Power Series Convergence and Representations

1. Example • Ex. For what values of x is the power series convergent? • Sol. By ratio test, the power series absolutely converges when and diverges when When we easily see that it diverges when x=4 and converges when x=1. Thus the power series converges for

2. Example • Ex. Find the domain of the Bessel function defined by • Sol. By ratio test, the power series absolutely converges for all x. In other words, The domain of the Bessel function is

3. Characteristic of convergence • Theorem For a given power series there are only three possibilities: (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 • The number R is called the radius of convergence of the power series. By convention, in case (i) the radius of convergence is R=0, and in case (ii)

4. Characteristic of convergence • The interval of convergence of a power series is the interval that consists of all x for which the series converges. • To find the interval of convergence, we need to determine whether the series converges or diverges at endpoints |x-a|=R. • Ex. Find the radius of convergence and interval of convergence of the series • Sol radius of convergence is 1/3. At two endpoints: diverge at 5/3, converge at 7/3. (5/3,7/3]

5. Radius and interval of convergence • From the above example, we found that the ratio test or the root test can be used to determine the radius of convergence. • Generally, by ratio test, if then • By root test, if then • Ex. Find the interval of convergence of the series • Sol. when |x|=1/e, the general does not have limit zero, so diverge. (-1/e,1/e)

6. Representations of functions as power series • We know that the power series converges to when –1<x<1. In other words, we can represent the function as a power series • Ex. Express as the sum of a power series and find the interval of convergence. • Sol. Replacing x by in the last equation, we have

7. Example • Ex. Find a power series representation for • Sol. The series converges when |-x/2|<1, that is |x|<2. So the interval of convergence is (-2,2). Question: find a power series representation for Sol.

8. Differentiation and integration • Theorem If the power series has radius of convergence R>0, then the sum function is differentiable on the interval and (i) (ii) The above two series have same radius of convergence R.

9. Example • The above formula are called term-by-term differentiationand integration. • Ex. Express as a power series and find the radius of convergence. • Sol. Differentiating gives By the theorem, the radius of convergence is same as the original series, namely, R=1.

10. Example • Ex. Find a power series representation of • Sol.

11. Example • Ex. Find a power series representation for and its radius of convergence. • Sol.

12. Taylor series • Theorem If f has a power series representation (expansion) at a, that is, if Then its coefficients are given by the formula • This is called the Taylor series of f at a (or about a)

13. Maclaurin series • The Taylor series of f at a=0 is called Maclaurin series • Ex. Find the Maclaurin series of the function and its radius of convergence.

14. Maclaurin series • Ex. Find the Maclaurin series for sinx. • Sol. So the Maclaurin series is

15. Important Maclaurin series • Important Maclaurin series and their convergence interval

16. Example • Ex. Find the Maclaurin series of • Sol.

17. Multiplication of power series • Ex. Find the first 3 terms in the Maclaurin series for • Sol I. Find and the Maclaurin series is found. • Sol II. Multiplying the Maclaurin series of and sinx collecting terms:

18. Division of power series • Ex. Find the first 3 terms in the Maclaurin series for • Sol I. Find and the Maclaurin series is found. • Sol II. Use long division

19. Application of power series • Ex. Find • Sol. Let then s=S(-1/2). To find S(x), we rewrite it as

20. Exercise • Ex. Find • Sol.

21. Application of power series • Ex. Find by the Maclaurin series expansion. • Sol. where in the last limit, we have used the fact that power series are continuous functions.

22. Homework 25 • Section 11.8: 10, 17, 24, 35 • Section 11.9: 12, 18, 25, 38, 39 • Section 11.10: 41, 47, 48