Final exam: Wednesday, March 22, 710pm in PCYNH 109. Bring ID!
Last Quiz 4: This Friday
Next: 11.10 Taylor and Maclaurin series
Next: 11.12 Applications of Taylor Polynomials
Midterm Letters:
A, 3238
B, 2631
C, 2025
D, 1419
Mean: 23.4, Standard Deviation: 7.8, High: 38, Low: 6.

Example 6.6.1
Suppose we have a degree
(cubic) polynomial
and
we know that
,
,
, and
.
Can we determine
? Answer: Yes!
We have
From what we mentioned above, we have:
Thus
Amazingly, we can use the idea of Example to compute power
series expansions of functions. E.g., we will show below that
Convergent series are determined by the values
of their derivatives.
Consider a general power series
We have
where for the last equality we use that
Remark 6.6.2
The definition of
is
(it's the empty product).
The empty sum is
0 and the empty product is
.
Theorem 6.6.3 (Taylor Series
)
If is a function that equals a power series centered
about , then that power series expansion is
Remark 6.6.4
WARNING: There are functions that have all derivatives defined, but do
not equal their Taylor expansion. E.g.,
for
and
. It's Taylor expansion is the
0 series (which
converges everywhere), but it is not the
0 function.
Definition 6.6.5 (Maclaurin Series)
A
Maclaurin series is just a Taylor series with
.
I will not use the term ``Maclaurin series'' ever again (it's common
in textbooks).
Example 6.6.6
Find the Taylor series for
about
.
We have
. Thus
for all
. Hence
What is the radius of convergence?
Use the ratio test:
Thus the radius of convergence is
.
Example 6.6.7
Find the Taylor series of
about
.
^{6.1} We have
To do this we have to
puzzle out a pattern:
First notice how the signs behave.
For
even,
and for
odd,
For
even we have
and for
odd we have
Finally,
Next we use the ratio test to compute the radius of convergence.
We have
which converges for each
. Hence
.
Example 6.6.8
Find the Taylor series for
about
.
We have
.
Thus from Example
(with infinite radius
of convergence) and that the Taylor
expansion is unique, we have
William Stein
20060315