Learning Objectives
- 5.6.1 Use the ratio test to determine absolute convergence of a series.
- 5.6.2 Use the root test to determine absolute convergence of a series.
- 5.6.3 Describe a strategy for testing the convergence of a given series.
In this section, we prove the last two series convergence tests: the ratio test and the root test. These tests are particularly nice because they do not require us to find a comparable series. The ratio test will be especially useful in the discussion of power series in the next chapter.
Throughout this chapter, we have seen that no single convergence test works for all series. Therefore, at the end of this section we discuss a strategy for choosing which convergence test to use for a given series.
Ratio Test
Consider a series From our earlier discussion and examples, we know that is not a sufficient condition for the series to converge. Not only do we need but we need quickly enough. For example, consider the series and the series We know that and However, only the series converges. The series diverges because the terms in the sequence do not approach zero fast enough as Here we introduce the ratio test, which provides a way of measuring how fast the terms of a series approach zero.
Theorem 5.16
Ratio Test
Let be a series with nonzero terms. Let
- If then converges absolutely.
- If or then diverges.
- If the test does not provide any information.
Proof
Let be a series with nonzero terms.
We begin with the proof of part i. In this case, Since there exists such that Let By the definition of limit of a sequence, there exists some integer such that
Therefore,
and, thus,
Since the geometric series
converges. Given the inequalities above, we can apply the comparison test and conclude that the series
converges. Therefore, since
where is a finite sum and converges, we conclude that converges.
For part ii.
Now suppose .
There exists some such that for all .
Then .
Now because for all , we know that .
It follows that .
By the divergence test, the series diverges.
diverges, and therefore the series diverges.
For part iii. we show that the test does not provide any information if by considering the For any real number
However, we know that if the diverges, whereas converges if
□
The ratio test is particularly useful for series whose terms contain factorials or exponentials, where the ratio of terms simplifies the expression. The ratio test is convenient because it does not require us to find a comparative series. The drawback is that the test sometimes does not provide any information regarding convergence.
Example 5.23
Using the Ratio Test
For each of the following series, use the ratio test to determine whether the series converges or diverges.
Solution
- From the ratio test, we can see that
Since
Since the series converges. - We can see that
Since the series diverges. - Since
we see that
Since the series converges.
Checkpoint 5.21
Use the ratio test to determine whether the series converges or diverges.
Root Test
The approach of the root test is similar to that of the ratio test. Consider a series such that for some real number Then for sufficiently large, Therefore, we can approximate by writing
The expression on the right-hand side is a geometric series. As in the ratio test, the series converges absolutely if and the series diverges if If the test does not provide any information. For example, for any p-series, we see that
To evaluate this limit, we use the natural logarithm function. Doing so, we see that
Using L’Hôpital’s rule, it follows that and therefore for all However, we know that the p-series only converges if and diverges if
Theorem 5.17
Root Test
Consider the series Let
- If then converges absolutely.
- If or then diverges.
- If the test does not provide any information.
The root test is useful for series whose terms involve exponentials. In particular, for a series whose terms satisfy then and we need only evaluate
Example 5.24
Using the Root Test
For each of the following series, use the root test to determine whether the series converges or diverges.
Solution
- To apply the root test, we compute
Since the series converges absolutely. - We have
Since the series diverges.
Checkpoint 5.22
Use the root test to determine whether the series converges or diverges.
Choosing a Convergence Test
At this point, we have a long list of convergence tests. However, not all tests can be used for all series. When given a series, we must determine which test is the best to use. Here is a strategy for finding the best test to apply.
Problem-Solving Strategy
Choosing a Convergence Test for a Series
Consider a series In the steps below, we outline a strategy for determining whether the series converges.
- Is a familiar series? For example, is it the harmonic series (which diverges) or the alternating harmonic series (which converges)? Is it a or geometric series? If so, check the power or the ratio to determine if the series converges.
- Is it an alternating series? Are we interested in absolute convergence or just convergence? If we are just interested in whether the series converges, apply the alternating series test. If we are interested in absolute convergence, proceed to step considering the series of absolute values
- Is the series similar to a or geometric series? If so, try the comparison test or limit comparison test.
- Do the terms in the series contain a factorial or power? If the terms are powers such that try the root test first. Otherwise, try the ratio test first.
- Use the divergence test. If this test does not provide any information, try the integral test.
Media
Visit this website for more information on testing series for convergence, plus general information on sequences and series.
Example 5.25
Using Convergence Tests
For each of the following series, determine which convergence test is the best to use and explain why. Then determine if the series converges or diverges. If the series is an alternating series, determine whether it converges absolutely, converges conditionally, or diverges.
Solution
- Step 1. The series is not a or geometric series.
Step 2. The series is not alternating.
Step 3. For large values of we approximate the series by the expression
Therefore, it seems reasonable to apply the comparison test or limit comparison test using the series Using the limit comparison test, we see that
Since the series diverges, this series diverges as well. - Step 1.The series is not a familiar series.
Step 2. The series is alternating. Since we are interested in absolute convergence, consider the series
Step 3. The series is not similar to a p-series or geometric series.
Step 4. Since each term contains a factorial, apply the ratio test. We see that
Therefore, this series converges, and we conclude that the original series converges absolutely, and thus converges. - Step 1. The series is not a familiar series.
