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

This preview shows pages 1–3. Sign up to view the full content.

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

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
| 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
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

### What students are saying

• As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

Kiran Temple University Fox School of Business ‘17, Course Hero Intern

• I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

Dana University of Pennsylvania ‘17, Course Hero Intern

• The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

Jill Tulane University ‘16, Course Hero Intern