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generally uniform; that is, it depends on x . In this regard, recall also a useful result given in Exercise 55.7.25 57.2. Radius of Convergence. The set S on which a power series is convergent is an important characteristic and its properties have to be studied. Lemma 8.3 (Properties of a Power Series) . (i) . If a power series c n x n converges when x = b = 0 , then it converges whenever | x | < | b | . (ii) . If a power series c n x n diverges when x = d = 0 , then it diverges whenever | x | > | d | . Proof. If c n b n converges, then, by the necessary condition for convergence, c n b n 0 as n → ∞ . This means, in particular, that, for ε = 1, there exists an integer N such that | c n b n | < ε = 1 for all n > N . Thus, for n > N , | c n x n | = c n b n x n b n = | c n b n | x b n < x b n . which shows that the series c n x n converges by comparison with the geometric series q n , where q = x/b and | x/b | < 1 or | x | < | b | .
136 8. SEQUENCES AND SERIES Suppose that c n d n diverges. If x is any number such that | x | > | d | , then c n x n cannot converge because, by part (i) of the lemma, the convergence of c n x n implies the convergence of c n d n . Therefore, c n x n diverges. This lemma allows us to establish the following description of the set S . Theorem 8.27 (Convergence Properties of a Power Series) . For a power series c n x n , there are only three possibilities: (i) The series converges only when x = 0 . (ii) The series converges for all x . (iii) There is a positive number R such that the series converges if | x | < R and diverges if | x | > R . Proof. Suppose that neither case 1 nor case 2 is true. Then there are numbers b = 0 and d = 0 such that the power series converges for x = b and diverges for x = d . By Lemma 8.3, the set of convergence S lies in the interval | x | ≤ | d | for all x S . This shows that | d | is an upper bound for the set S . By the completeness axiom, S has a least upper bound R = sup S . If | x | > R , then x S , and c n x n diverges. If | x | < R , then | x | is not an upper bound for S , and there exists a number b S such that b > | x | . Since b S , c n x n converges by Lemma 8.3. Theorem 8.27 shows that a power series converges in a single open interval ( R, R ) and diverges outside this interval. The set S may or may not include the points x = ± R . This question requires a special investigation just like in Example 8.38. So the number R is character- istic for convergence properties of a power series. Definition 8.11 (Radius of Convergence) . The radius of conver- gence of a power series c n x n is a positive number R > 0 such that the series converges in the open interval ( R, R ) and diverges outside it. A power series is said to have a zero radius of convergence , R = 0 , if it converges only when x = 0 . A power series is said to have an infinite radius of convergence , R = , if it converges for all values of x . The ratio or root test can be used to determine the radius of con- vergence.
57. POWER SERIES 137 Corollary 8.3 (Radius of Convergence of a Power Series) . Given a power series c n x n , if lim n →∞ | c n +1 | | c n | = α = R = 1 α , if lim n →∞ n | c n | = α = R = 1 α , where R = 0 if α = and R = if α = 0 .
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