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# 6.3 Convergence of Sequences

6.19   Definition ( converges to .)

Let be a sequence of real numbers, and let be a real number. We say that converges to if for every positive number there is a number in such that all of the numbers for which approximate with an error smaller than . We denote the fact that converges to by the notation

Thus " means:

For every there is a number in such that

Since

it follows immediately from the definition of convergence that

We will make frequent use of these equivalences.

6.20   Example. If then

Proof: Let be a generic element of . I must find a number such that

 (6.21)

whenever . Well, for all in

 (6.22)

Now for every in we have

and by the Archimedean property of there is some integer such that . For all we have

so by (6.22)

Hence by the definition of convergence we have
 (6.23)

A very similar argument can be used to show that
 (6.24)

6.25   Example. In the eighteenth century the rather complicated argument just given would have been stated as
If is infinitely large, then .
The first calculus text book (written by Guillaume François de l'Hôpital and published in 1696) sets forth the postulate
Grant that two quantities, whose difference is an infinitely small quantity, may be taken (or used) indifferently for each other: or (which is the same thing) that a quantity which is increased or decreased only by an infinitely small quantity, may be considered as remaining the same[35, page 314].
If is infinite, then is infinitely small, so , and similarly . Hence

There were numerous objections to this sort of reasoning. Even though , we do not have , since

It took many mathematicians working over hundreds of years to come up with our definition of convergence.

6.26   Theorem (Uniqueness theorem for convergence.) Let be a sequence of real numbers, and let be real numbers. Suppose

Then .

Proof: Suppose and . By the triangle inequality

 (6.27)

Let be a generic element of . Then is also in . Since , there is a number in such that
 (6.28)

Since there is a number in such that
 (6.29)

Let be the larger of and . If is a positive integer and then by (6.27), (6.28), and (6.29), we have

Since this holds for all in , we have .

6.30   Definition (Limit of a sequence.) Let be a sequence of real numbers. If there is a number such that , we write . The uniqueness theorem for convergence shows that this definition makes sense. If , we say is the limit of the sequence .

6.31   Definition (Convergent and divergent sequence.) Let be a sequence of real numbers. If there is a number such that , we say that is a convergent sequence. If there is no such number , we say that is a divergent sequence.

6.32   Example. It follows from example 6.20 that

for all in . Hence is a convergent sequence for each in .

The sequence is a divergent sequence. To see this, suppose there were a number such that .

Then we can find a number such that

In particular

(since is an integer greater than ). Hence, by the triangle inequality

i.e., which is false.

Since the assumption has led to a contradiction, it is false that .

6.33   Exercise. A Let be a sequence of real numbers, and let be a real number. Suppose that as gets larger and larger, gets nearer and nearer to , i.e., suppose that for all and in

Does it follow that converges to ?

6.34   Exercise. For each of the sequences below, calculate the first few terms, and make a guess as to whether or not the sequence converges. In some cases you will need to use a calculator. Try to explain the basis for your guess. (If you can prove your guess is correct, do so, but in several cases the proofs involve more mathematical knowledge than you now have.)

This problem was solved by Leonard Euler (1707-1783)[18, pp138-139].

This problem was solved by Jacob Bernoulli (1654-1705)[8, pp94-97].
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Next: 6.4 Properties of Limits. Up: 6. Limits of Sequences Previous: 6.2 Approximation   Index
Ray Mayer 2007-09-07