exsheet3b - n z n = if and only if lim n 1 z n = 0 . 5....

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Sivakumar M407 Example Sheet 3b 1. Suppose that z is a complex number and that is a fixed positive integer. Show that the series X n =0 n z n is absolutely convergent for | z | < 1 and divergent otherwise. 2. Show that the series X n =0 z n n ! is absolutely convergent for every complex number z . 3. Recall the following geometric series (from lecture): X n =0 z n = 1 1 - z , | z | < 1 . (3 . 1) Suppose that 0 < r < 1 and 0 θ < 2 π . Use equation (3.1) to show that X n =0 r n cos( ) = 1 - r cos θ 1 - 2 r cos θ + r 2 and X n =0 r n sin( ) = r sin θ 1 - 2 r cos θ + r 2 . 4. Suppose that { z n } is a sequence of nonzero complex numbers. Prove that lim
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Unformatted text preview: n z n = if and only if lim n 1 z n = 0 . 5. Show that a complex sequence is unbounded if and only if it has a subsequence which diverges to innity. 6. Suppose that R is a xed positive number. Describe the sets on the Riemann sphere which correspond to the following subsets of C : (i) C (0; R ) (ii) C \ D (0; R ) = { z C : | z | &gt; R } 1...
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