ࡱ> S\s@ bjbj "uu:~ppp,c,c,c,ce`fzzzWWW߷$RhpzUWzzzz(zp߷z߷gpf a,cPS00`, pSVaW,;$UY бYSequences 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  EMBED Equation.DSMT4 , and decreasing fractions  EMBED Equation.DSMT4 . 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,  EMBED Equation.DSMT4 . So  EMBED Equation.DSMT4 means the first term of some sequence.  EMBED Equation.DSMT4  is the tenth term of a sequence.  EMBED Equation.DSMT4  is the nth term or general term of a sequence.  EMBED Equation.DSMT4  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.  SHAPE \* MERGEFORMAT  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.  SHAPE \* MERGEFORMAT  Example 1. Let  EMBED Equation.DSMT4 . Does the sequence converge or diverge? We use a function f(x) that mimics the sequence, so we choose  EMBED Equation.DSMT4  EMBED Equation.DSMT4  We check  EMBED Equation.DSMT4 . From the chapter on limits, we recall that  EMBED Equation.DSMT4  Since  EMBED Equation.DSMT4 converged to 1, we know that the sequence  EMBED Equation.DSMT4 must also converge to 1. We have shown that converges.  Figure  SEQ Figure \* ARABIC 1 The graph of the sequence supports this result. The values of the sequence as indicated by the points seem to level off at  EMBED Equation.DSMT4 . 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  EMBED Equation.DSMT4 converge? Rewriting the sequence as a corresponding function of x, we get  EMBED Equation.DSMT4 . Applying the limit, we get  EMBED Equation.DSMT4  Since  EMBED Equation.DSMT4 is divergent,  EMBED Equation.DSMT4 is also divergent. Series A series is a sequence of partial sums. Symbolically, we write  EMBED Equation.DSMT4 . Let's examine a series term by term to see how the individual terms relate to each other. Example 3 Suppose that  EMBED Equation.DSMT4 , write the first five terms of  EMBED Equation.DSMT4   EMBED Equation.DSMT4  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  SHAPE \* MERGEFORMAT  Power Series In some of your earlier math classes you worked with a geometric sequences and series such as the sequence  EMBED Equation.DSMT4 and its related series  EMBED Equation.DSMT4  . 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  EMBED Equation.DSMT4 converges. We can rewrite this series as  EMBED Equation.DSMT4 which is a geometric series with a common ratio of EMBED Equation.DSMT4 . We recall the sum of any geometric series can be found by using the formula  EMBED Equation.DSMT4 . In this example  EMBED Equation.DSMT4 . The sum is  EMBED Equation.DSMT4 .  SHAPE \* MERGEFORMAT  Example 5. Check the series  EMBED Equation.DSMT4 for convergence. The first term is  EMBED Equation.DSMT4 and the common ratio is  EMBED Equation.DSMT4 . By the formula,  EMBED Equation.DSMT4  . This is a finite number so the series converges. Example 6. Does the series  EMBED Equation.DSMT4 converge? The first term is  EMBED Equation.DSMT4 , and the common ratio is  EMBED Equation.DSMT4 , 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  EMBED Equation.DSMT4  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  EMBED Equation.DSMT4 whenever EMBED Equation.DSMT4 . This expression is very powerful. It lets us describe a non-polynomial function as a polynomial function. As the figure shows, the partial sum,  EMBED Equation.DSMT4 , of the polynomial series converges very closely to  EMBED Equation.DSMT4  as long as  EMBED Equation.DSMT4 . It diverges quickly when x is outside the interval. As we increase the number of terms in partial sums, the convergence gets stronger.  