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# hmwk4 - Math 220A Complex Analysis Solutions to Homework#4...

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Math 220A Complex Analysis Solutions to Homework #4 Prof: Lei Ni TA: Kevin McGown Conway, Page 33, Problem 7. Show that the radius of convergence of the power series n =1 ( - 1) n n z n ( n +1) is 1, and discuss convergence for z = 1, - 1,and i . Proof. The sequence b n = 1 n 1 n ( n +1) is the subsequence of | a n | 1 /n which consists of exactly the nonzero terms. It is easy to see that lim n →∞ b n = 1. (For example, take logarithms and use results from calculus.) Since the lim sup of a sequence is its largest subsequential limit, we have lim sup n →∞ | a n | 1 /n = 1 and hence R = 1 - 1 = 1. At z = 1 and z = - 1, the series reduces to n =1 ( - 1) n n , which converges by the alternating series test. Of course the convergence here is conditional since 1 /n diverges. At z = i , we obtain the sum 1 - 1 2 - 1 3 + 1 4 + 1 5 - 1 6 - 1 7 + . . . . We consider two groupings of the above sum, both of which converge by the alternating series test: 1 - 1 2 + 1 3 + 1 4 + 1 5 - 1 6 + 1 7 + . . . , 1 - 1 2 - 1 3 - 1 4 + 1 5 - 1 6 - . . .

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hmwk4 - Math 220A Complex Analysis Solutions to Homework#4...

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