When discussing the Taylor series for sin(x), for example, we know the
series converges for all values of x and the convergence of the series
depends on the limit of (xn! going to zero for any fixed value of x as
n goes to infinity. Here are two interesting looks at what that
convergence means--from a technological look.
Pick several values of x. e.g. x = 5, x = 35, x = 100
For those values of x have the students find out how large n has to be
for the ratio (xn! to be less than 0.05. Hopefully one of your
students will have an HP calculator. I do not think the TI-85 will
handle the numbers. This brings up intersting discussion about what
such a limit means. It can also bring up great discussion about the
limitations of a calculator. It also brings home how "far out" you
have to go in a convergent series to get a good approximation for a
value not close to the "center" of the series. I have given this as a
homework exercise that we then use the next day to spark the discussion
of what the above limit means. Sometimes, I prove the limit out
formally and sometimes I do not.
Another study. Have students compare the graphs of sin(x) with the
graphs of one term of the Maclaurin Series, two terms, three terms,
4 terms,10 terms, etc.
Graphically, it becomes clear that the further the value for x is
from 0, the more terms have to be added to get a good approximation
from the partial series. This look coupled with the above exercise
prvides a good intuitive way to discuss series convergence and error.
2. Run a strand throughout the semester where the Taylor Series is one of
several ways to estimate a definite integral. Study Integral of ((ex)
First study: When discussing integration and methods of integration.
Graph f'(x) = (ex. From the graph, determine where
f(x) is increasing, decreasing, has horizontal tangents and is likely
to have vertical asymptotes. Sketch a possible rough graph when for
some initial condition, for example, f(1) = 0.
Second study: When discussing numeric integration--
Approximate The Integral of ((et) (from t = 1 to t = x) using
various values of x. On the TI-85, use the integral command.
Write a program to use the lists, or record the data in a chart on
paper. From the data, produce a graph of
f(x) = The Integral of ((et) (from t = 1 to t = x)
Third study: Approximate one of the chart values from step 1 using
Simpson's or Trapezoid rule.
Fourth study: Find the Taylor series for (ex and use it to
approximate The Integral of ((et) (from t = 1 to t = x) for
various values of x.
Martha
Martha Haehl
Maple Woods Community College
2601 N. E. Barry Rd.
Kansas City, Missouri 64156
(816)437-3147