Figure  SEQ Figure \* ARABIC 2 Although we use a polynomial to approximate the rational function, the expression  EMBED Equation.DSMT4 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  EMBED Equation.DSMT4 . Determine the interval of convergence. We rewrite the function as  EMBED Equation.DSMT4 . The ratio is -x. The expansion is given by  EMBED Equation.DSMT4 . The figure shows the relationship between f(x) and  EMBED Equation.DSMT4 on the interval  EMBED Equation.DSMT4 .  Figure  SEQ Figure \* ARABIC 3 After we have a power series representation, we can manipulate using techniques we have learned in algebra or calculus classes. Example 9. Since  EMBED Equation.DSMT4 , find a power series representation for  EMBED Equation.DSMT4 . We get a little sneaky here. Notice that  EMBED Equation.DSMT4 just happens to be the derivative of  EMBED Equation.DSMT4 . We can take the derivative of both sides of the equation.  EMBED Equation.DSMT4  As a result, we have a power series representation for a new function,  EMBED Equation.DSMT4 . We can use integration as well as differentiation to manipulate both sides of an equation. Example 10. The power series representation for  EMBED Equation.DSMT4 . Integrate both sides to determine a new power series.  EMBED Equation.DSMT4  We now have a power series representation for arctan x. Example 11. Find a power series representation for  EMBED Equation.DSMT4 . This problem looks similar to Example 8. Unfortunately, it is not quite the same. But we can rewrite our problem as  EMBED Equation.DSMT4 . We already have a power series representation for  EMBED Equation.DSMT4 . We can multiply both sides by x to get the new power series.  EMBED Equation.DSMT4  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.  SHAPE \* MERGEFORMAT  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  EMBED Equation.DSMT4 with the following characteristics at  EMBED Equation.DSMT4 .  EMBED Equation.DSMT4  We see immediately that EMBED Equation.DSMT4 . Since  EMBED Equation.DSMT4 . The other terms are a little harder to find. First, let's take the first four derivatives of P(x).  EMBED Equation.DSMT4  We can evaluate each derivative at x = 0 and set it equal to the given value.  EMBED Equation.DSMT4  With the value for each constant now determined, we can write our customized polynomial:  EMBED Equation.DSMT4 . Example 13. Given EMBED Equation.DSMT4 , construct a polynomial in the form  EMBED Equation.DSMT4 centered about  EMBED Equation.DSMT4 . Since we are centered about  EMBED Equation.DSMT4 , we need to rewrite the polynomial as  EMBED Equation.DSMT4 . First, we evaluate the function at  EMBED Equation.DSMT4 . Then we find the first four derivatives and evaluate each at  EMBED Equation.DSMT4 .  EMBED Equation.DSMT4   EMBED Equation.DSMT4  From our work in Example 12 we can find the derivatives of the polynomial  EMBED Equation.DSMT4  When we evaluate each derivative at  EMBED Equation.DSMT4 , each factor  EMBED Equation.DSMT4 will become zero. We can find each of the constants.  EMBED Equation.DSMT4  We complete the problem by substituting the values for the constants. The answer is  EMBED Equation.DSMT4 . This polynomial does not equal  EMBED Equation.DSMT4 . 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  EMBED Equation.DSMT4 . Example 12 was expanded about  EMBED Equation.DSMT4 . Any Taylor polynomial expanded about  EMBED Equation.DSMT4 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.  SHAPE \* MERGEFORMAT  Example 14. Find the Taylor Series expansion for  EMBED Equation.DSMT4 . First, we find all the coefficients by using the definition at  EMBED Equation.DSMT4   EMBED Equation.DSMT4  You can see that the higher-order derivatives of the EMBED Equation.DSMT4 will repeat in the preceding pattern. The Taylor Series expansion of EMBED Equation.DSMT4 is given by  EMBED Equation.DSMT4  Test Tip. It is easy to mix up the position of the terms. The first term is  EMBED Equation.DSMT4 . 