# Taylor Series

 Final exam: Wednesday, March 22, 7-10pm 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, 32-38 B, 26-31 C, 20-25 D, 14-19 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:

 for any fixed

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 2006-03-15