1. 9.2 day 2 Maclaurin Series Liberty Bell, Philadelphia, PA
Greg Kelly, Hanford High School, Richland, Washington
Photo by Vickie Kelly, 2003

2. There are some Maclaurin series that occur often enough that they should be memorized. They are on your formula sheet, but today we are going to look at where they come from.
Maclaurin Series:
(generated by f at )

3. List the function and its
derivatives.

4. a = 1 and r = x. List the function and its derivatives.
Evaluate column one
for x = 0.
This is a geometric series with
a = 1 and r = x.

5. We could generate this same series for with polynomial long division:

6. This is a geometric series with
a = 1 and r = -x.

7. We wouldn’t expect to use the previous two series to evaluate the functions, since we can evaluate the functions directly.
They do help to explain where the formula for the sum of an infinite geometric comes from.
We will find other uses for these series, as well.
A more impressive use of Taylor series is to evaluate transcendental functions.

8. Both sides are even functions.
Cos (0) = 1 for both sides.

9. Both sides are odd functions.
Sin (0) = 0 for both sides.

10. If we start with this function:
and substitute for , we get:
This is a geometric series with a = 1 and r = -x2.
If we integrate both sides:
This looks the same as the series for sin (x), but without the factorials.

12. We have saved the best for last!

14. An amazing use for infinite series:
Substitute xi for x.
Factor out the i terms.

15. p This is the series for cosine. This is the series for sine. Let
This amazing identity contains the five most famous numbers in mathematics, and shows that they are interrelated. p