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  EMBED Equation.DSMT4  which makes them Maclaurin Series. To convert to a general Taylor Series about  EMBED Equation.DSMT4 , substitute  EMBED Equation.DSMT4 . The first term will no longer be  EMBED Equation.DSMT4 . Seven Well-known Power Series (with Intervals of Convergence)  EMBED Equation.DSMT4   EMBED Equation.DSMT4   EMBED Equation.DSMT4   EMBED Equation.DSMT4   EMBED Equation.DSMT4   EMBED Equation.DSMT4   EMBED Equation.DSMT4  Example 15. Find a power series representation for  EMBED Equation.DSMT4 . We will use the power series for EMBED Equation.DSMT4 , substituting  EMBED Equation.DSMT4 .  EMBED Equation.DSMT4  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.  SHAPE \* MERGEFORMAT  The function  EMBED Equation.DSMT4 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 EMBED Equation.DSMT4 . Estimate the error bound in the approximation. Using only the first three terms means we want a third order Taylor polynomial which gives us EMBED Equation.DSMT4 . Evaluating the polynomial at x = 0.1, we get  EMBED Equation.DSMT4  The next term in the series, the error bound is less than  EMBED Equation.DSMT4 . The maximum Error Bound is less than  EMBED Equation.DSMT4 . Example 17. Use a power series to find  EMBED Equation.DSMT4 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  EMBED Equation.DSMT4  Summing the first three terms of the answer, we get  EMBED Equation.DSMT4 . The answer to four decimal places is 0.3103. The error bound is less than  EMBED Equation.DSMT4 . 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  EMBED Equation.DSMT4 . 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. n5505005000Sn1.18571.20191.20205491.2020569 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.  SHAPE \* MERGEFORMAT  We apply the Integral Test to our example. The improper integral associated with  EMBED Equation.DSMT4  which we know converges by the p-series rule we developed in the last chapter. Therefore, the series  EMBED Equation.DSMT4 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  EMBED Equation.DSMT4 converged if  EMBED Equation.DSMT4 . The Integral Test for Convergence allows us to formalize the rule for series.  SHAPE \* MERGEFORMAT  Example 18. Does  EMBED Equation.DSMT4 converge? Beginning the sequence at n=4 is okay. If a series converges on the interval  EMBED Equation.DSMT4 , 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 EMBED Equation.DSMT4 , we can safely say the series will converge. The next test is used to confirm divergence. The nth Term Test for Divergence  SHAPE \* MERGEFORMAT  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  EMBED Equation.DSMT4 diverges. As n gets larger,  EMBED Equation.DSMT4 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.  SHAPE \* MERGEFORMAT  As before, if the series  EMBED Equation.DSMT4 converges to a limit, it "squeezes" all series smaller that it to the same limit. If the series  EMBED Equation.DSMT4 diverges, it "pushes" all series that are larger away from a limit. Example 20. Show that  EMBED Equation.DSMT4 converges for all real values of x. There are no negative terms in the series and we know that  EMBED Equation.DSMT4 . We recognize  EMBED Equation.DSMT4 as the Taylor Series for  EMBED Equation.DSMT4 , which we know converges. Therefore,  EMBED Equation.DSMT4 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.  SHAPE \* MERGEFORMAT  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.  SHAPE \* MERGEFORMAT  Example 21. Determine if the series  EMBED Equation.DSMT4 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.  