Sequences and Series**



Sequences and Series (BC Only)

Sequences and series is one of the most important topics to be tested on the BC Exam. It is not listed in the topics for the AB Exam. There is a great deal of material in this section. The Test Development Committee always includes a good selection of topics about sequences and series in the BC portion of the AP Exam. Because of the amount of material, not all topics are tested each year. However, as a student, you don't know which items will be covered and which one won't. So, you have the responsibility of being familiar with entire section.

In this chapter we will review sequences and explain how series differ from sequences. Then we will look at Power Series, Taylor Series, and Maclaurin Series. We will next discuss Taylor Polynomials along with their error bounds. Finally, we will review a variety of techniques to determine if a series converges or diverges. We will place the emphasis on the understanding rather than the manipulation.

Sequences

A sequence is a list of numbers. You have worked with sequences throughout your math classes. We speak of the even numbers {2, 4, 6,…}, alternating powers of 2 [pic], and decreasing fractions [pic]. All of the lists follow some pattern. Sometimes the pattern is easy to spot; other times it is more difficult to determine. The domain of a sequence is the set of positive integers. These integers are not the value of the sequence. Rather, they are the means by which we order the sequence. We identify the first term, the second term, the third term, and so on. We label the terms for convenience using the notation, [pic]. So [pic]means the first term of some sequence. [pic] is the tenth term of a sequence. [pic] is the nth term or general term of a sequence. [pic] would mean the third term of some sequence that is different from the sequence a.

Limit or Convergence of a Sequence

As we work with sequences, we want to find out if the sequence converges or diverges. We use the following definition to help us.

[pic]

To establish convergence of a series, we often look at functions based on x that mimic the sequence. We then see if the function in x converges or diverges and apply the results to the sequence that is based on n. The following rule summarizes this statement.

[pic]

Example 1.

Let [pic]. Does the sequence [pic]converge or diverge?

We use a function f(x) that mimics the sequence, so we choose [pic][pic] We check [pic]. From the chapter on limits, we recall that [pic] Since [pic]converged to 1, we know that the sequence [pic]must also converge to 1. We have shown that [pic]converges.

[pic]

Figure 1

The graph of the sequence supports this result. The values of the sequence as indicated by the points seem to level off at [pic].

Test Tip. Although a graphing calculator can be used to investigate convergence of a sequence, the graph itself can never be considered as proof. As we stated in the Rule of Three, we can investigate graphically, numerically, or analytically. But we can only verify analytically.

Example 2.

Does the sequence [pic]converge?

Rewriting the sequence as a corresponding function of x, we get [pic]. Applying the limit, we get

[pic]

Since [pic]is divergent, [pic]is also divergent.

Series

A series is a sequence of partial sums. Symbolically, we write [pic]. Let's examine a series term by term to see how the individual terms relate to each other.

Example 3

Suppose that [pic], write the first five terms of [pic]

[pic]

It is very easy to mix sequences with series. There is a lot of vocabulary that you need to understand and keep straight in your mind. The following list is given to explain the differences in vocabulary

[pic]

Power Series

In some of your earlier math classes you worked with a geometric sequences and series such as the sequence [pic]and its related series [pic] . In this section, we want to review a geometric series with constants; then we will extend it to variables.

Example 4.

Determine if the series [pic]converges.

We can rewrite this series as [pic]which is a geometric series with a common ratio of[pic]. We recall the sum of any geometric series can be found by using the formula [pic]. In this example [pic]. The sum is [pic].

[pic]

Example 5.

Check the series [pic]for convergence.

The first term is [pic]and the common ratio is [pic]. By the formula, [pic] .

This is a finite number so the series converges.

Example 6.

Does the series [pic]converge?

The first term is [pic], and the common ratio is [pic], which is greater than 1. Since the ratio is greater than 1, the series diverges.

Example 7.

Write the expansion of a geometric series whose ratio is x.

We replace r with x which gives us the sum [pic]

Representing Function by Series

Example 7 illustrates the reason we are so concerned about the geometric series. When x was used in place of the constant, the right side became a polynomial expression that is equivalent to [pic]whenever[pic]. This expression is very powerful. It lets us describe a non-polynomial function as a polynomial function.

