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Testing for Convergence & Divergence by Comparing Series

Testing for Convergence & Divergence by Comparing Series
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  • 0:01 Comparing Series
  • 1:11 Testing for Convergence
  • 2:29 Testing for Divergence
  • 3:37 Example
  • 4:20 Lesson Summary
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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.

Comparing Series

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.

Testing for Convergence

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.

series

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:

series

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:

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.

Testing for Divergence

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.

series

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:

series

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