Calculus II For Dummies, 2nd Edition (2012)

Part IV. Infinite Series

Chapter 12. Where Is This Going? Testing for Convergence and Divergence

In This Chapter

Understanding convergence and divergence

Using the nth-term test to prove that a series diverges

Applying the versatile integral test, ratio test, and root test

Testing for convergence and divergence is The Main Event in your Calculus II study of series. Recall from Chapter 11 that when a series converges, it can be evaluated as a real number. However, when a series diverges, it can’t be evaluated as a real number, because it either explodes to positive or negative infinity or fails to settle in on a single value.

In Chapter 11, I give you two tests for determining whether specific types of series (geometric series and p-series) are convergent or divergent. In this chapter, I give you seven more tests that apply to a much wider range of series.

The first of these is the nth-term test, which is sort of a no-brainer. With this test under your belt, I move on to two comparison tests: the direct comparison test and the limit comparison test. These tests are what I call one-way tests; they provide an answer only if the series passes the test but not if the series fails it. Next, I introduce three two-way tests, which provide one answer if the series passes the test and the opposite answer if the series fails it. These tests are the integral test, the ratio test, and the root test.

Finally, I introduce you to alternating series, in which terms are alternately positive and negative. I contrast alternating series with positive series, which are the series you’re already familiar with, and I show you how to turn a positive series into an alternating series and vice versa. Then I show you how to prove whether an alternating series is convergent or divergent by using the alternating series test. To finish up, I introduce you to the important concepts of absolute convergence and conditional convergence. Whew. You better get started!

Starting at the Beginning

When testing for convergence or divergence, don’t get too hung up on where the series starts. For example:

This is just a harmonic series with the first 1,000 terms lopped off:

These fractions may look tiny, but the harmonic series diverges (see Chapter 11), and removing a finite number of terms from the beginning of this series doesn’t change this fact.

The lesson here is that, when you’re testing for convergence or divergence, what’s going on at the beginning of the series is irrelevant. Feel free to lop off the first few billion or so terms of a series if it helps you to prove that the series is convergent or divergent.

Similarly, in most cases you can add on a few terms to a series without changing whether it converges or diverges. For example:

You can start this series anywhere from n = 2 to n = 999 without changing the fact that it diverges (because it’s a harmonic series). Just be careful, because

if you try to start the series from n = 1, you’re adding the term , which is a big no-no. However, in most cases you can extend an infinite series without causing problems or changing the convergence or divergence of the series.

Although eliminating terms from the beginning of a series doesn’t affect whether the series is convergent or divergent, it does affect the sum of a convergent series. For example:

Lopping off the first few terms of this series — say, 1, , and — doesn’t change the fact that it’s convergent. But it does change the value that the series converges to. For example:

Using the nth-Term Test for Divergence

The nth-term test for divergence is the first test that you need to know. It’s easy, and it enables you to identify lots of series as divergent.

If the limit of sequence {a_n} doesn’t equal 0, then the series Σ a_n is divergent.

To show you why this test works, I define a sequence that meets the necessary condition — that is, a sequence that doesn’t approach 0:

Notice that the limit of the sequence is 1 rather than 0. So here’s the related series:

Because this series is the sum of an infinite number of terms that are very close to 1, it naturally produces an infinite sum, so it’s divergent.

The fact that the limit of a sequence {a_n} equals 0 doesn’t necessarily imply that the series Σ a_n is convergent.

For example, the harmonic sequence approaches 0, but (as I demonstrate in Chapter 11) the harmonic series is divergent.

When testing for convergence or divergence, always perform the nth-term test first. It’s a simple test, and plenty of teachers test for it on exams because it’s easy to grade but still catches the unwary student. Remember: If the defining sequence of a series doesn’t approach 0, the series diverges; otherwise, you need to move on to other tests.

Let Me Count the Ways

Tests for convergence or divergence tend to fall into two categories: one-way tests and two-way tests. I explain both in the following sections.