As the figure shows, the partial sum, [pic], of the polynomial series converges very closely to [pic] as long as [pic]. It diverges quickly when x is outside the interval. As we increase the number of terms in partial sums, the convergence gets stronger.

[pic]

Figure 2

Although we use a polynomial to approximate the rational function, the expression [pic]is not a polynomial. Polynomials have finite degrees and are not subject to divergence.

Example 8.

Use the formula for the geometric series to find a power series representation for [pic]. Determine the interval of convergence.

We rewrite the function as [pic]. The ratio is -x. The expansion is given by

[pic].

The figure shows the relationship between f(x) and [pic]on the interval [pic].

[pic]

Figure 3

After we have a power series representation, we can manipulate using techniques we have learned in algebra or calculus classes.

Example 9.

Since [pic], find a power series representation for [pic].

We get a little sneaky here. Notice that [pic]just happens to be the derivative of [pic]. We can take the derivative of both sides of the equation.

[pic]

As a result, we have a power series representation for a new function, [pic].

We can use integration as well as differentiation to manipulate both sides of an equation.

Example 10.

The power series representation for [pic]. Integrate both sides to determine a new power series.

[pic]

We now have a power series representation for arctan x.

Example 11.

Find a power series representation for [pic].

This problem looks similar to Example 8. Unfortunately, it is not quite the same. But we can rewrite our problem as [pic]. We already have a power series representation for [pic]. We can multiply both sides by x to get the new power series.

[pic]

As you can imagine, there are many, many functions for which we could use one of these techniques to develop a power series representation. Because you may be wondering which one to use, consider this idea. You have only three tools at your disposal. You can differentiate a power series, integrate a power series, or multiply (or divide) a power series by a variable. Always try one of these techniques to the compact side of a known expression to see if you can obtain an expression you need. If the left side works out properly, do the same thing to the right side.

The power series that we developed in the preceding examples all had similar features. They all started with increasing powers of x, the defining characteristic of a power series.

[pic]

These definitions are very powerful. They allow us to custom design any function we want as a polynomial representation.

Taylor Series

Example 12.

Construct a polynomial [pic]with the following characteristics at [pic].

[pic]

We see immediately that[pic]. Since [pic]. The other terms are a little harder to find. First, let's take the first four derivatives of P(x).

[pic]

We can evaluate each derivative at x = 0 and set it equal to the given value.

[pic]

With the value for each constant now determined, we can write our customized polynomial:

[pic].

Example 13.

Given[pic], construct a polynomial in the form [pic]centered about [pic].

Since we are centered about [pic], we need to rewrite the polynomial as [pic]. First, we evaluate the function at [pic]. Then we find the first four derivatives and evaluate each at [pic].

[pic]

[pic]

From our work in Example 12 we can find the derivatives of the polynomial

[pic]

When we evaluate each derivative at [pic], each factor [pic]will become zero. We can find each of the constants.

[pic]

We complete the problem by substituting the values for the constants. The answer is [pic].

This polynomial does not equal [pic]. It is a very close approximation over the interval of convergence (which we have not yet found).

Examples 12 and 13 are examples of Taylor Polynomials. Since both polynomials are written to the fourth power, they are referred to as fourth order Taylor Polynomials. Example 13 was expanded about [pic]. Example 12 was expanded about [pic]. Any Taylor polynomial expanded about [pic]is called a Maclaurin polynomial.

In examples 12 and 13, our work was to find the value of each coefficient. We could work a problem step by step each time. However, mathematicians long ago discovered the relationship between the coefficient and the degree of the term.

[pic]

Example 14.

Find the Taylor Series expansion for [pic].

First, we find all the coefficients by using the definition at [pic]

[pic]

You can see that the higher-order derivatives of the[pic]will repeat in the preceding pattern. The Taylor Series expansion of[pic]is given by

[pic]

Test Tip. It is easy to mix up the position of the terms. The first term is [pic]. We always begin our counting at n = 0.

Earlier in the chapter, we talked about the many different functions that could be represented by a power series expansion. There are seven power series expansions that you should know. Many AP questions have their beginnings in these power series. We will present them centered about [pic] which makes them Maclaurin Series. To convert to a general Taylor Series about [pic], substitute [pic]. The first term will no longer be [pic].

