Copyright

Convergent Sequence: Definition, Formula & Examples

An error occurred trying to load this video.

Try refreshing the page, or contact customer support.

Coming up next: Using a Graph to Define Limits

You're on a roll. Keep up the good work!

Take Quiz Watch Next Lesson
 Replay
Your next lesson will play in 10 seconds
  • 0:04 Sequences and Limits
  • 1:51 Convergent Sequences &…
  • 3:38 Examples
  • 4:49 Lesson Summary
Add to Add to Add to

Want to watch this again later?

Log in or sign up to add this lesson to a Custom Course.

Login or Sign up

Timeline
Autoplay
Autoplay
Speed

Recommended Lessons and Courses for You

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.

Sequences 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
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:

  • a1, a2, a3, . . .

Where an is the nth 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:

  • a4 = 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, an 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:


convseq2


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.

Convergent Sequences & Formulas

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 an 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:

  1. Find a formula for the nth term, or an, of the sequence.
  2. 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:


convseq3


Looking at the sequence this way, we look for ways to write the an 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 nth term is equal to 10,000/n. Ah-ha! We have a formula for the nth term of the sequence:

  • an = 10,000/n

To unlock this lesson you must be a Study.com Member.
Create your account

Register to view this lesson

Are you a student or a teacher?

Unlock Your Education

See for yourself why 30 million people use Study.com

Become a Study.com member and start learning now.
Become a Member  Back
What teachers are saying about Study.com
Try it risk-free for 30 days

Earning College Credit

Did you know… We have over 160 college courses that prepare you to earn credit by exam that is accepted by over 1,500 colleges and universities. You can test out of the first two years of college and save thousands off your degree. Anyone can earn credit-by-exam regardless of age or education level.

To learn more, visit our Earning Credit Page

Transferring credit to the school of your choice

Not sure what college you want to attend yet? Study.com has thousands of articles about every imaginable degree, area of study and career path that can help you find the school that's right for you.

Create an account to start this course today
Try it risk-free for 30 days!
Create An Account
Support