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AP Calculus AB & BC: Help and Review17 chapters | 160 lessons

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Lesson Transcript

Instructor:
*Laura Pennington*

Laura has taught collegiate mathematics and holds a master's degree in pure mathematics.

This lesson will explore sequences that are convergent. We will define convergent sequences, and look at how to determine if a given sequence is convergent using formulas and limits.

Suppose that you just won a jet ski worth $10,000 on a game show. Congratulations! After the show, you go home and look up the make and model of your new jet ski online to learn everything about it. You find a chart giving the forecasted value, due to depreciation, of the jet ski at the beginning of each year that passes.

Year | 1 | 2 | 3 | 4 | 5 | â€¦ |

Value | $10000 | $5000 | $3333.33 | $2500 | $2000 | . . . |

Consider just the list of the values:

- 10000, 5000, 3333.33, 2500, 2000, . . .

This is an example of a sequence in mathematics. A **sequence** is a list of numbers in a specific order and takes on the following form:

*a*1,*a*2,*a*3, . . .

Where *a**n* is the *n*th term of the sequence. For instance, in our values sequence, 2500 is the value of the jet ski at the beginning of the fourth year after winning the jet ski. Mathematically speaking, 2500 is the fourth term in the sequence, so we have the following:

*a*4 = 2500

Notice that the values in the sequence get lower and lower each year due to depreciation. If we were to continue the sequence, we would find that this pattern continues. In other words, *a**n* approaches zero as *n* approaches infinity (gets larger and larger).

This phenomenon also has mathematical significance. When dealing with sequences, we call a number that the terms of the sequence approach the **limit** of the sequence, and we use this notation:

These limits are a broad subject that would take a number of lessons to cover. They play an important part in identifying characteristics of a sequence, so in order to properly explore these characteristics, any limits needed in this lesson will be given.

When a sequence has a limit that exists, we say that the sequence is a **convergent sequence**. Not all sequences have a limit that exists. For instance, consider the sample sequence of the counting numbers:

- 1, 2, 3, 4, . . .

If we continue this sequence, the terms just get larger and larger, so *a**n* approaches infinity as *n* approaches infinity. Therefore, the terms do not approach a number, because infinity is not a number. Thus, the sequence doesn't have a limit and is not convergent.

Okay, so some sequences are convergent, and some aren't, but how do we determine which is the case for a given sequence? It all comes down to two steps:

- Find a formula for the
*n*th term, or*a**n*, of the sequence. - Find the limit of that formula as
*n*approaches infinity. If the limit exists, the sequence is convergent. If not, the sequence is not convergent.

The tricky part is that first step. Some formulas for sequences are obvious, but some are not. Consider our value sequence again. At first glance, you may not be able to recognize a formula, but take a look at each term written a little differently:

Looking at the sequence this way, we look for ways to write the *a**n* in terms of *n*. Do you see how we can do this?

Notice that the first term is equal to 10,000/1, the second term is equal to 10,000/2, and so on. Continuing this, we have that the *n*th term is equal to 10,000/*n*. Ah-ha! We have a formula for the *n*th term of the sequence:

*a**n*= 10,000/*n*

The second step is to find the limit of 10,000/*n* as *n* approaches infinity.

Sure enough, we see that the limit of the sequence is zero, as we suspected. Since the limit of the sequence exists, the sequence is convergent.

Let's consider some more examples!

Consider the following sequence:

- 1, 4, 9, 16, 25, . . .

We want to determine if the sequence is convergent or not, so we just follow our steps. The first of which is to find a formula for the *n*th term of the sequence. Do you see it? If not, let's look at each term like we did before.

We see that the *n*th term of the sequence can be represented as *n*2, so we have the following:

*a**n*=*n*2

Now, we just find the limit!

We get that the limit of the sequence is infinity, which is not a number, so the sequence is not convergent.

One more example! Same question, different sequence:

- 1/2, 2/3, 3/4, 4/5, 5/6, . . .

We want to determine if this sequence is convergent. First, we find our formula for *a**n* by observing the terms one by one and looking for patterns.

We get that the *n*th term is equal to *n* / (*n* + 1).

*a**n*=*n*/ (*n*+ 1)

Now, we find the limit of *n* / (*n* + 1) as *n* approaches infinity.

We get that the limit of the sequence is 1, which is a number, so the limit exists and the sequence is convergent. I think we're getting the hang of this!

A **sequence** is a list of numbers in a specific order:

*a*1,*a*2,*a*3, â€¦

where *a**n* is the *n*th term of the sequence.

We call a number that the terms of the sequence approach a **limit** of the sequence. Not all sequences have a limit that exists. When a sequence does have a limit that is a number and exists, we call it a **convergent sequence**. To determine if a given sequence is convergent, we use the following two steps:

- Find a formula for the
*n*th term, or*a**n*, of the sequence. - Find the limit of that formula as
*n*approaches infinity. If the limit exists, the sequence is convergent. If not, the sequence is not convergent.

Being able to determine if a sequence is convergent or not really helps us to analyze the sequence and what it represents in a real-life situation, so let's store this process in our math toolbox for future use!

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AP Calculus AB & BC: Help and Review17 chapters | 160 lessons

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