Seven Well-known Power Series (with Intervals of Convergence)

[pic]

[pic]

[pic]

[pic]

[pic]

[pic]

[pic]

Example 15.

Find a power series representation for [pic].

We will use the power series for[pic], substituting [pic].

[pic]

Taylor Series Formula with Remainder

We can write any Taylor Series as a two-part definition. It is the nth-degree Taylor polynomial plus any remainder after the nth-degree term. The remainder is actually the error that occurs when using a Taylor polynomial to approximate a function. We are able to connect both the polynomial and the error involved in using the Taylor polynomial as an approximation for the series.

[pic]

The function [pic]is the error term or the remainder of order n. Some texts refer to this as the Lagrange form of the remainder which can be used to find the Lagrange error bounds.

This formula is especially useful if we are working with an alternating convergent series. Since the remaining terms converge and always switch between positive and negative values, the remainder will always be less that the value of the (n+1)th term.

Example 16.

Use the first three nonzero terms to estimate the value of[pic]. Estimate the error bound in the approximation.

Using only the first three terms means we want a third order Taylor polynomial which gives us[pic]. Evaluating the polynomial at x = 0.1, we get [pic]

The next term in the series, the error bound is less than [pic]. The maximum Error Bound is less than [pic].

Example 17.

Use a power series to find [pic]correct to four decimal places. Determine the maximum error bound.

We aren't told how many terms to use. For convenience, let's use the first four non-zero terms to see what happens. If needed, we can take additional terms to improve the accuracy of our answer.

We can the power series representation for sin(x) to write

[pic]

Summing the first three terms of the answer, we get [pic]. The answer to four decimal places is 0.3103. The error bound is less than [pic].

Power series can give very accurate approximations within just a very few terms. To reach the same degree of accuracy in Example 17 using the Riemann Sum would require dividing the interval into many, many subintervals which would require more calculations than we did.

Tests for Convergence

Power series are useful when we know they converge. They allow to us decide the degree of accuracy we want. Series that do not converge are of little use to us. We need to be able to check for convergence for any functions that can be represented by power series. This section will explain and illustrate the most useful tests. There is no special order to the presentation of the tests. We are showing them in a convenient manner.

The Integral Test

Consider the series

[pic].

There is no convenient formula that gives the sum like the Geometric Series formula did. We can generate a table of values to see if convergence seems likely.

|n |5 |50 |500 |5000 |

|Sn |1.1857… |1.2019… |1.2020549… |1.2020569… |

We remember that a table is not proof of convergence, but it seems reasonable in this case that convergence occurs in this series. We still need to verify the result analytically. We will use the Integral Test to do this.

[pic]

We apply the Integral Test to our example. The improper integral associated with [pic] which we know converges by the p-series rule we developed in the last chapter. Therefore, the series [pic]converges.

Once again we raise the warning. Although we know the series converges, we don't know the actual value for the limit of the series.

As we learned in the review of the p-series rule, any improper integral in the form [pic]converged if [pic]. The Integral Test for Convergence allows us to formalize the rule for series.

[pic]

Example 18.

Does [pic]converge?

Beginning the sequence at n=4 is okay. If a series converges on the interval [pic], it will surely converge on some smaller interval that starts to the right of n = 1. It is not really necessary for f to be always decreasing as long as f is ultimately decreasing on the infinite interval. This example fits the p-series format. Since[pic], we can safely say the series will converge.

The next test is used to confirm divergence.

The nth Term Test for Divergence

[pic]

This statement is logical. If the terms of a sequence do not converge, it is impossible for the sequence of partial sums generated by that sequence to converge. Usually, we check for convergence, not divergence. But if we know that a series is clearly divergent, we do not have to apply any of the tests for convergence.

Example 19.

Verify that [pic]diverges.

As n gets larger, [pic]increases without bound. Since the sequence does not limit to zero, the associated series must diverge.

Direct Comparison Test

The Direct Comparison Test for series is similar to the Direct Comparison Test for improper integrals we studied in the last chapter.

[pic]

As before, if the series [pic]converges to a limit, it "squeezes" all series smaller that it to the same limit. If the series [pic]diverges, it "pushes" all series that are larger away from a limit.

Example 20.

Show that [pic]converges for all real values of x.

