exsheet3a

exsheet3a - Sivakumar Example Sheet 3a 1. Suppose that θ...

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Unformatted text preview: Sivakumar Example Sheet 3a 1. Suppose that θ is a ﬁxed real number. Evaluate the following limits: cos(nθ) + i sin(nθ) n θ (ii) lim n 1 − cos − i sin n→∞ n (i) lim n→∞ M407 θ n 2. Show that the limit of a convergent sequence of complex numbers is unique. 3. Suppose that S is a nonempty (proper) subset of the complex plane. Prove that the following statements are equivalent: (i) S is closed. (ii) If {zn } is any sequence in S and lim zn = z , then z ∈ S . n→∞ Deﬁnition. A sequence {zn } of complex numbers is said to be Cauchy if for every exists a positive integer N such that |zn − zm | < for every m, n ≥ N . > 0 there Theorem. Every Cauchy sequence of real numbers converges (to a real number). 4. Use the preceding theorem to show that every Cauchy sequence of complex numbers is also convergent (to some complex number). 5. Suppose that {zn } is a complex sequence, and let w be a ﬁxed complex number. Show that the following are equivalent: (a) {zn } does not converge to w. (b) There exists a positive number |znk − w| ≥ 0 0 and a subsequence {znk : k ∈ N} of {zn } such that for every k . 6. Use Question 5 to show that the sequence {in : n ∈ N} does not converge to any complex number. 1 ...
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This note was uploaded on 03/08/2010 for the course MATH 407 taught by Professor Staff during the Fall '08 term at Texas A&M.

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