EMBED Equation.DSMT4  EMBED Equation.DSMT4  Find the limit of the ratio. We did not use absolute value signs here, since all terms were already positive.  EMBED Equation.DSMT4  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 EMBED Equation.DSMT4 . 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.  SHAPE \* MERGEFORMAT  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  EMBED Equation.DSMT4  All the terms are positive and we see that the function is somewhat like a p-series. As n grows very large,  EMBED Equation.DSMT4 . So we will compare our series with the series  EMBED Equation.DSMT4  EMBED Equation.DSMT4 .  EMBED Equation.DSMT4  Since the limit is greater than one and the series EMBED Equation.DSMT4  diverges, the series  EMBED Equation.DSMT4  must also diverge. Example 23. Check the convergence of EMBED Equation.DSMT4  Ignoring the constant in the denominator, we see that  EMBED Equation.DSMT4   EMBED Equation.DSMT4 . We will try the LCT.  EMBED Equation.DSMT4  The limit is a constant and  EMBED Equation.DSMT4 converges, so  EMBED Equation.DSMT4 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.  SHAPE \* MERGEFORMAT  We can visualize the reason this test works. Suppose we have the alternating series  EMBED Equation.DSMT4 . As the figure shows, the first sum moves to right  EMBED Equation.DSMT4 , then moves back  EMBED Equation.DSMT4 , then moves right  EMBED Equation.DSMT4 , back  EMBED Equation.DSMT4 , 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.  Figure 5 Example 24. Determine if  EMBED Equation.DSMT4 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  EMBED Equation.DSMT4 ? - yes 3. Does  EMBED Equation.DSMT4 ? - no, the limit is  EMBED Equation.DSMT4 . By the nth term test, we see the series diverges. Example 25. Does  EMBED Equation.DSMT4 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  EMBED Equation.DSMT4 ? - This is difficult to tell by inspection. So we get sneaky.  EMBED Equation.DSMT4  The answer to the second condition is yes. 3. Is  EMBED Equation.DSMT4 ? -- 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.  SHAPE \* MERGEFORMAT  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.  SHAPE \* MERGEFORMAT  Example 26. Given the series  EMBED Equation.DSMT4 , find the interval of convergence and the radius of convergence. Use the Ratio Test.  EMBED Equation.DSMT4  Find the limit.  EMBED Equation.DSMT4  Set the absolute value of the limit less than 1.  EMBED Equation.DSMT4  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  EMBED Equation.DSMT4  Case 2. Check left endpoint x = -3. When x = -3, the series becomes  EMBED Equation.DSMT4  Since the power series does not converge at either endpoint, the interval of convergence is EMBED Equation.DSMT4 . The radius of convergence is the distance from the center of the interval to one endpoint. In this example the center is at EMBED Equation.DSMT4 . The distance from  EMBED Equation.DSMT4  to  EMBED Equation.DSMT4 is 5. So, 5 is the radius of convergence centered at  EMBED Equation.DSMT4 . Example 27. Find the interval of convergence for EMBED Equation.DSMT4 . Find the limit using the Ratio Test.  EMBED Equation.DSMT4  Since the limit is zero, the series converges absolutely for all values of x. The interval of convergence is  EMBED Equation.DSMT4 . 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  EMBED Equation.DSMT4  Use the Ratio Test to establish the limit, then check the limit against the intervals of convergence.  EMBED Equation.DSMT4  The series will converge when  EMBED Equation.DSMT4 . Next we check endpoints for convergence. Case 1. x = 1. When x = 1, the series becomes  EMBED Equation.DSMT4  Case 2. x = -1. When x = -1, the series becomes  EMBED Equation.DSMT4  Therefore, the interval of convergence is  EMBED Equation.DSMT4 . 