1. Taylor and Maclaurin Series
Lesson 9.10

2. Convergent Power Series Form
Consider representing f(x) by a power series
For all x in open interval I
Containing c
Then

3. Taylor Series
If a function f(x) has derivatives of all orders at x = c, then the series is called the Taylor series for f(x) at c.
If c = 0, the series is the Maclaurin series for f .

4. Taylor Series
This is an extension of the Taylor polynomials from section 9.7
We said for f(x) = sin x, Taylor Polynomial of degree 7

5. Guidelines for Finding Taylor Series
Differentiate f(x) several times
Evaluate each derivative at c
Use the sequence to form the Taylor coefficients
Determine the interval of convergence
Within this interval of convergence, determine whether or not the series converges to f(x)

6. Try It Try for at x = 0 Differentiate several times Evaluate at x = 0
Develop the general term of the series
Check for interval of convergence

7. Series for Composite Function
What about when f(x) = cos(x2)?
Note the series for cos x
Now substitute x2 in for the x's

8. Binomial Series Consider the function
This produces the binomial series
We seek a Maclaurin series for this function
Generate the successive derivatives
Determine
Now create the series using the pattern

9. Binomial Series We note that
Thus Ratio Test tells us radius of convergence R = 1
Series converges to some function in interval -1 < x < 1

10. Combining Power Series
Consider
We know
So we could multiply and collect like terms

11. Assignment
Lesson 9.10
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