
You may need to experiment with the ranges of the error plots to get the most
information out of each one.
Repeat step 3 for n = 10, 20,
and 40. What do you think is the interval of convergence of the
Taylor polynomials for g(x)? Explain.
Use what you have learned in this part to describe and carry out a pencil-and-paper
calculation of ln(9/4) that is accurate to within 0.01.
[Caution: This requires some thought before you start to calculate.] You may
use your computer algebra system (or a calculator) to do the arithmetic, but
it should be arithmetic that you could do by hand if you had to. Use your
CAS to check your answer.
