*Yuanxin (Amy) Yang Alcocer*Show bio

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

Lesson Transcript

Instructor:
*Yuanxin (Amy) Yang Alcocer*
Show bio

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

Watch this video lesson, and you'll learn how you can use the ratio test to help you determine whether a particular series converges or diverges. You will also learn what to watch out for when using the ratio test.

A very useful test that can help you work with series problems is the **ratio test**. The ratio test helps you to determine whether a particular series converges or diverges. You will need to know this ratio test as you will most likely see this on the standardized tests that you will take as you progress through your schooling and as you prepare for college. Knowing this ratio test will help you to easily answer the problems that ask you whether a certain series converges or not. The ratio test is this:

As you can see, to use the test for a particular series *a*` n`, you take the limit as

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The convergence and divergence rules for this ratio test are these:

1. If *L* < 1, then the series converges and is convergent.

2. If *L* > 1, then the series diverges and is divergent.

3. If *L* = 1, then the series may be divergent or may be convergent.

For example, if after you take the limit, you get 3 for your answer, then that tells you that your series is divergent because 3 is greater than 1. Now, if you get 10/19 for an answer, then your series is convergent.

Let's take a look at a couple of examples of this ratio test in action now.

First, let's look at this series.

The problem asks you to determine whether the series is divergent or convergent. Using the ratio test, you need to write the *n* + 1 term over the *n* term and take the limit as *n* approaches infinity. The *n* term is the series itself. To find the *n* + 1 term, all you need to do is to plug in an *n* + 1 wherever you see an *n* in the series itself:

Even though the ratio test says that you need to place the *n* + 1 term over the *n* term, to make it easier for yourself, you can also multiply the *n* + 1 term by 1 over the *n* term. Remember that when you divide 1 by a fraction, the fraction inverts where the numerator and denominator switch places.

Now comes the fun part. You now need to evaluate this expression and see what you can cancel. As you can see, if you are not careful, you can make a mistake that will give you a wrong answer. So, when you look at what you can cancel, do it very carefully. Canceling the terms, you get this:

Now, taking the limit of this as *n* approaches infinity, you get 8/16. Don't forget, you are taking the limit of an absolute value. So, if you get -8/16, the absolute value of this is 8/16. Since this is less than one, this series converges.

Let's look at one more series.

Determine whether this series converges or not.

You use the ratio test again here. You want to take the limit of the *n* + 1 term over the *n* term. Here again, the *n* term is the series itself. The *n* + 1 term is the series term where you plug in *n* + 1 wherever you see *n*.

Multiplying this with 1 over the *n* term or *n* over 3 to the *n*th power and then eliminating like terms, you get this:

Taking the limit of this as *n* approaches infinity, you get 3. 3 is larger than 1, so this series diverges.

Let's review what you've learned. The **ratio test** helps you to determine whether a particular series converges or diverges. The ratio test is this.

The convergence and divergence rules for this ratio test are these:

1. If *L* < 1, then the series converges and is convergent.

2. If *L* > 1, then the series diverges and is divergent.

3. If *L* = 1, then the series may be divergent or may be convergent.

To find the *n* + 1 term, all you need to do is to plug in *n* + 1 wherever you see *n*.

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