One-way tests

A one-way test allows you to draw a conclusion only when a series passes the test, not when it fails. Typically, passing the test means that a given condition has been met.

The nth-term test is a perfect example of a one-way test: If a series passes the test — that is, if the limit of its defining sequence doesn’t equal 0 — the series is divergent. But if the series fails the test, you can draw no conclusion.

Later in this chapter, you discover two more one-way tests: the direct comparison test and the limit comparison test.

Two-way tests

A two-way test allows you to draw one conclusion when a series passes the test and the opposite conclusion when a series fails the test. As with a one-way test, passing the test means that a given condition has been met. Failing the test means that the negation of that condition has been met.

For example, the test for geometric series is a two-way test (see Chapter 11 to find out more about testing geometric series for convergence and divergence). If a series passes the test — that is, if r falls in the open set (–1, 1) — the series is convergent. And if the series fails the test — that is, if r ≤ –1 or r ≥ 1 — the series is divergent.

Similarly, the test for p-series is also a two-way test (see Chapter 11 for more on this test).

Keep in mind that no test — even a two-way test — is guaranteed to give you an answer. Think of each test as a tool. If you run into trouble trying to cut a piece of wood with a hammer, it’s not the hammer’s fault: You just chose the wrong tool for the job. Likewise, if you can’t find a clever way to demonstrate either the condition or its negation required by a specific test, you’re out of luck. In that case, you may need to use a different test that’s better suited to the problem.

Later in this chapter, I show you three more two-way tests: the integral test, the ratio test, and the root test.

Choosing Comparison Tests

Comparison tests allow you to use stuff that you know to find out stuff that you want to know. The stuff that you know is more eloquently called a benchmark series — a series whose convergence or divergence you’ve already proven. The stuff that you want to know is, of course, whether an unfamiliar series converges or diverges.

As with the nth-term test, comparison tests are one-way tests: When a series passes the test, you prove what you’ve set out to prove (that is, either convergence or divergence). But when a series fails the test, the result of a comparison test in inconclusive.

In this section, I show you two basic comparison tests: the direct comparison test and the limit comparison test.

Getting direct answers with the direct comparison test

You can use the direct comparison test to prove either convergence or divergence, depending on how you set up the test.

To prove that a series converges:

1. Find a benchmark series that you know converges.

2. Show that each term of the series you’re testing is less than or equal to the corresponding term of the benchmark series.

To prove that a series diverges:

1. Find a benchmark series that you know diverges.

2. Show that each term of the series you’re testing is greater than or equal to the corresponding term of the benchmark series.

For example, suppose that you’re asked to determine whether the following series converges or diverges:

It’s hard to tell just by looking at it whether this particular series is convergent or divergent. However, it looks a bit like a p-series with p = 2:

You know that this p-series converges (see Chapter 11 if you’re not sure why), so use it as your benchmark series. Now your task is to show that every term in the series you’re testing is less than the corresponding term of the benchmark series:

This looks good, but to complete the proof formally, here’s what you want to show:

To see that this statement is true, notice that the numerators are the same, but the denominator (n² + 1) is greater than n². So the function is less than , which means that every term in the test series is less than the corresponding term in the convergent benchmark series. Therefore, both series are convergent.

As another example, suppose that you want to test the following series for convergence or divergence:

This time, the series reminds you of the trusty harmonic series, which you know is divergent:

Using the harmonic series as your benchmark, compare the two series term by term:

Again, you have reason to be hopeful, but to complete the proof formally, you want to show the following:

This time, notice that the denominators are the same, but the numerator 3 is greater than the numerator 1. So the function is greater than .

Again, you’ve shown that every term in the test series is greater than the corresponding term in the divergent benchmark series, so both series are divergent.

