Next: 8.2 Continuity Up: 8. Continuity Previous: 8. Continuity   Index

# 8.1 Compositions with Sequences

8.1   Definition (Composition) Let be a complex sequence. Let be a function such that , and for all . Then the composition is the sequence such that for all . If is a sequence, I will often write instead of . Then

8.2   Examples. If

and for all , then

Figure a) below shows representations of and where

 (8.3)

I leave it to you to check that , and , and . Figure b) shows representations for and where

and is defined as in (8.3). Here it is easy to check that . From the figure, doesn't appear to converge.

8.4   Exercise. A Let be a non-zero complex number with . Let

Under what conditions on does converge? What does it converge to? (Your answer should show that the sequence from the previous example does not converge.)

8.5   Definition (Complex function.) By a complex function I will mean a function whose domain is a subset of , and whose codomain is . I will consider functions from to to be complex functions by identifying a function with a function in the expected manner.

Next: 8.2 Continuity Up: 8. Continuity Previous: 8. Continuity   Index