11.2 Properties of Power Series

1. 11.2 Properties of Power Series Math 6B Calculus II

2. Power Series • Power Series centered at x = 0. • Power Series centered at x = a.

3. The Function 1/(1 – x) • The function 1/(1 – x) can be rewritten into a power series.

4. How to Test a Power Series for Convergence • Use the Ratio test (or Root test) to find the interval where the series converges absolutely. Ordinarily this is an open interval or a – R < x < a + R • If the interval of absolute convergence is finite, test for convergence or divergence at each endpoint. Use an appropriate test.

5. How to Test a Power Series for Convergence • If the interval of absolute convergence is a – R < x < a + R, the series diverges for

6. The Radius and Interval of Convergence Possible behavior of • There is a positive number R (also known as the radius of convergence) such that the series diverges for all but converges for . The series may or may not converge at either endpoint x = a – R and x = a + R.

7. The Radius and Interval of Convergence • The series converges absolutely for every x • The series converges at x = a and diverges everywhere else. (R = 0 )

8. Combining Power Series Suppose the power series and converge absolute to and respectively, on an interval I. • Sum and difference: The power series converges absolutely to on I. • Multiplication by a power: The power series converges absolutely to on I, provided m is an integer such that for all terms of the series

9. Combining Power Series • Composition: If , where m is a positive integer and b is a real number, the power series converges absolutely to the composite function , for all x such that h(x) is in I.

10. Differentiation and Integration of a Power Series

11. Differentiation and Integration of a Power Series The radii of convergence of the power series in Equations (i) and (ii) are both R.