As a third example, suppose that you’re asked to show whether this series is convergent or divergent:

In this case, multiplying out the denominators is a helpful first step:

Now the series looks a little like a p-series with p = 2, so make this your benchmark series:

The benchmark series converges, so you want to show that every term of the test series is less than the corresponding term of the benchmark. This looks likely because:

However, to convince the professor, you want to show that every term of the test series is less than the corresponding term:

As with the first example in this section, the numerators are the same, but the denominator of the test series is greater than that of the benchmark series. So the test series is, indeed, less than the benchmark series, which means that the test series is also convergent.

Testing your limits with the limit comparison test

As with the direct comparison test, the limit comparison test works by choosing a benchmark series whose behavior you know and using it to provide information about a test series whose behavior you don’t know.

Here’s the limit comparison test: Given a test series Σ a_n and a benchmark series Σ b_n, find the following limit:

If this limit evaluates as a positive number, then either both series converge or both diverge.

As with the direct comparison test, when the test succeeds, what you learn depends on what you already know about the benchmark series. If the benchmark series converges, so does the test series. However, if the benchmark series diverges, so does the test series.

Remember, however, that this is a one-way test: If the test fails, you can draw no conclusion about the test series.

The limit comparison test is especially good for testing infinite series based on rational expressions. For example, suppose that you want to see whether the following series converges or diverges:

When testing an infinite series based on a rational expression, choose a benchmark series that’s proportionally similar — that is, whose numerator and denominator differ by the same number of degrees.

In this example, the numerator is a first-degree polynomial, and the denominator is a second-degree polynomial (for more on polynomials, see Chapter 2). So the denominator is one degree greater than the numerator. Therefore, I choose a benchmark series that’s proportionally similar — the trusty harmonic series:

Before you begin, take a moment to get clear on what you’re testing, and jot it down. In this case, you know that the benchmark series diverges. So if the test succeeds, you prove that the test series also diverges. (If it fails, however, you’re back to square one because this is a one-way test.)

Now set up the limit (by the way, it doesn’t matter which series you put in the numerator and which in the denominator):

At this point, you just crunch the numbers:

Notice at this point that the numerator and denominator are both second-degree polynomials. Now, as you apply L’Hopital’s Rule (taking the derivative of both the numerator and denominator; see Chapter 2), watch what happens:

As if by magic, the limit evaluates to a positive number, so the test succeeds. Therefore, the test series diverges. Remember, however, that you made this magic happen by choosing a benchmark series in proportion to the test series.

Another example should make this crystal clear. Discover whether this series is convergent or divergent:

When you see that this series is based on a rational expression, you immediately think of the limit comparison test. Because the denominator is two degrees higher than the numerator, choose a benchmark series with the same property:

Before you begin, jot down the following: The benchmark converges, so if the test succeeds, the test series also converges. Next, set up your limit:

Now just solve the limit:

Again, the numerator and denominator have the same degree, so you’re on the right track. Now solving the limit is just a matter of grinding through a few iterations of L’Hopital’s Rule:

The test succeeds, so the test series converges. And again, the success of the test was prearranged because you chose a benchmark series in proportion to the test series.

Two-Way Tests for Convergence and Divergence

Earlier in this chapter, I give you a variety of tests for convergence or divergence that work in one direction at a time. That is, passing the test gives you an answer, but failing it provides no information.

The tests in this section all have one important feature in common: Regardless of whether the series passes or fails, whenever the test gives you an answer, that answer always tells you whether the series is convergent or divergent.

Integrating a solution with the integral test

Just when you thought that you wouldn’t have to think about integration again until two days before your final exam, here it is again. The good news is that the integral test gives you a two-way test for convergence or divergence.

Here’s the integral test:

For any series of the form consider its associated integral

If this integral converges, the series also converges; however, if this integral diverges, the series also diverges.

In most cases, you use this test to find out whether a series converges or diverges by testing its associated integral. Of course, changing the series to an integral makes all the integration tricks that you already know and love available to you.

For example, here’s how to use the integral test to show that the harmonic series is divergent. First, the series:

The integral test tells you that this series converges or diverges depending on whether the following definite integral converges or diverges:

To evaluate this improper integral, express it as a limit, as I show you in Chapter 9:

This is simple to integrate and evaluate:

Because the limit explodes to infinity, the integral doesn’t exist. Therefore, the integral test tells you that the harmonic series is divergent.

