Power Series

1. Power Series Lesson 9.8

2. Definition • A power series centered at 0 has the form • Each is a fixed constant • The coefficient of

3. Example • A geometric power series • Consider for which real numbers x does S(x) converge? • Try x = 1, x = ½ • Converges for |x| < 1 • Limit is

4. An e Example • It is a fact … (later we see why) • The right side is a power series • We seek the values of x for which the series converges

5. An e Example • We use the ratio test • Thus since 0 < 1, series converges absolutely for all values of x • Try evaluating S(10), S(20), S(30)

6. Power Series and Polynomials • Consider that power series are polynomials • Unending • Infinite-degree • The terms are power functions • Partial sums are ordinary polynomials

7. Choosing Base Points • Consider • These all represent the same function • Try expanding them • Each uses different base point • Can be applied to power series

8. Choosing Base Points • Given power series • Written in powers of x and (x – 1) • Respective base points are 0 and 1 • Note the second is shift to right • We usually treat power series based at x = 0

9. Definition • A power series centered at c has the form • This is also as an extension of a polynomial in x

10. Examples • Where are these centered, what is the base point?

11. Power Series as a Function • Domain is set of all x for which the power series converges • Will always converge at center c • Otherwise domain could be • An interval (c – R, c + R) • All reals c c

12. Example • Consider • What is the domain? • Think of S(x) as a geometric series • a = 1 • r = 2x • Geometric series converges for |r| < 1

13. Finding Interval of Convergence • Often the ratio test is sufficient • Consider • Show it converges for x in (-1, 1)

14. Finding Interval of Convergence • Ratio test • As k gets large, ratio tends to |x| • Thus for |x| < 1 the series is convergent

15. Convergence of Power Series For the power series centered at cexactly one of the following is true • The series converges only for x = c • There exists a real number R > 0 such that the series converges absolutely for |x – c| < R and diverges for |x – c| > R • The series converges absolutely for all x

16. Example • Consider the power series • What happens at x = 0? • Use generalized ratio test for x ≠ 0 • Try this

17. Dealing with Endpoints • Consider • Converges trivially at x = 0 • Use ratio test • Limit = | x | … converges when | x | < 1 • Interval of convergence -1 < x < 1

18. Dealing with Endpoints • Now what about when x = ± 1 ? • At x = 1, diverges by the divergence test • At x = -1, also diverges by divergence test • Final conclusion, convergence set is (-1, 1)

19. Try Another • Consider • Again use ratio test • Should get which must be < 1or -1 < x < 5 • Now check the endpoints, -1 and 5

20. Power Assignment • Lesson 9.8 • Page • Exercises 1 – 33 EOO