1. 9.1 B Power Series

2. This series would converge of course provided that … Write f (x) as a series: This looks like the sum of… A Geometric Series: in which a = 1 and r = x And it would converge to the value of at x

3. The coefficients c 0, c 1, c 2 …c n are constants. Their value determines the uniqueness of the series. Another example: r =  x Essentially, this is a polynomial with a degree of… Now let’s graph the function and some partial sums  This is called a Power Series

4. This is called a Power Series Let’s graph the function and some partial sums:

5. This is called a Power Series Let’s graph the function and some partial sums: Where does the polynomial seem to overlap with f (x) ? Near x = 0.

6. This is called a Power Series Let’s graph the function and some partial sums: What seems to happen as we add terms?

7. This is called a Power Series Let’s graph the function and some partial sums: What seems to happen as we add terms? The overlap increases.

8. This is called a Power Series Let’s graph the function and some partial sums: What x values can the overlap never reach and why? x = ± 1 Remember…  1 < x < 1

9. Interval of Convergence This is called a Power Series Let’s graph the function and some partial sums: Because the series only converges for these values of x, it is called the Remember…  1 < x < 1 This power series is “centered” at x = 0.

10. This is called a Power Series Use a Geometric Series to represent this: Remember that this is how you made horizontal shifts to the right in Algebra and Trig a = 1 r = 1 – x

11. This is called a Power Series Use a Geometric Series to represent this: What is the interval of convergence here? (Geometric Series)

12. Use to write a power series for ln x What about this C ? We do know that if we plug 1 in for x Solving will give us C = −1 And so… 0 Remember:

13. In upcoming sections, you will learn more about why we use power series to represent functions we already know. This power series is “centered” at x = 1.