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, dont 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. If you can see quickly that  EMBED Equation.DSMT4 , check for divergence. If the series fits the form  EMBED Equation.DSMT4 , it is a p-series. If  EMBED Equation.DSMT4 , the series converges. If  EMBED Equation.DSMT4 , the series diverges. If the series fits the form  EMBED Equation.DSMT4 , then try the Alternating Series Test. If the series fits the form  EMBED Equation.DSMT4 , then it is a geometric series. If  EMBED Equation.DSMT4 , the series converges. If  EMBED Equation.DSMT4 , the series diverges. If a series that has factorials or other products, try the Ratio Test. 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  EMBED Equation.DSMT4 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  EMBED Equation.DSMT4  If there is a function  EMBED Equation.DSMT4 that corresponds to  EMBED Equation.DSMT4 , then try the Integral Test on the improper integral  EMBED Equation.DSMT4 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. Dont 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.  EMBED Equation.DSMT4  Since the  EMBED Equation.DSMT4 , check for divergence. Example 30.  EMBED Equation.DSMT4   EMBED Equation.DSMT4 is an algebraic function; we can compare it to a convenient p-series. But first we need to manipulate it.  EMBED Equation.DSMT4 . The comparison series should be  EMBED Equation.DSMT4 . Example 31.  EMBED Equation.DSMT4  This is an alternating series; use the Alternating Series Test. Example 32.  EMBED Equation.DSMT4  The presence of factorials suggests that we try the Ratio Test. Example 33.  EMBED Equation.DSMT4  This series fits very nicely with  EMBED Equation.DSMT4  use the Integral Test. Example 34.  EMBED Equation.DSMT4  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.  EMBED Equation.DSMT4  2.  EMBED Equation.DSMT4  3.  EMBED Equation.DSMT4  4.  EMBED Equation.DSMT4  Check the following series for convergence. If the series contains negative terms, determine if it is absolutely convergent, conditionally convergent, or divergent. 5.  EMBED Equation.DSMT4  6.  EMBED Equation.DSMT4  7.  EMBED Equation.DSMT4  8.  EMBED Equation.DSMT4  9.  EMBED Equation.DSMT4  10.  EMBED Equation.DSMT4  11.  EMBED Equation.DSMT4  12.  EMBED Equation.DSMT4  13.  EMBED Equation.DSMT4  14.  EMBED Equation.DSMT4  15.  EMBED Equation.DSMT4  16.  EMBED Equation.DSMT4  17.  EMBED Equation.DSMT4  18.  EMBED Equation.DSMT4  Write the first four terms and the general term for a Maclaurin Series representation for the following functions. 19.  EMBED Equation.DSMT4  20.  EMBED Equation.DSMT4  21.  EMBED Equation.DSMT4  22.  EMBED Equation.DSMT4  23.  EMBED Equation.DSMT4  24.  EMBED Equation.DSMT4  25.  EMBED Equation.DSMT4  Find the first four nonzero terms and the general term for the Taylor Series generated by  EMBED Equation.DSMT4  26.  EMBED Equation.DSMT4  27.  EMBED Equation.DSMT4  28.  EMBED Equation.DSMT4  29.  EMBED Equation.DSMT4  30. Let  EMBED Equation.DSMT4  be the fourth-order Taylor polynomial for the function  EMBED Equation.DSMT4 centered at  EMBED Equation.DSMT4 . (a) Find  EMBED Equation.DSMT4  (b) Write the third-order Taylor polynomial for  EMBED Equation.DSMT4 and use it to approximate  EMBED Equation.DSMT4 . (c) Is it possible to find the exact value of  EMBED Equation.DSMT4  from the information given. Explain. (d)  EMBED Equation.DSMT4  Write the fourth-order Taylor polynomial representation for  EMBED Equation.DSMT4  at x = -3. 31.  EMBED Equation.DSMT4  (a) Write the first four terms and the general term of the Taylor Series generated by  EMBED Equation.DSMT4 . (b) Use the results from part (a) to write the first four terms and general term for  EMBED Equation.DSMT4 . 