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Algebra II Textbook26 chapters | 256 lessons

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Lesson Transcript

Instructor:
*Yuanxin (Amy) Yang Alcocer*

Amy has a master's degree in secondary education and has taught math at a public charter high school.

Watch this video lesson to learn how to calculate the sum of an infinite geometric series. Learn about the interesting thing that happens when your common ratio is less than one.

A **geometric series** is a sequence of numbers where each number is the previous multiplied by a constant, the common ratio. These geometric series go on forever, but most times we are only interested in finding the sum of the beginning part of the series.

For example, say you wanted to spread the word about this huge pool party that you are having at your dream house by the ocean. You send your party invite to five of your friends. Then you ask these five friends to each send the invite to five more people.

Now 25 new people will have an invitation. If these 25 people send the invite to five more people each, your invite will have reached 125 new people. You can see that you only need to add up the first few numbers to get to a really large number for your pool party.

Sometimes, though, you want to see what kind of numbers you get when you add up the infinite series. For example, say that you have a pie and you slice your pie in half. You take one of those slices and slice it in half. You take one of these slices and slice it in half. You keep repeating. Imagine doing this an infinite number of times.

Now you want to add up your pie slices to see how much pie you have. We call this 'when *n* goes to infinity' since *n* tells us the number of terms we are talking about. So, if *n* is infinity, then we are talking about all the terms in our infinite series. In our case, it is all the slices that we have.

It is the sum that we will be talking about in this video lesson. I will show you a formula you can use when your common ratio is within a certain range. So, let's get going!

If your common ratio is less than 1 or greater than -1, but not 0, then you can use this formula to calculate the sum for your infinite geometric series:

The *r* is our common ratio, and the *a* is the beginning number of our geometric series. To use this formula, our *r* has to be between -1 and 1, but it cannot be 0. Nor can it be -1 or 1.

So, if our *r* is 1/2, 1/4, 1/3, etc., or even -1/2, -1/4, -1/3, then we can use this formula. If our *r* is outside these limits, if it is greater than or equal to 1 or less than or equal to -1, then the sum of the infinite geometric series cannot be evaluated.

Let's try using this formula with our pie example. With our first cut, we set aside half of our pie. So, our beginning number, our *a*, is 1/2. Our *r* is 1/2 since we are slicing our pie in half every time.

Let's see what kind of answer we get. We plug in our 1/2 for *a* and our 1/2 for *r*. Now we evaluate. 1 minus 1/2 is 1/2. 1/2 divided by 1/2 is 1. So, our answer is 1.

Adding up all our slices, beginning with our half slice, gives us a whole pie. That makes sense since we are simply cutting our one pie down into very tiny slices.

Let's try one more example. See if you can calculate it yourself as we go. Our first term is 1/3 and our common ratio is 1/4. Since our common ratio is between -1 and 1 and is not 0, we can use our formula.

We plug in 1/3 for *a* and 1/4 for *r*. 1 minus 1/4 is 3/4. 1/3 divided by 3/4 is 4/9. So, this infinite geometric series with a beginning term of 1/3 and a common ratio of 1/4 will have an infinite sum of 4/9.

What have we learned? We learned that a **geometric series** is a sequence of numbers where each number is the previous multiplied by a constant, the common ratio.

To find the sum of the infinite geometric series, we can use the formula *a* / (1 - *r*) if our *r*, our common ratio, is between -1 and 1 and is not 0. Our *a* in this formula is our beginning term.

We use this formula by plugging in our beginning term, our *a*, and our common ratio, our *r*, and evaluating. If our *r* is outside this range, if it is greater than 1 or less than -1, then the sum of the infinite geometric series cannot be evaluated.

You'll have the ability to do the following after this lesson:

- Define geometric series and common ratio
- Identify the formula for finding the infinite geometric series
- Explain when you can use this formula and how to calculate it

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Algebra II Textbook26 chapters | 256 lessons

- Introduction to Sequences: Finite and Infinite 4:57
- How to Use Factorial Notation: Process and Examples 4:40
- How to Use Series and Summation Notation: Process and Examples 4:16
- Arithmetic Sequences: Definition & Finding the Common Difference 5:55
- How and Why to Use the General Term of an Arithmetic Sequence 5:01
- The Sum of the First n Terms of an Arithmetic Sequence 6:00
- Understanding Arithmetic Series in Algebra 6:17
- Working with Geometric Sequences 5:26
- How and Why to Use the General Term of a Geometric Sequence 5:14
- The Sum of the First n Terms of a Geometric Sequence 4:57
- Understand the Formula for Infinite Geometric Series 4:41
- Using Sigma Notation for the Sum of a Series 4:44
- Mathematical Induction: Uses & Proofs 7:48
- How to Find the Value of an Annuity 4:49
- How to Use the Binomial Theorem to Expand a Binomial 8:43
- Special Sequences and How They Are Generated 5:21
- Go to Algebra II: Sequences and Series

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