Unformatted text preview: S is contained in S . (Note: Proving that the statements are equivalent entails doing the following: assume (i) and prove that (ii) holds; then assume (ii) and prove that (i) holds.) 2. ( 10 marks ) Let ( z n : n ∈ N ) be a sequence of complex numbers. Prove the following state-ments: (i) If lim n →∞ | z n | 1 /n = L , and 0 ≤ L < 1, then the series ∞ ∑ n =1 z n is absolutely convergent . (ii) If lim n →∞ | z n | 1 /n = L , and L > 1, or if lim n →∞ | z n | 1 /n = + ∞ , then lim n →∞ | z n | = ∞ ; in particular the series ∞ ∑ n =1 z n diverges. (Model your argument after the proof of Theorem 1.3.10 in the book.)...
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- Fall '08
- Addition, Complex number, Closed set, lim |zn |1/n