32.  EMBED Equation.DSMT4 . (a) Write the first four terms and the general term of the Taylor Series generated by  EMBED Equation.DSMT4 . (b) What is the interval of convergence of the series found in part (a). Show your reasoning. 33. Let  EMBED Equation.DSMT4 be a function with derivatives of all orders with the following properties:  EMBED Equation.DSMT4 . (a) Write a third-order Taylor polynomial for  EMBED Equation.DSMT4 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  EMBED Equation.DSMT4  at x = 6? 34. The first two terms for the power series for  EMBED Equation.DSMT4 . For what values of x will the error bound be less than EMBED Equation.DSMT4 . Explain your reasoning. Limit of a Sequence A sequence  EMBED Equation.DSMT4 has a limit L, or converges to L, if  EMBED Equation.DSMT4 . If such a number L does not exist, the sequence has no limit or diverges. Let  EMBED Equation.DSMT4 be a sequence with  EMBED Equation.DSMT4  and suppose that  EMBED Equation.DSMT4 exists for every real number EMBED Equation.DSMT4 . Then the following cases apply. Case 1.  EMBED Equation.DSMT4  Case 2.  EMBED Equation.DSMT4  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.  EMBED Equation.DSMT4  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  EMBED Equation.DSMT4  We can use  EMBED Equation.DSMT4 notation to write this in a short form,  EMBED Equation.DSMT4 . This series converges if  EMBED Equation.DSMT4 and diverges if  EMBED Equation.DSMT4 . Definition of a Power Series An expression in the form  EMBED Equation.DSMT4 , which expands as  EMBED Equation.DSMT4  is a power series centered at  EMBED Equation.DSMT4 . This is known as a Maclaurin Series. An expression in the form  EMBED Equation.DSMT4 , which expands as  EMBED Equation.DSMT4  is a power series centered at  EMBED Equation.DSMT4 . This is known as a Taylor Series. Taylor Series at x = a Let  EMBED Equation.DSMT4 be a function with derivatives of all orders on some open interval containing a. The Taylor Series expansion about  EMBED Equation.DSMT4 is  EMBED Equation.DSMT4  which is equal to  EMBED Equation.DSMT4 . The partial sum given by  EMBED Equation.DSMT4  is called a Taylor polynomial of order k at  EMBED Equation.DSMT4 . Taylor's Formula with Remainder Let  EMBED Equation.DSMT4 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  EMBED Equation.DSMT4  EMBED Equation.DSMT4  The Integral Test for Convergence Suppose that  EMBED Equation.DSMT4 is a continuous, positive, decreasing function on the interval  EMBED Equation.DSMT4  and let  EMBED Equation.DSMT4 . Then the series defined by  EMBED Equation.DSMT4 is convergent if and only if the improper integral  EMBED Equation.DSMT4 is convergent. In other words, If  EMBED Equation.DSMT4 is convergent, then  EMBED Equation.DSMT4 is convergent. or If  EMBED Equation.DSMT4 is divergent, then  EMBED Equation.DSMT4 is divergent. p-series Test for Convergence The p-series  EMBED Equation.DSMT4  converges if  EMBED Equation.DSMT4 and diverges if  EMBED Equation.DSMT4 . The nth Term Test for Divergence The series  EMBED Equation.DSMT4 diverges if  EMBED Equation.DSMT4 is not equal to zero of fails to exist. The Direct Comparison Test for Series Let  EMBED Equation.DSMT4 be a series with no negative terms Case 1. If  EMBED Equation.DSMT4 for all  EMBED Equation.DSMT4 and if  EMBED Equation.DSMT4 converges, then  EMBED Equation.DSMT4 also converges. Case 2. If  EMBED Equation.DSMT4 for all  EMBED Equation.DSMT4 and if  EMBED Equation.DSMT4 diverges, then  EMBED Equation.DSMT4 also diverges. Absolute Convergence If the series  EMBED Equation.DSMT4 converges, then  EMBED Equation.DSMT4 must also converge and  EMBED Equation.DSMT4 is said to converge absolutely. The Ratio Test for Absolute Convergence Let  EMBED Equation.DSMT4 be a series with positive terms and let  EMBED Equation.DSMT4 . 