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AP Calculus BC: Exam Prep24 chapters | 169 lessons | 9 flashcard sets

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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.

Comparing a series that you are given to a series that you know the answer to can help you to answer questions quickly and easily. Learn how to do so for convergence and divergence testing.

In this lesson, we'll see how we can compare series to help us determine whether a series is convergent or divergent. We define a **series** as the sum of a sequence of terms. We usually use the summation notation to show a series.

When we are working with series, if we know whether another series is convergent or divergent, then by comparing that series to the one we have to find if it meets certain criteria, we can easily and quickly determine whether our series is convergent or divergent.

In this lesson on testing for convergence and divergence, the types of series we'll be working with are the infinite ones. These are series that go on forever, such as the example we just saw that goes from *n* = 1 to infinity. When an infinite series is **convergent**, it means that the series reaches a finite value. When the infinite series is **divergent**, it means that the series does not reach a finite value. Divergent series usually diverge to either negative infinity or positive infinity. This means that as our *n* increases in value, the series grows towards either positive infinity or negative infinity.

To test for convergence when comparing series, we use this rule:

- For two series,
*a sub n*and*b sub n*, where all terms are greater or equal to 0 and all the terms of*a sub n*are less than or equal to all the corresponding terms of*b sub n*, then the series*a sub n*is convergent if the series*b sub n*is convergent.

What this is saying is that if we have two series where all the terms are positive and where the terms of one series is always smaller than the other, then the smaller series will converge if the larger is convergent. Also note that both series must begin and end at the same values. If one series goes from *n* = 1 to infinity, then the other series must also.

Let's see how we can use this rule.

Given this series, determine whether it converges:

This series will converge if we can find another series that converges whose terms are all larger than our terms. Well, thinking about all the series that we know of, we think of this series:

This series has terms that are larger than the corresponding terms of our series. We know this particular series converges. So, since this series is larger than our series, our series also converges because this larger series converges.

Now, the test of divergence is very similar to that of convergence:

- For two series,
*a sub n*and*b sub n*, where all terms are greater or equal to 0 and all the terms of*a sub n*are less than or equal to all the corresponding terms of*b sub n*, then if the series*a sub n*is divergent, then series*b sub n*will also be divergent.

What this is saying is that if the smaller series diverges, then the larger series will also diverge.

Let's take a look at this test in action.

Determine whether this series diverges:

This series will diverge if we can find another series that is smaller and diverges as well. Looking at this series and looking back on the other series we've studied, we remember that the series 1 / *n* has terms that are all smaller than our series. Does this series diverge? If so, then our series diverges as well.

From what we know of this series, we know that it diverges. This is actually a harmonic which diverges as a series. So, because this smaller series diverges, our series also diverges.

Let's look at one more example:

Does this series converge or diverge?

Looking at this series, we see that the exponent in the numerator is higher than the exponent in the denominator. So, this could mean that this series diverges. To know for sure, we need to find another smaller series that is divergent. Then if this smaller series diverges, our series will also diverge. One series that is smaller than our series is the series *n*^3 / *n*^2. By not having the -10, the new series is smaller.

This series simplifies to just *n*, which diverges. Since this series is smaller and diverges, then our larger series will also diverge.

Let's review what we've learned. A **series** is the sum of a sequence of terms. An infinite series is one that goes up to infinity. When an infinite series is **convergent**, it means that the series reaches a finite value. When the infinite series is **divergent**, it means that the series does not reach a finite value.

We can use the comparison test to test whether a particular series is convergent or divergent:

- For two series,
*a sub n*and*b sub n*, where all terms are greater or equal to 0 and all the terms of*a sub n*are less than or equal to all the corresponding terms of*b sub n*, then the series*a sub n*is convergent if the series*b sub n*is convergent. - For two series,
*a sub n*and*b sub n*, where all terms are greater or equal to 0 and all the terms of*a sub n*are less than or equal to all the corresponding terms of*b sub n*, then if the series*a sub n*is divergent, then series*b sub n*will also be divergent.

Summarizing the above, we can say: if a larger series converges, then a smaller series will also converge; and if a smaller series diverges, then a larger series will also diverge.

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AP Calculus BC: Exam Prep24 chapters | 169 lessons | 9 flashcard sets

- What is Expanded Form in Math? - Definition & Examples 4:09
- Convergence & Divergence of a Series: Definition & Examples
- Harmonic Series in Math: Definition & Formula 4:19
- How to Apply the Ratio Test for Convergence & Divergence 4:24
- Testing for Convergence & Divergence by Comparing Series 5:38
- Go to Series of Constants

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