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Unformatted text preview: rem 7.17 of Walter Rudin’s Principles of Mathematical Analysis .) 3. With the aid of series prove that the function f ( z ) = ± e z1 z if z 6 = 0 1 if z = 0 is entire. 4. (a) Prove that the power series ∑ ∞ n =0 a n ( zz ) n and the corresponding series of derivatives ∑ ∞ n =1 na n ( zz ) n1 have the same radius of convergence. (Hint: Note that it suﬃces to consider where the series converge absolutely; thus you may use the ratio test for real series.) (b) Prove that the series ∑ ∞ n =1 z n n 2 converges at all points inside and on its circle of convergence. Prove that this is not true for its series of derivatives....
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This note was uploaded on 02/21/2012 for the course MATH 118 taught by Professor Forde during the Fall '09 term at University of Houston.
 Fall '09
 Forde
 Integrals, Limits

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