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Lecture 29 – Power Series Def: The power series centered at x = a:

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Presentation on theme: "Lecture 29 – Power Series Def: The power series centered at x = a:"— Presentation transcript:

1 Lecture 29 – Power Series Def: The power series centered at x = a:
x is the variable and the c’s are constants (coefficients)

2 For any power series, exactly one of the following is true:

3 Example 1 – Radius and Interval of Convergence
Ratio Test: Series converges for

4 Example 2 – Radius and Interval of Convergence
Ratio Test: Series converges for

5 Example 3 – Radius and Interval of Convergence
Ratio Test:

6 Example 3 – continued (testing endpoints)

7 Example 4 – Radius and Interval of Convergence
Root Test:

8 Example 4 – continued (testing endpoints)

9 Example 5 – Radius and Interval of Convergence
Geometric Series:

10 Example 5 – continued – what is the converging value?
Geometric Series:

11 Lecture 30 – More Power Series
The geometric series: As a power series with a = 1, r = x and cn = 1 for all n: In other words, the function f(x) can be written as a power series. with

12 Example 1 Create new power series for other functions through: and
sum, difference, multiplication, division, composition and differentiation and integration Example 1 with

13 Example 2 with

14 Consider the graphs:

15 Example 3 with

16 Need to solve for C. Set x = 0 to get:
Test endpoints???

17 Example 4 with

18 Need to solve for C. Set x = 0 to get:
Test endpoints???


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