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

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

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)

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]

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)

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

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

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.

Example The above formula are called term-by-term differentiation and 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.

Example Ex. Find a power series representation of Sol.

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

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)

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.

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

Important Maclaurin series Important Maclaurin series and their convergence interval

Example Ex. Find the Maclaurin series of Sol.

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:

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

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

Exercise Ex. Find Sol.

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.

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