ch8 - Chapter Eight Series 8.1 Sequences The basic...

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Chapter Eight Series 8.1. Sequences. The basic definitions for complex sequences and series are essentially the same as for the real case. A sequence of complex numbers is a function g : Z C from the positive integers into the complex numbers. It is traditional to use subscripts to indicate the values of the function. Thus we write g n z n and an explicit name for the sequence is seldom used; we write simply z n to stand for the sequence g which is such that g n z n . For example, i n is the sequence g for which g n i n . The number L is a limit of the sequence z n if given an   0, there is an integer N such that | z n L |   for all n N . If L is a limit of z n , we sometimes say that z n converges to L . We frequently write lim z n L . It is relatively easy to see that if the complex sequence z n u n iv n converges to L , then the two real sequences u n and v n each have a limit: u n converges to Re L and v n converges to Im L . Conversely, if the two real sequences u n and v n each have a limit, then so also does the complex sequence u n iv n . All the usual nice properties of limits of sequences are thus true: lim z n w n lim z n lim w n ; lim z n w n lim z n lim w n ; and lim z n w n lim z n lim w n . provided that lim z n and lim w n exist. (And in the last equation, we must, of course, insist that lim w n 0.) A necessary and sufficient condition for the convergence of a sequence a n is the celebrated Cauchy criterion : given   0, there is an integer N so that | a n a m |   whenever n , m N . A sequence f n of functions on a domain D is the obvious thing: a function from the positive integers into the set of complex functions on D . Thus, for each z D , we have an ordinary sequence f n z  . If each of the sequences f n z  converges, then we say the sequence of functions f n converges to the function f defined by f z lim f n z  . This pretty obvious stuff. The sequence f n is said to converge to f uniformly on a set S if given an   0, there is an integer N so that | f n z f z |   for all n N and all z S . Note that it is possible for a sequence of continuous functions to have a limit function that is not continuous. This cannot happen if the convergence is uniform. To see this, suppose the sequence f n of continuous functions converges uniformly to f on a domain D , let z 0 D ,and let   0. We need to show there is a so that | f z 0 f z |   whenever 8.1
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| z 0 z | . Let’s do it. First, choose N so that | f N z f z | 3 . We can do this because of the uniform convergence of the sequence f
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