Chapter7 - Chapter 7 Power Series It is a pain to think...

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Chapter 7 Power Series It is a pain to think about convergence but sometimes you really have to. Sinai Robins 7.1 Sequences and Completeness As in the real case (and there will be no surprises in this chapter of the nature ‘real versus complex’), a (complex) sequence is a function from the positive (sometimes the nonnegative) integers to the complex numbers. Its values are usually denoted by a n (as opposed to, say, a ( n )) and we commonly denote the sequence by ( a n ) n =1 , ( a n ) n 1 , or simply ( a n ). The notion of convergence of a sequence is based on the following sibling of Definition 2.1. Definition 7.1. Suppose ( a n ) is a sequence and a C such that for all ± > 0, there is an integer N such that for all n N , we have | a n - a | < ± . Then the sequence ( a n ) is convergent and a is its limit , in symbols lim n →∞ a n = a. If no such a exists then the sequence ( a n ) is divergent . Example 7.2. lim n →∞ i n n = 0: Given ± > 0, choose N > 1 . Then for any n N , ± ± ± ± i n n - 0 ± ± ± ± = ± ± ± ± i n n ± ± ± ± = | i | n n = 1 n 1 N < ±. Example 7.3. The sequence ( a n = i n ) diverges: Given a C , choose ± = 1 / 2. We consider two cases: If Re a 0, then for any N , choose n N such that a n = - 1. (This is always possible since a 4 k +2 = i 4 k +2 = - 1 for any k 0.) Then | a - a n | = | a + 1 | ≥ 1 > 1 2 . If Re a < 0, then for any N , choose n N such that a n = 1. (This is always possible since a 4 k = i 4 k = 1 for any k > 0.) Then | a - a n | = | a - 1 | ≥ 1 > 1 2 . 68
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CHAPTER 7. POWER SERIES 69 The following limit laws are the relatives of the identities stated in Lemma 2.4. Lemma 7.4. Let ( a n ) and ( b n ) be convergent sequences and c C . (a) lim n →∞ a n + c lim n →∞ b n = lim n →∞ ( a n + cb n ) . (b) lim n →∞ a n · lim n →∞ b n = lim n →∞ ( a n · b n ) . (c) lim n →∞ a n lim n →∞ b n = lim n →∞ ± a n b n ² . In the quotient law we have to make sure we do not divide by zero. Moreover, if f is continuous at a then lim n →∞ f ( a n ) = f ( a ) if lim n →∞ a n = a, where we require that a n be in the domain of f . The most important property of the real number system is that we can, in many cases, determine that a sequence converges without knowing the value of the limit . In this sense we can use the sequence to define a real number. Definition 7.5. A Cauchy sequence is a sequence ( a n ) such that lim n →∞ | a n +1 - a n | = 0 . We say a metric space X (which for us means Z , Q , R , or C ) is complete if, for any Cauchy sequence ( a n ) in X , there is some a X such that lim n →∞ a n = a . In other words, completeness means Cauchy sequences are guaranteed to converge. For example, the rational numbers are not complete: we can take a Cauchy sequence of rational numbers getting arbitrarily close to 2, which is not a rational number. However, each of Z , R , and C is complete.
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Chapter7 - Chapter 7 Power Series It is a pain to think...

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