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  EMBED Equation.DSMT4  for all  EMBED Equation.DSMT4 . Case 1. If  EMBED Equation.DSMT4 then  EMBED Equation.DSMT4 both converge or both diverge. Case 2. If  EMBED Equation.DSMT4  converges, then  EMBED Equation.DSMT4 converges. Case 3. If  EMBED Equation.DSMT4 diverges, then  EMBED Equation.DSMT4 diverges. The Alternating Series Test If the series  EMBED Equation.DSMT4  satisfies all three of the following conditions: 1. Each term  EMBED Equation.DSMT4 is positive 2.  EMBED Equation.DSMT4  3.  EMBED Equation.DSMT4  then the series is convergent. The Ratio Test for Absolute Convergence Let  EMBED Equation.DSMT4 be a series with positive terms and let  EMBED Equation.DSMT4 . 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  EMBED Equation.DSMT4  2. Set the absolute value of the limit less than 1 and solve for the open interval. 3. 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FMathType 5.0 Equation MathType EFEquation.DSMT49q$DSMT5WinAllBasicCodePagesTimes New RomanSymbolCourier NewMT Extra!/'_!!/G_Ole JCompObjKiObjInfoMEquation Native NAPAPAE%B_AC_A %!AHA_D_E_E_A   12 FMathType 5.0 Equation MathType EFEquation.DSMT49qtDSMT5WinAllBasicCodePagesTimes New RomanSymbolCourier NewMT Extra!/'_!!/G__1121088897FaaOle RCompObjSiObjInfoUEquation Native V_1121088911FaaOle ZCompObj[iAPAPAE%B_AC_A %!AHA_D_E_E_A   13 FMathType 5.0 Equation MathType EFEquation.DSMT49qdDSMT5WinAllBasicCodePagesTimes New RomanSymbolCourier NewMT Extra!/'_!!/G_ObjInfo]Equation Native ^_1121088926FaaOle bAPAPAE%B_AC_A %!AHA_D_E_E_A   14 FMathType 5.0 Equation MathType EFEquation.DSMT49qlDSMT5WinAllBasicCodePagesTimes New RomanSymbolCourier NewMT Extra!/'_!!/G_CompObjciObjInfoeEquation Native f_1121089556FaaAPAPAE%B_AC_A %!AHA_D_E_E_A   15 FMathType 5.0 Equation MathType EFEquation.DSMT49qVDSMT5WinAllBasicCodePagesTimes New RomanSymbolCourier NewMT Extra!/'_!!/G_Ole jCompObjkiObjInfomEquation Native nrAPAPAE%B_AC_A %!AHA_D_E_E_A  p"-1() n"-1  3n4n++2 n==1" " FMathType 5.0 Equation MathType EFEquation.DSMT49q_1121090847FaaOle tCompObjuiObjInfowEquation Native x_1121090874FaaOle }CompObj~ilDSMT5WinAllBasicCodePagesTimes New RomanSymbolCourier NewMT Extra!/'_!!/G_APAPAE%B_AC_A %!AHA_D_E_E_A  a n e"a n++1 FMathType 5.0 Equation MathType EFEquation.DSMT49qObjInfoEquation Native E_1121090951FaaOle )$DSMT5WinAllBasicCodePagesTimes New RomanSymbolCourier NewMT Extra!/'_!!/G_APAPAE%B_AC_A %!AHA_D_E_E_A  lim n!"  3n4n++2==0CompObj iObjInfo Equation Native _1121091806 Faa FMathType 5.0 Equation MathType EFEquation.DSMT49q,DSMT5WinAllBasicCodePagesTimes New RomanSymbolCourier NewMT Extra!/'_!!/G_APAPAE%B_AC_A %!AHA_D_E_E_A   34Ole CompObj iObjInfoEquation Native  FMathType 5.0 Equation MathType EFEquation.DSMT49qplDSMT5WinAllBasicCodePagesTimes New RomanSymbolCourier NewMT Extra!/'_!!/G_APAPAE%B_AC_A %!AHA_D_E_E_A  p"-1() n"-1  n 3 n 4 ++1 n==1" " FMathType 5.0 Equation MathType EFEquation.DSMT49q ,DSMT5WinAllBasicCodePagesTimes New RomanSymbolCourier NewMT Extra!/'_!!/G__1126031063FaaOle CompObjiObjInfoEquation Native  _1121092939 yFuauaOle CompObjiAPAPAE%B_AC_A %!AHA_D_E_E_A  We consider the related function f(x)== x 3 x 4 ++1.  If this function decreases, we know the sequence also decreases.  We can check for decreasing function by looking at the derivative of f(x).f(x)== 3x 2 "-x 6 x 4 ++1() 2 .   The denominator is always positive.  We need to see when 3x 2 "-x 6 <<0.  Factoring,we get x 2 (3"-x 4 ).  x 2  is always positive.  (3"-x 4 )<<0!x>> 3 4 .  Therefore, for all n>>2, f(x) is decreasing. 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