Step 2. It is not an alternating series.
Step 3. There is no obvious series with which to compare this series.
Step 4. There is no factorial. There is a power, but it is not an ideal situation for the root test.
Step 5. To apply the divergence test, we calculate that
Therefore, by the divergence test, the series diverges. - Step 1. This series is not a familiar series.
Step 2. It is not an alternating series.
Step 3. There is no obvious series with which to compare this series.
Step 4. Since each term is a power of we can apply the root test. Since
by the root test, we conclude that the series converges.
Checkpoint 5.23
For the series determine which convergence test is the best to use and explain why.
In Table 5.3, we summarize the convergence tests and when each can be applied. Note that while the comparison test, limit comparison test, and integral test require the series to have nonnegative terms, if has negative terms, these tests can be applied to to test for absolute convergence.
Series or Test | Conclusions | Comments |
---|---|---|
Divergence Test For any series evaluate |
If the test is inconclusive. | This test cannot prove convergence of a series. |
If the series diverges. | ||
Geometric Series |
If the series converges to |
Any geometric series can be reindexed to be written in the form where is the initial term and is the ratio. |
If the series diverges. | ||
p-Series |
If the series converges. | For we have the harmonic series |
If the series diverges. | ||
Comparison Test For with nonnegative terms, compare with a known series |
If for all and converges, then converges. | Typically used for a series similar to a geometric or -series. It can sometimes be difficult to find an appropriate series. |
If for all and diverges, then diverges. | ||
Limit Comparison Test For with positive terms, compare with a series by evaluating |
If is a real number and then and both converge or both diverge. | Typically used for a series similar to a geometric or -series. Often easier to apply than the comparison test. |
If and converges, then converges. | ||
If and diverges, then diverges. | ||
Integral Test If there exists a positive, continuous, decreasing function such that for all evaluate |
and both converge or both diverge. | Limited to those series for which the corresponding function can be easily integrated. |
Alternating Series |
If for all and then the series converges. | Only applies to alternating series. |
Ratio Test For any series with nonzero terms, let |
If the series converges absolutely. | Often used for series involving factorials or exponentials. |
If the series diverges. | ||
If the test is inconclusive. | ||
Root Test For any series let |
If the series converges absolutely. | Often used for series where |
If the series diverges. | ||
If the test is inconclusive. |
Student Project
Series Converging to and
Dozens of series exist that converge to or an algebraic expression containing Here we look at several examples and compare their rates of convergence. By rate of convergence, we mean the number of terms necessary for a partial sum to be within a certain amount of the actual value. The series representations of in the first two examples can be explained using Maclaurin series, which are discussed in the next chapter. The third example relies on material beyond the scope of this text.
- The series
was discovered by Gregory and Leibniz in the late This result follows from the Maclaurin series for We will discuss this series in the next chapter.
- Prove that this series converges.
- Evaluate the partial sums for
- Use the remainder estimate for alternating series to get a bound on the error
- What is the smallest value of that guarantees Evaluate
- The series
has been attributed to Newton in the late The proof of this result uses the Maclaurin series for
- Prove that the series converges.
- Evaluate the partial sums for
- Compare to for and discuss the number of correct decimal places.
- The series
was discovered by Ramanujan in the early William Gosper, Jr., used this series to calculate to an accuracy of more than million digits in the At the time, that was a world record. Since that time, this series and others by Ramanujan have led mathematicians to find many other series representations for and
- Prove that this series converges.
- Evaluate the first term in this series. Compare this number with the value of from a calculating utility. To how many decimal places do these two numbers agree? What if we add the first two terms in the series?
- Investigate the life of Srinivasa Ramanujan and write a brief summary. Ramanujan is one of the most fascinating stories in the history of mathematics. He was basically self-taught, with no formal training in mathematics, yet he contributed in highly original ways to many advanced areas of mathematics.
Section 5.6 Exercises
Use the ratio test to determine whether converges, where is given in the following problems. State if the ratio test is inconclusive.
Use the root test to determine whether converges, where is as follows.
For this exercise, let n start at 2.
In the following exercises, use either the ratio test or the root test as appropriate to determine whether the series with given terms converges, or state if the test is inconclusive.
(Hint: Compare to
Use the ratio test to determine whether converges, or state if the ratio test is inconclusive.
Use the root and limit comparison tests to determine whether converges.
In the following exercises, use an appropriate test to determine whether the series converges.
(Hint:
(Hint:
The following series converge by the ratio test. Use summation by parts, to find the sum of the given series.
(Hint: Take and
The kth term of each of the following series has a factor Find the range of for which the ratio test implies that the series converges.
Does there exist a number such that converges?
Suppose that For which values of must converge?
Suppose that for all where is a fixed real number. For which values of is guaranteed to converge?
Suppose that for all Can you conclude that converges?
Let where is the greatest integer less than or equal to Determine whether converges and justify your answer.
The following advanced exercises use a generalized ratio test to determine convergence of some series that arise in particular applications when tests in this chapter, including the ratio and root test, are not powerful enough to determine their convergence. The test states that if then converges, while if then diverges.
Let Explain why the ratio test cannot determine convergence of Use the fact that is increasing to estimate
Let Show that For which does the generalized ratio test imply convergence of (Hint: Write as a product of factors each smaller than
Let Show that as