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# 7.2 Convergence

7.6   Definition (Convergent sequence.) Let be a complex sequence, and let . We will say converges to and write if for every disc there is a number such that

We say converges if there is some such that . We say diverges if and only if does not converge.

It appears from figure a 7.1 that for every disc centered at the terms of the sequence eventually get into ; i.e., it appears that . Similarly, it appears that .

From figure b, it appears that there are numbers such that , and . You should be able to put your finger on and , and maybe to guess what their exact values are. We will return to these examples later.

Let . The figure below represents the sequence . It appears from the figure that there is no number such that . The following theorem shows that this is the case. (Note that .)

7.7   Theorem. Let satisfy and . Then diverges.

Proof: Suppose that and . Then for all ,

 (7.8)

Now suppose, to get a contradiction, that there is a number such that . Then corresponding to the disc , there is a number such that

In particular,

so

By the triangle inequality,

Combining this result with (7.8), we get

so . This contradiction shows that diverges.

We can also show that constant sequences converge.

7.9   Theorem. Let . Then the constant sequence converges to .

Proof: Let . Let be a disc centered at . Then

Hence, .

For purposes of calculation it is sometimes useful to rephrase the definition of convergence. Since the disc is determined by its radius , and for all , , we can reformulate definition 7.6 as

7.10   Definition (Convergence.) Let be a sequence in , and let . Then if and only if for every there is some such that

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