Chapter 8 Infinite Series

Section 8.3 Power Series

Up to this point we have dealt with infinite series whose terms were fixed numbers. We broaden our perspective now to consider series whose terms are variables. The simplest kind is known as a power series, and the main question will be determining the set of values of the variable for which the series is convergent. Definition 8.3.1 Let be a sequence of real numbers. The series is called a power series. The number an is called the nth coefficient of the series. Example 8.3.2 Consider the power series whose coefficients are all equal to 1:  xn. This is the geometric series, and it converges iff | x | < 1.

Theorem 8.3.3 Let  an xn be a power series and let  = lim sup | an |1/n. Define R by Then the series converges absolutely whenever | x | < R and diverges whenever | x | > R. (When R = + , we take this to mean that the series converges absolutely for all real x. When R = 0, then the series converges only at x = 0.) Proof: Let bn = an xn and apply the root test (Theorem 8.2.8). If lim sup | an |1/n =   , we have  = lim sup | bn |1/n = lim sup | an xn |1/n = | x | lim sup | an |1/n = | x | . Thus if  = 0, then  = 0 < 1, and the series converges (absolutely) for all real x. If 0 <  < + , then the series converges when | x |  < 1 and diverges when | x |  > 1. That is,  an xn converges when | x | < 1/ = R and diverges when | x | > 1/ = R. If  = + , then for x  0 we have  = + , so  an xn diverges when | x | > 0 = R. Certainly, the series will converge when x = 0, for then all the terms except the first are zero. 

Theorem 8.3.4 (Ratio Criterion) From Theorem 3 we see that the set of values C for which a power series converges will either be {0}, , or a bounded interval centered at 0. The R that is obtained in the theorem is referred to as the radius of convergence and the set C is called the interval of convergence. We think of {0} as an interval of zero radius and as an interval of infinite radius. When R = + , we may denote the interval of convergence by ( , ). Notice that when R is a positive real number, the theorem says nothing about the convergence or divergence of the series at the endpoints of the interval of convergence. It is usually necessary to check the endpoints individually for convergence using one of the other tests in Section 8.2. The following Ratio Criterion is also useful in determining the radius of convergence. (Ratio Criterion) The radius of convergence R of a power series  an xn is equal to limn   | an/an  +  1 |, provided that this limit exists. Theorem 8.3.4 Note that this ratio is the reciprocal of the ratio in the ratio test for sequences (Theorem 8.2.7.)

Example 8.3.5 (a) The interval of convergence for the geometric series  xn is ( –1, 1). (b) For the series we have so the radius of convergence is 1. When x = 1, we get the divergent harmonic series: When x = – 1, we get the convergent alternating harmonic series: So the interval of convergence is [ – 1, 1). (c) For the series we have so the radius of convergence is 1. When x = 1, we get the convergent p-series with p = 2: When x = – 1, the series is also absolutely convergent since So the interval of convergence is [ – 1, 1].

Example 8.3.7 Example 8.3.9 (a) For the series we have Thus the radius of convergence is + , and the interval of convergence is . (b) For the series , it is easier to use the root formula: Thus R = 0 and the interval of convergence is {0}. Example 8.3.9 Consider the series Letting y = x2, we may apply the ratio criterion to the series and obtain Since it also converges when y =  3 but diverges when y = 3, its interval of convergence is [ 3, 3). Thus the series in y converges when | y | < 3. But y = x2, so the original series has as its interval of convergence. Note that is not included, because this corresponds to y = +3.

Sometimes we wish to consider more general power series of the form where x0 is a fixed real number. By making the substitution y = x – x0, we can apply the familiar techniques to the series If we find that the series in y converges when | y | < R, we conclude that the original series converges when | x – x0 | < R. Example 8.3.10 For the series we have R = 1, so it converges when | x – 1 | < 1. That is, when 0 < x < 2. Since it diverges at x = 0 and x = 2, the interval of convergence is (0, 2). In fact, using the formula for the geometric series, we see that for all x in (0, 2) the sum of the series is equal to