Sect. 9-8 Power Series

1. Sect. 9-8 Power Series

2. Power Series A power series is a series of the form where x is a variable and the cn’s are constants called the coefficients of the series.

3. Power Series General Form With a radius of convergence R > 0 Ratio Test best for determining convergence or divergence of a power series

4. Examples of Power Series • The following power series is centered at 0. • The following power series is centered at –1. • The following power series is centered at 1.

5. 1. • For what values does the series converge? b) For what values does the series diverge?

6. Convergence of a Power Series For a given power series there are only three possibilities for convergence: (i) The series converges only when x=a. (ii) The series converges for all x. (iii) There is a positive number R such that the series converges if and diverges if Where R is the radius of convergence of the power series.

7. The Interval of Convergence Is the interval that consists of all values of x for which the series converges. In case (i) (The series converges only when x = a) the interval consists of just a single point a. In case (ii) (The series converges for all x) the interval is (-∞,∞).

8. The Interval of Convergence In case (iii) note that the inequality │x-a│<R can be rewritten as a – R < x < a + R . Thus in case (iii) there are four possibilities for the interval of convergence: * Endpoint Convergence

9. The Radius of Convergence

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