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7.3 Null Sequences

Sequences that converge to are simpler to work with than general sequences, and many of the convergence theorems for general sequences can be easily deduced from the properties of sequences that converge to . In this section we will just consider sequences that converge to .

7.11   Definition (Null sequence.) Let be a sequence in . We will say is a null sequence if and only if for every there is some such that for every , .

By comparing this definition with definition 7.10, you see that

Definition 7.11 is important. You should memorize it.

7.12   Definition (Dull sequence.) Let be a sequence in . We say is a dull sequence if and only if there is some such that for every in , and for every .

The definitions of null sequence and dull sequence use the same words, but they are not in the same order, and the definitions are not equivalent.

If satisfies condition (7.12), then whenever ,

If , this condition would say , which is false. Hence if , then ; i.e., if , then . Hence a dull sequence has the property that there is some such that for all . Thus every dull sequence is a null sequence. The sequence

is a dull sequence, but

is not a dull sequence. In the next theorem we show that is a null sequence, so null sequences are not necessarily dull.

7.13   Theorem. . For all , is a null sequence

Proof: Let . By the Archimedean property for , there is an such that . Then for all ,

so for all .

The difference between a null sequence and a dull sequence is that the  " in the definition of null sequence can (and usually does) depend on , while the " in the definition of dull sequence depends only on . To emphasize that depends on (and also on ), I will often write or instead of .

Here is another reformulation of the definition of null sequence.

7.14   Definition (Precision function.) Let be a complex sequence. Then is a null sequence if and only if there is a function such that

I will call such a function a precision function for .

This formulation shows that in order to show that a sequence is a null sequence, you need to find a function such that

In the proof of theorem 7.13, for the sequence we had

This description for could be made more precise, but it is good enough for our purposes.

7.15   Theorem. If , then the constant sequence is not a null sequence.

Proof: If , then . Suppose, to get a contradiction, that is a null sequence. Then there is a number such that for all . Then for all ,

 (7.16)

If then (7.16) is false and this shows that is not a null sequence.

7.17   Theorem (Comparison theorem for null sequences.) Let be
complex sequences. Suppose that is a null sequence and that

Then is a null sequence.

Proof: Since is a null sequence, there is a function such that for all ,

Then

Hence, we can let .

7.18   Example. We know that for all , so for all . Since is a null sequence, it follows from the comparison theorem that is a null sequence. Also, since for all , we see that is a null sequence.

7.19   Theorem (Root theorem for null sequences.) Let be a null sequence, and let . Then is a null sequence where for all .

Scratchwork: Let . I want to find so that for all and all ,

i.e.

i.e.

This suggests that I should take .

Proof: Let be a null sequence in and let be a precision function for . Define by for all . Then for all ,

Hence is a precision function for .

7.20   Examples. Let . Then is a null sequence in , so it follows that is a null sequence.

Consider the sequence . For all ,

Hence , so it follows from the comparison theorem that is a null sequence.

Since is a null sequence, it follows from the root theorem that is a null sequence. Now , so so

for all , and by another comparison test, is a null sequence. Since , it follows that is a null sequence for all with .

You probably suspect that is a null sequence for all with . This is correct, but we will not prove it yet.

7.21   Exercise. A Show that the geometric sequences

and
that are sketched above 7.1 are in fact null sequences.

7.22   Exercise. A Which, if any, of the sequences below are null sequences? Justify your answers.
a)
b)
c)

7.23   Entertainment. Show that

is a null sequence. (If you succeed, you will probably find a proof that is a null sequence whenever .) NOTE: If you use calculator operations, then is not a null sequence on most calculators.

It follows from remark 5.38 that we can add, subtract and multiply complex sequences, and that the usual associative, commutative, and distributive laws hold. If and then and . If then the constant sequences satisfy

7.24   Exercise. A Which of the field axioms are satisfied by addition and multiplication of sequences? Does the set of complex sequences form a field? (You know that the associative, distributive and commutative laws hold, so you just need to consider the remaining axioms.

7.25   Notation. If is a complex sequence, we define sequences , , , and by

7.26   Theorem. Let be a complex null sequence. Then , , and are all null sequences.

Proof: All four results follow by the comparison theorem. We have, for all :

Next: 7.4 Sums and Products Up: 7. Complex Sequences Previous: 7.2 Convergence   Index