As another example, suppose that you want to discover whether the following series is convergent or divergent:

Notice that this series starts at n = 2, because n = 1 would produce the term .

To use the integral test, transform the sum into this definite integral, using 2 as the lower limit of integration:

Again, rewrite this improper integral as the limit of an integral (see Chapter 9):

To solve the integral, use the following variable substitution:

So you can rewrite the integral as follows:

Note that as the variable changes from x to u, the limits of integration change from 2 and c to ln 2 and ln c. This change arises when I plug the value x = 2 into the equation u = ln x, so u = ln 2. (For more on using variable substitution to evaluate definite integrals, see Chapter 5.)

At this point, you can evaluate the integral:

You can see without much effort that as c approaches infinity, so does ln c, and the rest of the expression doesn’t affect this. Therefore, the series that you’re testing is divergent.

Rationally solving problems with the ratio test

The ratio test is especially good for handling series that include factorials. Recall that the factorial of a counting number, represented by the symbol !, is that number multiplied by every counting number less than itself. For example:

5! = 5 · 4 · 3 · 2 · 1 = 120

Flip to Chapter 2 for some handy tips on factorials that may help you in this section.

To use the ratio test, take the limit (as n approaches ∞) of the (n + 1)th term divided by the nth term of the series:

At the risk of destroying all the trust that you and I have built between us over these pages, I must confess that there are not two, but three possible outcomes to the ratio test:

If this limit is less than 1, the series converges.

If this limit is greater than 1, the series diverges.

If this limit equals 1, the test is inconclusive.

But I’m sticking to my guns and calling this a two-way test, because — depending on the outcome — it can potentially prove either convergence or divergence.

For example, suppose that you want to find out whether the following series is convergent or divergent:

Before you begin, expand the series so you can get an idea of what you’re working with. I do this in two steps to make sure the arithmetic is correct:

To find out whether this series converges or diverges, set up the following limit:

As you can see, I place the function that defines the series in the denominator. Then I rewrite this function, substituting n + 1 for n, and I place the result in the numerator. Now evaluate the limit:

At this point, to see why the ratio test works so well for exponents and factorials, factor out a 2 from 2ⁿ⁺¹ and an n + 1 from (n + 1)!:

This trick allows you to simplify the limit greatly:

Because the limit is less than 1, the series converges.

Rooting out answers with the root test

The root test works best with series that have powers of n in both the numerator and denominator.

To use the root test, take the limit (as n approaches ∞) of the nth root of the nth term of the series:

As with the ratio test, even though I call this a two-way test, there are really three possible outcomes:

If the limit is less than 1, the series converges.

If the limit is greater than 1, the series diverges.

If the limit equals 1, the test is inconclusive.

For example, suppose that you want to decide whether the following series is convergent or divergent:

This would be a very hairy problem to try to solve using the ratio test. To use the root test, take the limit of the nth root of the nth term:

At first glance, this expression looks worse than what you started with. But it begins to look better when you separate the numerator and denominator into two roots:

Now a lot of cancellation is possible:

Suddenly, the problem doesn’t look so bad. The numerator and denominator both approach ∞, so apply L’Hopital’s Rule:

Because the limit is less than 1, the series is convergent.

Looking at Alternating Series

Each of the series that I discuss earlier in this chapter (and most of those in Chapter 11) have one thing in common: Every term in the series is positive. So each of these series is a positive series. In contrast, a series that has infinitely many positive and infinitely many negative terms is called an alternating series.

Most alternating series flip back and forth between positive and negative terms so that every odd-numbered term is positive and every even-numbered term is negative, or vice versa. This feature adds another spin onto the whole question of convergence and divergence. In this section, I show you what you need to know about alternating series.