There are no negative terms in the series and we know that [pic]. We recognize [pic]as the Taylor Series for [pic], which we know converges. Therefore, [pic]also converges by the Direct Comparison Test.

Test Tip. When verifying convergence of a series, always state the test you are using. It helps the reader of the question determine if you understand what you are doing.

[pic]

The Ratio Test

The Ratio Test is one of the most powerful tests we have to check convergence of series. It is often use in conjunction with absolute convergence.

The following form of the Ratio Test can be used to check for absolute convergence.

[pic]

Example 21.

Determine if the series [pic]is convergent.

We see that for all values of n, each term will be positive. We will apply the Ratio Test. We will take this explanation slowly, so you can follow the algebra.

[pic][pic]

Find the limit of the ratio. We did not use absolute value signs here, since all terms were already positive.

[pic]

Since L < 1, the series converges absolutely, so it is convergent.

The Limit Comparison Test

We found the p-series Test to be very useful in checking convergence of series in the form[pic]. Unfortunately, not all series fit this pattern exactly. We can test series with p-series of similar dimension. We use the Limit Comparison Test to do this.

[pic]

The Limit Comparison Test is not quite the same as the Direct Comparison Test. That test compares one series with a bounding series. The Limit Comparison Test checks the limit of the ratio between two series.

Example 22.

Determine if [pic]

All the terms are positive and we see that the function is somewhat like a p-series. As n grows very large, [pic]. So we will compare our series with the series [pic][pic].

[pic]

Since the limit is greater than one and the series[pic] diverges, the series [pic] must also diverge.

Example 23.

Check the convergence of[pic]

Ignoring the constant in the denominator, we see that [pic] [pic]. We will try the LCT.

[pic]

The limit is a constant and [pic]converges, so [pic]converges.

The Alternating Series Test

When the terms of the sequence alternate between positive and negative values, we can use the Alternating Series Test to check for convergence.

[pic]

We can visualize the reason this test works. Suppose we have the alternating series [pic]. As the figure shows, the first sum moves to right [pic], then moves back [pic], then moves right [pic], back [pic], and so on. As this process continues forever, the sums will congregate about a specific value.

We can also see this in a series plot.

[pic]

Figure 5

Example 24.

Determine if [pic]converges or diverges.

This is an alternating series, so we try the Alternating Series Test. Checking each condition, we get

1. Are all terms positive? - yes

2. Is [pic]? - yes

3. Does [pic]? - no, the limit is [pic]. By the nth term test, we see the series diverges.

Example 25.

Does [pic]converge or diverge? Show your reasoning.

You should recognize this as an alternating series. The Alternating Series Test is reasonable to use here. We check each of the conditions

1. Are all terms positive? - yes

2. Is [pic]? - This is difficult to tell by inspection. So we get sneaky.

[pic]

The answer to the second condition is yes.

3. Is [pic]? -- yes.

Since all three conditions have been met, the series is convergent by the Alternating Series Test.

Intervals of Convergence on Power Series

In the list of power series representation for seven common functions, we also listed the interval of convergence. In this section we want to see how those intervals relate to a specific power series and whether the interval is open or closed. We will demonstrate a series of steps you can follow to get the solution. This method uses the Ratio Test.

[pic]

Notice that the series converges when the limit is less than one, diverges when the limit is greater than one, and is inconclusive when the limit is equal to one.

[pic]

Example 26.

Given the series [pic], find the interval of convergence and the radius of convergence.

Use the Ratio Test.

[pic]

Find the limit.

[pic]

Set the absolute value of the limit less than 1.

[pic]

We still need to check the endpoints of the interval for convergence. We substitute these values into the series and check for convergence.

Case 1. Check right endpoint x = 7. When x = 7, the series becomes

[pic]

Case 2. Check left endpoint x = -3. When x = -3, the series becomes

[pic]

Since the power series does not converge at either endpoint, the interval of convergence is[pic]. The radius of convergence is the distance from the center of the interval to one endpoint. In this example the center is at[pic]. The distance from [pic] to [pic]is 5. So, 5 is the radius of convergence centered at [pic].

Example 27.

Find the interval of convergence for[pic].

Find the limit using the Ratio Test.

[pic]

Since the limit is zero, the series converges absolutely for all values of x. The interval of convergence is [pic]. It doesn't make sense to talk about the radius of convergence for this interval.

