232sheet6 - ³ X k b k ³ ³ is bounded for all n ∈ N 5 Investigate the convergence of ∞ X k =1 z k k 6 Find the domain of convergence of the

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B U Department of Mathematics Math 232 Introduction To Complex Analysis Spring 2008 Exercise Sheet 6 1 One last word on M¨obius transformations 1. Let C 1 , C 2 be two circles in C and f be a M¨ obius transformation given by f ( z ) = z +2 z - 5 i . Describe the image under f of C 1 C 2 when a) C 1 C 2 = , 5 i / C 1 C 2 b) C 1 C 2 = , 5 i C 1 c) C 1 C 2 = {- 2 } , 5 i / C 1 C 2 d) C 1 C 2 = {- 2 } , 5 i C 2 e) C 1 C 2 = {- 2 , 0 } , 5 i / C 1 C 2 f) C 1 C 2 = {- 2 , 0 } , 5 i C 1 g) C 1 C 2 = { 5 i } h) C 1 C 2 = { 5 i, - 2 } . 2. Find the limit of the sequence z n = 1 n + 2 n 2 n + 1 . 3. Investigate the convergence of the following sequences. a) z n = i n n b) z n = (1 + i ) - n c) z n = n 2 + in n 2 + i d) z n = i n e) z n = (1 + i ) n f) z n = ( - 1) n n 2 - in n 2 + 1 g) z n = i 2 + ± 3 - 4 i 6 ² n 4. Prove that the series X k a k b k converges if X k | a k - a k +1 | converges, lim n →∞ a n = 0 and ³
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Unformatted text preview: ³ X k b k ³ ³ is bounded for all n ∈ N . 5. Investigate the convergence of ∞ X k =1 z k k ! . 6. Find the domain of convergence of the series ∞ X k =1 z k k . 7. Expand the function f ( z ) = 1+2 z z 2 + z 3 into a series involving powers of z . 8. Prove that ∞ X k =1 ´ √ n + 1-√ n µ diverges. Observe that this is a divergent series with general term tending to . You can show it. Notes 1 Please visit: http://www.math.boun.edu.tr/deptcourses/math232...
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This note was uploaded on 11/08/2011 for the course MATH 232 taught by Professor Gurel during the Spring '11 term at Boğaziçi University.

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