Calculus II (MAT 146) Dr. Day Wednes day November 20, 2013

1. Calculus II (MAT 146)Dr. Day Wednesday November 20, 2013 • Transforming Series into Functions • Power Series • Coefficients • Applying the Ratio Test to Power Series • Radius of Convergence • Interval of Convergence • Assignments and Announcements MAT 146

2. Use the Ratio Test to Determine Convergence MAT 146

3. Determine the 5th partial sum for each series. MAT 146

4. Power Series • x is a variable. • The cn’s are constants, called the coefficients. • For any fixed value of x, we can test the series for convergence. MAT 146

5. Power Series • The sum of the series is a function with domain the set of all x values for which the series converges. • The function seems to be a polynomial, except it has an infinite number of terms. MAT 146

6. Power Series: Example • If we let cn = 1 for all n, we get a familiar series: • This geometric series has common ratio x and we know the series converges for |x| < 1. • We also know the sum of this series: MAT 146

7. Generalized Power Series • This is called: • a power series in (x – a), or • a power series centered at a, or • a power series about a. MAT 146

8. Power Series Convergence • For what values of x does this series converge? • Determine its Radius of Convergence and its Interval of Convergence. MAT 146

9. Power Series Convergence • For what values of x does this series converge? • Determine its Radius of Convergence and its Interval of Convergence. MAT 146

10. Power Series Convergence • For what values of x does this series converge? • Use the Ratio Test to determine values of x that result in a convergent series. MAT 146

11. Power Series Convergence • For what values of x does this series converge? • Use the Ratio Test to determine values of x that result in a convergent series. MAT 146

12. Power Series Convergence • For what values of x does this series converge? • Determine its Radius of Convergence and its Interval of Convergence. MAT 146

13. Power Series Convergence • For what values of x does this series converge? • Determine its Radius of Convergence and its Interval of Convergence. MAT 146

14. Power Series Convergence • For what values of x does this series converge? • Determine its Radius of Convergence and its Interval of Convergence. MAT 146

15. Power Series Convergence • For what values of x does this series converge? • Determine its Radius of Convergence and its Interval of Convergence. MAT 146

16. Geometric Power Series • If we let cn = 1 for all n, we get a familiar series: • This geometric series has common ratio x and we know the series converges for |x| < 1. • We also know the sum of this series: MAT 146

17. Geometric Power Series MAT 146

18. Geometric Power Series MAT 146

19. Geometric Power Series MAT 146

20. Assignments Homework Tasks • Today: Quiz #8 returned, solutions online • This Week: 11.8 (Today) • After Break: 11.9 (Wed 12/4) and 11.10 (Fri 12/6) • By the Day of Semester Exam: Four Review Assignments Also • Test #4: Tomorrow and Fri (11/21 and 11/22) • Help Session Tonight: STV 214, 6:15 pm – 7:30 pm • Check Your Semester Exam Schedule! • Semester Extra-Credit Option: Due December 2 MAT 146