Example 28.

Find the interval of convergence for the series defined by [pic]

Use the Ratio Test to establish the limit, then check the limit against the intervals of convergence.

[pic]

The series will converge when [pic]. Next we check endpoints for convergence.

Case 1. x = 1. When x = 1, the series becomes

[pic]

Case 2. x = -1. When x = -1, the series becomes

[pic]

Therefore, the interval of convergence is [pic]. The radius of convergence is 1.

Strategies for Testing Series

We have reviewed several tests for verifying convergence and divergence of series. The difficult job you have as a student is deciding which one to use. There are no specific rules to follow, but the following guidelines can help. As you look at the strategies, don’t try each test until you find one that works. Instead, try to classify the series according to its form. Then find the guideline that fits the form.

1. If you can see quickly that [pic], check for divergence.

2. If the series fits the form [pic], it is a p-series. If [pic], the series converges. If [pic], the series diverges.

3. If the series fits the form [pic], then try the Alternating Series Test.

4. If the series fits the form [pic], then it is a geometric series. If [pic], the series converges. If [pic], the series diverges.

5. If a series that has factorials or other products, try the Ratio Test.

6. If a series fits a form similar to a p-series or geometric series, then try one of the comparison tests (Direct or Limit). If [pic]is a rational function or an algebraic function of n, try to compare it with a p-series. The comparison tests apply to series with positively valued terms. If any terms are negative, we can use a comparison test with [pic]

7. If there is a function [pic]that corresponds to [pic], then try the Integral Test on the improper integral [pic]if the antiderivative is easy to do.

Not all situations can be described in exactly one of these guidelines. In some ways, it is like factoring or integrating. You may need to do some algebra manipulations on the expressions to write them in a more convenient form. Don’t get too stressed about these rules. The questions on the AP Exam tend to be fairly standard. You are being tested to see if you know the various tests for series, not if you know some obscure tricks that just happen to make the problem work.

For the following examples, indicate which test would be most appropriate. Do not work the problems out.

Example 29.

[pic]

Since the [pic], check for divergence.

Example 30.

[pic]

[pic]is an algebraic function; we can compare it to a convenient p-series. But first we need to manipulate it. [pic]. The comparison series should be [pic].

Example 31.

[pic]

This is an alternating series; use the Alternating Series Test.

Example 32.

[pic]

The presence of factorials suggests that we try the Ratio Test.

Example 33.

[pic]

This series fits very nicely with [pic] use the Integral Test.

Example 34.

[pic]

This series is a rational function that is similar to a p-series. Use the Direct Comparison Test.

Now You Try It

Determine if the sequence converges or diverges. If the sequence is convergent, give the limit

1. [pic]

2. [pic]

3. [pic]

4. [pic]

Check the following series for convergence. If the series contains negative terms, determine if it is absolutely convergent, conditionally convergent, or divergent.

5. [pic]

6. [pic]

7. [pic]

8. [pic]

9. [pic]

10. [pic]

11. [pic]

12. [pic]

13. [pic]

14. [pic]

15. [pic]

16. [pic]

17. [pic]

18. [pic]

Write the first four terms and the general term for a Maclaurin Series representation for the following functions.

19. [pic]

20. [pic]

21. [pic]

22. [pic]

23. [pic]

24. [pic]

25. [pic]

Find the first four nonzero terms and the general term for the Taylor Series generated by [pic]

26. [pic]

27. [pic]

28. [pic]

29. [pic]

30. Let [pic] be the fourth-order Taylor polynomial for the function [pic]centered at [pic].

(a) Find [pic]

(b) Write the third-order Taylor polynomial for [pic]and use it to approximate [pic].

(c) Is it possible to find the exact value of [pic] from the information given. Explain.

(d) [pic] Write the fourth-order Taylor polynomial representation for [pic] at x = -3.

31. [pic]

(a) Write the first four terms and the general term of the Taylor Series generated by [pic].

(b) Use the results from part (a) to write the first four terms and general term for [pic].

32. [pic].

(a) Write the first four terms and the general term of the Taylor Series generated by [pic].

(b) What is the interval of convergence of the series found in part (a). Show your reasoning.

33. Let [pic]be a function with derivatives of all orders with the following properties: [pic].

(a) Write a third-order Taylor polynomial for [pic]at x = 6.