Eyeballing two forms of the basic alternating series

The most basic alternating series comes in two forms. In the first form, the odd-numbered terms are negated; in the second, the even-numbered terms are negated.

Without further ado, here’s the first form of the basic alternating series:

As you can see, in this series the odd terms are all negated. And here’s the second form, whose even terms are negated:

Obviously, in whichever form it takes, the basic alternating series is divergent because it never converges on a single sum but instead jumps back and forth between two sums for all eternity. Although the functions that produce these basic alternating series aren’t of much interest by themselves, they get interesting when they’re multiplied by an infinite series.

Making new series from old ones

You can turn any positive series into an alternating series by multiplying the series by (–1)ⁿ or (–1)^n–1. For example, here’s an old friend, the harmonic series:

To negate the odd terms, multiply by (–1) ⁿ:

To negate the even terms, multiply by (–1)^n–1:

Alternating series based on convergent positive series

If you know that a positive series converges, any alternating series based on this series also converges. This simple rule allows you to list a ton of convergent alternating series. For example:

The first series is an alternating version of a geometric series with r = . The second is an alternating variation on the familiar p-series with p = 2. The third is an alternating series based on a series that I introduce in the earlier section “Rationally solving problems with the ratio test.” In each case, the nonalternating version of the series is convergent, so the alternating series is also convergent.

To see why this works, consider the first of these three series, and calculate the first few partial sums:

Notice that the partial sums for this series alternately increase and decrease. Additionally, because the terms of the original series approach 0, the partial sums tend to alternate less and less erratically — that is, they hone in on a specific value. You may not know how to calculate this value, but you can still state that such a value exists, so the series is convergent.

As you see in the next section, “Checking out the alternating series test,” this is a specific case of a broader test for convergence. For now, just remember that if a positive series converges, the alternating version of this series also converges.

Checking out the alternating series test

As I discuss in the previous section, when you know that a positive series is convergent, you can assume that any alternating series based on that series is also convergent. In contrast, some divergent positive series become convergent when transformed into alternating series.

Fortunately, I can give you a simple test to decide whether an alternating series is convergent or divergent.

An alternating series converges if these two conditions are met:

1. Its defining sequence converges to zero — that is, it passes the nth-term test.

2. Its terms are nonincreasing (ignoring minus signs) — that is, each term is less than or equal to the term before it.

These conditions are fairly easy to test for, making the alternating series test one of the easiest tests in this chapter. For example, here are three alternating series:

Just by eyeballing them, you can see that each of these series meets both criteria of the alternating series test, so they’re all convergent. Notice, too, that in each case, the positive version of the same series is divergent. This underscores an important point: When a positive series is convergent, an alternating series based on it is also necessarily convergent; but when a positive series is divergent, an alternating series based on it may be either convergent or divergent.

Technically speaking, the alternating series test is a one-way test: If the series passes the test — that is, if both conditions hold — the series is convergent. However, if the series fails the test — that is, if either condition isn’t met — you can draw no conclusion.

In practice, however — and I’m going out on a thin mathematical limb here — I’d say that when a series fails the alternating series test, you have strong circumstantial evidence that the series is divergent.

Why do I say this? First of all, notice that the first condition is the good old-fashioned nth-term test. If any series fails this test, you can just chuck it on the divergent pile and get on with the rest of your day.

Second, it’s rare when a series — any series — meets the first condition but fails to meet the second condition. Sure, it happens, but you really have to hunt around to find a series like that. And even when you find one, the series usually settles down into an ever-decreasing pattern fairly quickly.

For example, take a look at the following alternating series:

Clearly, this series passes the first condition of the alternating series test — the nth-term test — because the denominator explodes to infinity at a much faster rate than the numerator.

What about the second condition? Well, the first three terms are increasing (disregarding sign), but beyond these terms the series settles into an ever-decreasing pattern. So you can chop off the first few terms and express the same series in a slightly different way:

This version of the series passes the alternating series test with flying colors, so it’s convergent. Obviously, adding a few constants to this series doesn’t make it divergent, so the original series is also convergent.

