A10 - MATH185: Assignment 10 Throughout this homework let U...

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MATH185: Assignment 10 Throughout this homework let U be an open subset of C : 1. Recall that a sequence of functions f n : U C is uniformly Cauchy on U iff lim N →∞ ( sup {| f n ( w ) - f m ( w ) | : n, m N and w U } ) = 0 (a) Show that a sequence of functions f n is Uniformly Cauchy on U if and only if there is a function g : U C such that f n converges uniformly to g on U . (b) Use part ( a ) to show that, if the series n =1 | f n ( z ) | converges uniformly on U then the series n =1 f n ( z ) converges uniformly on U (Recall that a series converges unformly if it converges to and moreover the sequence of partial sums converges uniformly to the limit) 2. The following is a very useful method to show that a series of functions converges uniformly on U : Suppose that there exists a sequence of positive real numbers c n such that for n >> 0 and for all z U we have | f n ( z ) | ≤ c n . Show that if n =1 c n converges then n =1 f n
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A10 - MATH185: Assignment 10 Throughout this homework let U...

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