(b) Use the polynomial found in part (a) to approximate f(6.15).

(c) What is the second-order Taylor polynomial for [pic] at x = 6?

34. The first two terms for the power series for [pic]. For what values of x will the error bound be less than[pic]. Explain your reasoning.

-----------------------

Limit of a Sequence

A sequence [pic]has a limit L, or converges to L, if

[pic].

If such a number L does not exist, the sequence has no limit or diverges.

Let [pic]be a sequence with [pic] and suppose that [pic]exists for every real number[pic]. Then the following cases apply.

Case 1. [pic]

Case 2. [pic]

Sequence an infinite set of ordered numbers. 1, 4, 9, 16, 25,… is the sequence of perfect squares.

Series the sum of all the terms of a sequence. 1 + 4 + 9 + 16 + 25 + … is the series of perfect squares.

Partial sum the sum of the first n terms of a sequence. [pic] is the third partial sum of the series of perfect squares.

n the term index used to calculate the term value of a sequence or a series. It is always an integer greater than or equal to 1.

Geometric Series

A geometric series is written in the form [pic]

We can use [pic]notation to write this in a short form,

[pic].

This series converges if [pic]and diverges if [pic].

Definition of a Power Series

An expression in the form [pic], which expands as

[pic]

is a power series centered at [pic]. This is known as a Maclaurin Series.

An expression in the form [pic], which expands as

[pic]

is a power series centered at [pic]. This is known as a Taylor Series.

Taylor Series at x = a

Let [pic]be a function with derivatives of all orders on some open interval containing a. The Taylor Series expansion about [pic]is

[pic]

which is equal to [pic].

The partial sum given by

[pic]

is called a Taylor polynomial of order k at [pic].

Taylor's Formula with Remainder

Let [pic]have all order of derivatives in an interval between a and x. If x is any number in the interval, then there is a number c between a and x such that

[pic][pic]

The Integral Test for Convergence

Suppose that [pic]is a continuous, positive, decreasing function on the interval [pic] and let [pic]. Then the series defined by [pic]is convergent if and only if the improper integral [pic]is convergent. In other words,

If [pic]is convergent, then [pic]is convergent.

or

If [pic]is divergent, then [pic]is divergent.

p-series Test for Convergence

The p-series [pic] converges if [pic]and diverges if [pic].

The nth Term Test for Divergence

The series [pic]diverges if [pic]is not equal to zero of fails to exist.

The Direct Comparison Test for Series

Let [pic]be a series with no negative terms

Case 1. If [pic]for all [pic]and if [pic]converges, then [pic]also converges.

Case 2. If [pic]for all [pic]and if [pic]diverges, then [pic]also diverges.

Absolute Convergence

If the series [pic]converges, then [pic]must also converge and [pic]is said to converge absolutely.

The Ratio Test for Absolute Convergence

Let [pic]be a series with positive terms and let [pic].

Case 1. If L < 1, the series is absolutely convergent.

Case 2. If L >1, the series is divergent.

Case 3. If L = 1, the test is inconclusive. Apply a different test.

The Limit Comparison Test (LCT)

Let [pic] for all [pic].

Case 1. If [pic]then [pic]both converge or both diverge.

Case 2. If [pic] converges, then [pic]converges.

Case 3. If [pic]diverges, then [pic]diverges.

The Alternating Series Test

If the series [pic]

satisfies all three of the following conditions:

1. Each term [pic]is positive

2. [pic]

3. [pic]

then the series is convergent.

The Ratio Test for Absolute Convergence

Let [pic]be a series with positive terms and let [pic].

Case 1.: If L < 1, the series is absolutely convergent.

Case 2. If L >1, the series is divergent.

Case 3. If L = 1, the test is inconclusive. Apply a different test.

How to Test a Power Series for Convergence

1. Use the Ratio Test to find [pic]

2. Set the absolute value of the limit less than 1 and solve for the open interval.

3. Evaluate the series when x equals the right endpoint of the open interval. If this is a finite value, the series converges at the right endpoint. Otherwise, the interval is open on the right side.

4. Evaluate the series when x equals the left endpoint of the open interval. If this is a finite value, the series converges at the left endpoint. Otherwise, the interval is open on the left side.

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