So when you’re testing an alternating series, here’s what you do:

1. Test for the first condition — that is, apply the nth-term test.

If the series fails, it’s divergent, so you’re done.

2. If the series passes the nth-term test, test for the second condition — that is, see whether its terms eventually settle into a constantly-decreasing pattern (ignoring their sign, of course).

In most cases, you’ll find that a series that meets the first condition also meets the second, which means that the series is convergent.

In the rare cases when an alternating series meets the first condition of the alternating series test but doesn’t meet the second condition, you can draw no conclusion about whether that series converges or diverges.

These cases really are rare, but I show you one so you know what to do in case your professor decides to get cute on an exam:

Both of these series meet the first criteria of the alternating series test but fail to meet the second, so you can draw no conclusion based on this test. In fact, the first series is convergent and the second is divergent. Spend a little time studying them and I believe that you’ll see why. (Hint: Try to break each series apart into two separate series.)

Understanding absolute and conditional convergence

In the previous two sections, I demonstrate this important fact: When a positive series is convergent, an alternating series based on it is also necessarily convergent; but when a positive series is divergent, an alternating series based on it may be either convergent or divergent.

So for any alternating series, you have three possibilities:

An alternating series is convergent, and the positive version of that series is also convergent.

An alternating series is convergent, but the positive version of that series is divergent.

An alternating series is divergent, so the positive version of that series must also be divergent.

The existence of three possibilities for alternating series makes a new concept necessary: the distinction between absolute convergence and conditional convergence.

Table 12-1 tells you when an alternating series is absolutely convergent, conditionally convergent, or divergent.

Table 12-1 Understanding Absolute and Conditional Convergence of Alternating Series

Here are a few examples of alternating series that are absolutely convergent:

I pulled these three examples from “Alternating series based on convergent positive series” earlier in this chapter. In each case, the positive version of the series is convergent, so the related alternating series must be convergent as well. Taken together, these two facts mean that each series converges absolutely.

And here are a few examples of alternating series that are conditionally convergent:

I pulled these examples from “Checking out the alternating series test” earlier in this chapter. In each case, the positive version of the series diverges, but the alternating series converges (by the alternating series test). So each of these series converges conditionally.

Finally, here are a couple of examples of alternating series that are divergent:

As you can see, the first two series fail the nth-term test, which is also the first condition of the alternating series test, so these two series diverge. As for the third series, it’s basically a divergent harmonic series minus a convergent geometric series — that is, a divergent series with a finite number subtracted from it — so the entire series diverges.

Testing alternating series

Suppose that somebody (like your professor) hands you an alternating series that you’ve never seen before and asks you to determine whether it’s absolutely convergent, conditionally convergent, or divergent. Here’s what you do:

1. Apply the alternating series test.

In most cases, this test tells you whether the alternating series is convergent or divergent:

a. If it’s divergent, you’re done! (The alternating series is divergent.)

b. If it’s convergent, the series is either absolutely convergent or conditionally convergent. Proceed to Step 2.

c. If the alternating series test is inconclusive, you can’t rule any option out. Proceed to Step 2.

2. Rewrite the alternating series as a positive series by:

a. Removing (–1)ⁿ or (–1)^n–1 when you’re working with sigma notation.

b. Changing the minus signs to plus signs when you’re working with expanded notation.

3. Test this positive series for convergence or divergence by using any of the tests in this chapter or Chapter 11:

a. If the positive series is convergent, the alternating series is absolutely convergent.

b. If the positive series is divergent and the alternating series is convergent, the alternating series is conditionally convergent.

c. If the positive series is divergent but the alternating series test is inconclusive, the series is either conditionally convergent or divergent, but you still can’t tell which.

In most cases, you’re not going to get through all these steps and still have a doubt about the series. In the unlikely event that you do find yourself in this position, see whether you can break the alternating series into two separate series — one with positive terms and the other with negative terms — and study these two series for whatever clues you can.