Absolute and Conditional Convergence (O

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ABSOLUTE AND CONDITIONAL CONVERGENCE Elizabeth Wood In the previous set of notes, we investigated the alternating series. We learned a test that we could use to determine if this type of series converges or diverges. Now the question is how can we determine if both the positive term series and the related alternating series converge or diverge or if only one of them converges. To talk about this we must define two terms. FACT: ABSOLUTE CONVERGENCE

A series  a n converges absolutely if the series of the absolute values,  |a n | converges. This means that if the positive term series converges, then both the positive term series and the alternating series will converge. FACT: A series that converges, but does not converge absolutely, converges conditionally. This means that the positive term series diverges, but the alternating series converges. EXAMPLE 1: Does the following series converge absolutely, converge conditionally, or diverge?

SOLUTION: Let us look at the positive term series for this given series.

This is a geometric series with ratio, r = 4/5, which is less than 1. Therefore this series converges, and the given series converges absolutely.

FACT:

This fact is also called the absolute convergence test.

EXAMPLE 2: Does the following series converge absolutely, converge conditionally, or diverge?

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SOLUTION: Let us look at the positive term series for the given series.

What do we know about this series? Well, this is the harmonic series and it diverges, so the given series will not converge absolutely. Now we must determine if the given series will converge conditionally or diverge. To do this, we will have to look at the alternating series. To do this, we must use the alternating series test. If you need to review this test, refer back to supplemental notes 24.

u n > 0 for all n  1, so the first condition of this test is satisfied. Now I must determine if the second condition is satisfied. This is easy to see. As n gets larger, the fraction 1/ n gets smaller. So u n  u n + 1 and the second condition is true. Now let us determine if the third condition is satisfied.

The third condition holds, so the alternating series converges, and the given series converges conditionally.

So here are the steps you will need to follow when determining absolute convergence, conditional convergence or divergence of a series. Look at the positive term series first. If the positive term

A. If it converges, then the given series converges absolutely.

B. If the positive term series diverges, use the alternating series test to determine if the alternating series converges. If this series converges, then the given series converges conditionally. If the alternating series diverges, then the given series diverges.

EXAMPLE 3: Does the following series converge absolutely, converges conditionally, or diverges?

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SOLUTION: Here is the positive term series.

I am going to use the ratio test to determine the convergence of this series. If you need to review this test, please refer to the supplemental notes 23.

So the positive term series diverges by the ratio test, and the given series does not converge absolutely. Therefore, we will have to look at the alternating series to determine if it converges or not.

u n is positive for n  1, so the first condition is satisfied. Now to determine if the second condition is holds. To help me determine this, I will plot the first 5 terms of this sequence.

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© Elizabeth Wood (http://faculty.eicc.edu/bwood/) Saylor.org Used by Permission. 3 of 6 Therefore, the third condition is not satisfied because the terms of this sequence are increasing. In fact 4 n grows faster than n 2. So the alternating series diverges, and the given series also diverges.

EXAMPLE 4: Does the following series converge absolutely, converge conditionally, or diverge?

SOLUTION: Since the cos n  is the alternating term, the positive term series is the harmonic series. Remember that the harmonic series diverges, so the given series does not converge absolutely. Now to determine the convergence of the alternating series.

The third condition holds, so the alternating series converges, and the given series converges conditionally.

EXAMPLE 5: Does the following series converge absolutely, converge conditionally, or diverge?

SOLUTION: Here is the positive term series.

Since nth term is raised to the nth power, I will use the nth-root test to determine convergence or divergence of this series.

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The positive term series converges by the nth-root test. Therefore, the given series converges absolutely.

EXAMPLE 6: Does the following series converge absolutely, converge conditionally, or diverge?

SOLUTION: Here is the positive term series.

I will use the integral test to determine the convergence or divergence of this series.

The positive series diverges by the integral test. Therefore, the original series does not converge absolutely. Now we have to determine if this series will converge conditionally.

u n > 0 for all n  1, so the first condition of this test is satisfied. Now I must determine if the second condition is satisfied. This is not easy to determine by listing terms, so I am going to plot the first 20 terms of this sequence.

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Notice that the second condition holds, so all I have to do is to determine if the third condition holds.

The third condition holds, therefore the alternating series converges, and the given series converges conditionally.

These two concepts are going to be very important in the next topics that we will be discussing in the next few sets of supplemental notes. Where we are heading is the idea of using a series to approximate a function that we cannot integrate directly. So we will need to know where the series approximation converges absolutely, converges conditionally, or diverges.

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