13.4 Limits of Infinite Sequences. Objective To find or estimate the limit of an infinite sequence or to determine that the limit does not exist. Sequences of Numbers. Definition. 1. Examples. 2. 3. Limits of Sequences. Definition.

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13.4 Limits of Infinite Sequences

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Objective To find or estimate the limit of an infinite sequence or to determine that the limit does not exist

Sequences of Numbers

Definition

1

Examples

2

3

Limits of Sequences

Definition

If a sequence has a finite limit, then we say that the sequence is convergent, or that it converges, or it has limit. Otherwise it diverges, or is divergent, or has no limit.

Impartant Note

1. The limit is the intention that a sequence is trying to reach. The sequence itself may physically reach the limit value, or may not physically reach but intend to reach the limit value.

Limits of Sequences

Definition

If a sequence has a finite limit, then we say that the sequence is convergent, or that it converges, or it has limit. Otherwise it diverges, or is divergent, or has no limit.

Impartant Note

2. The wording “arbitrarily close” means we can find the large enough index number so that the distance between the value of an and the limit value L,or |an–L|, can be ALL (still infinite many)smaller than the pre-setting error value.

Limits of Sequences

Definition

If a sequence has a finite limit, then we say that the sequence is convergent, or that it converges, or it has limit. Otherwise it diverges, or is divergent, or has no limit.

Impartant Note

3. The wording “as the index n grows” means n increases without bound = “n goes to infinity”. The infinity is neither a specific large finite number nor a place.

Limits of Sequences

Definition

If a sequence has a finite limit, then we say that the sequence is convergent, or that it converges, or it has limit. Otherwise it diverges, or is divergent, or has no limit.

Impartant Note

4. The notation for “A finite number L is the limit of the sequence {an}” is

Example 1 Given the sequence:

This sequence can reach the value 1 since a1 = 1 and it also has the intention to reach the limit value 1 when n grows.

Example 2 Given the sequence:

and the error 10-100.

Let the distance between the value of an and the limit value L be |an–L|, we want to know starting what index and after this distance can be smaller than the pre-setting error.

Solving this inequality, we can easily get n > 10100. This means although the distance between the first term a1 = 1 to the intention value 1 is ZERO which satisfies the pre-setting error, the term 2 to term 10100 do not meet the pre-setting error. However, those an for these growing indices (or n > 10100) all do.

Example 3 Given the sequence:

and the error 10-100.

Let the distance between the value of an and the limit value L is smaller than the pre-setting error.

We have already had from index n > 10100. This means although the distance between the first term a1 = 1 and the 106th term to the intention value 1 is 99 which does satisfies the pre-setting error, however, those an for these growing indices (or n > 10100) all do.

Example 4 Given the sequence:

Find the limit of the sequence.

When the values of ans are increasing without bound, not approaching or reaching a finite number as index n increases, then we say “the limit of the sequence is infinity”. Then we write

Limits of Sequences

Definition

If a sequence has a finite limit, then we say that the sequence is convergent, or that it converges, or it has limit. Otherwise it diverges, or is divergent, or has no limit.

Examples

5

0.

Limits of Sequences

6

The sequence (1,-2,3,-4,…) diverges.

7

Computing Limits of Sequences

The phrase “inserting n = ∞” is not an accurate wording. Since we have not systematically learned the concept of “Limit”, this phrase may give us a short cut to calculate a lot of limit questions.

1

0

n2

Examples

Examples

8

8

9

This example shows that if we meet a sequence of type , we may consider to divide the variable n or n2, so the

limit can be found easily.

Finding Patterns & Limits

Examples

10

Describe a pattern for each sequence. Write a formula for the n-th term and find the limit.

y = L is a horizontal asymptote when sequence converges to L.

Write the first 5 terms for the sequence:

Examples

11

Graph on a number line

As a function

The terms in this sequence get closer and closer to 1. The limit of this sequence is 1. Or, the sequence CONVERGES to 1.

Write the first 5 terms for the sequence:

Examples

12

Graph on a number line

As a function

The terms in this sequence swing to -1 and 1 and do not get close to any single value. So the sequence diverges. The limit does not exist!

A sequence that diverges (see P. 495)

Examples

12

The terms in this sequence condense to -1 and 1 and do not get close to any single value. So the sequence diverges. The limit does not exist!

Write the terms for an= 3 and find the limit

Examples

13

The terms are 3, 3, 3, …, 3, …

Graph on a number line

As a function

The sequence converges to 3. This sequence is called constant sequence which is always converges. It physically reach its intention infinite many times.