Section 10.5 Reading Assignment: Absolute Convergence; The Ratio and Root Tests

Answer Only Exercise 1, 2, and 3 by using a screenshot provided Calculus Pearson textbook. Make sure you read these three questions very carefully and see on what it is asking for and what is really about. Please be very careful with this assignment.

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Section 10.5 Reading Assignment: Absolute Convergence; The Ratio and Root Tests Instructions. Read through this assignment and complete the three exercises below by reading the appropriate passages of the textbook. Exercise 1. Read Example 1, part a (p. 612). Explain how the "corresponding series of absolute values" (p. 612) changes the original series in this specific case. Explain this without using the phrase "absolute value" for this. Consider the role of subtraction and addition when writing out the series. Notice the "Caution" note in the sidebar on p. 612. We will talk about conditionally convergent series next section, which are series that converge but don't converge absolutely. Be sure to carefully read Theorem 13 - The Ratio Test (p. 613) and Theorem 14 - The Root Test (p. 615). Pay special attention to the cases. Limit is less than 1 means converges absolutely, limit larger than 1 means diverges, and limit equal to 1 is inconclusive. An inconclusive result means a different test should be used. Inconclusive is generally not an acceptable final answer on assignments or exams." An interesting fact is that the ratio and root test will result in the same limit. The choice between the two tests is just a matter of which one is easier to compute. Notice that we don't really have a method for taking limits of the nth root of factorials, so any factorials are a good sign to use the ratio test. Exercise 2. Read Example 2(b) (p. 614). How does the term (2n + 2)! expand in this example? Use the definition of factorial as the product of the first positive integers to explain why this expansion is true.Factorials do not work algebraically in the way that some people want it to as this example shows. It's worth being careful whenever you have a complicated expression in a factorial. Remembering the meaning of factorial can help with this. Exercise 3. Read Example 4 (p. 616). Explain how the limit lim Vn = 1 helps with the n-+00 computation for parts (a) and (b). Be sure to explain to explain the algebraic work for computing related limits in this example. It might be worth exploring the limit of nth roots of various expressions to use the root test. Notice that the nth root of any polynomial or rational function will approach 1 as n goes to infinity. Similarly, the nth root of a logarithm will be approach 1 as n goes to infinity as well. Though not knowing is often an acceptable answer in reality, there are plenty of series whose convergence are open questions to this day. Assignments and exams will definitely have known answers though. We can use asymptotic equality to compute this, but this is considered an advanced technique (Stirling's approximation), so we will avoid it. This can be seen by combining asymptotic equality with the observation in Exercise 3.Chapter 10 Infinite Sequences and Series 612 Chapter 10 Infinite Sequences and Series diverges (since [r) = 5/4 > 1). In series (1), there is some cancelation in the partial sums, which may be assisting the convergence property of the series. However, if we make all of the terms positive in series (1) to form the new series 5+3+5 we see that it still converges. For a general series with both positive and negative terms, we can apply the tests for convergence studied before to the series of absolute values of its terms. In doing so, we are led naturally to the following concept. DEFINITION A series _a, converges absolutely (is absolutely convergent) if the corresponding series of absolute values, _ a,|, converges. So the geometric series (1) is absolutely convergent. We observed, too, that it is also con- vergent. This situation is always true: An absolutely convergent series is convergent as well, which we now prove. Caution THEOREM 12-The Absolute Convergence Test Be careful when using Theorem 12. A convergent series need not converge It Elal converges, then Ed, converges. absolutely, as you will see in the next section. Proof For each n. -asasad, so 0santa seal If Ex Ja, converges, then Ex 2/a,,| converges and, by the Direct Comparison Test, the nonnegative series 20 (a, + a,) converges. The equality a,, = (a, + [a,|) - a. now lets us express Yo" 210. as the difference of two convergent series: Ean = E (a, + lal - la,l) = E(a, + lal) - Elal. Therefore, 24, converges.EXAMPLE 1 This example gives two series that converge absolutely. (a) For $1+ = 1- 4 + 16 + ..., the corresponding series of absolute values is the convergent series iMe = 1 + 4 + + 16 + The original series converges because it converges absolutely. (b) For sinn _ sin 1 _ sin 2 _ sin 3 4 9 + .-., which contains both positive and negative terms, the corresponding series of absolute values is sinn sin 1 sin 2 iME + . . . 4 which converges by comparison with _ _ (1/w') because | sinn| = 1 for every n. The original series converges absolutely; therefore it converges.Chapter 10 Infinite Sequences and Series 10.5 Absolute Convergence; The Ratio and Root Tests 613 The Ratio Test The Ratio Test measures the rate of growth (or decline) of a series by examining the ratio a,+1/0. For a geometric series > or", this rate is a constant ((art Mar") = r), and the series converges if and only if its ratio is less than 1 in absolute value. The Ratio Test is a powerful rule extending that result. THEOREM 13-The Ratio Test Let _ a, be any series and suppose that p is the Greek lowercase letter tho, lim = P. which is pronounced "row." Then (a) the series converges absolutely if p 1 or p is infinite, (c) the test is inconclusive if p = 1. Proof (a) p a,, is absolutely convergent. (b) 1 1 or p is infinite, (c) the test is inconclusive if p = 1.Proof (a) p 1, so that |al > 1 for n > M. The terms of the series do not converge to zero. The series di- verges by the nth-Term Test. (c) p = 1. The series )_, (1) and _ (1 ) show that the test is not conclusive when p = 1. The first series diverges and the second converges, but in both cases Va-1.Chapter 10 Infinite Sequences and Series 616 Chapter 10 Infinite Sequences and Series J n/2". # odd EXAMPLE 3 Consider again the series with terms a, = 11/20 * even. Does Za, converge? Solution We apply the Root Test, finding that Mall = [ Vn/2. n odd 1/2. n even. Therefore. is Val syn Since Vn - 1 (Section 10.1, Theorem 5), we have lim,_Va,| = 1/2 by the Sandwich Theorem. The limit is less than 1, so the series converges absolutely by the Root Test. I EXAMPLE 4 Which of the following series converge, and which diverge? (a) Ma (b) (c) Solution We apply the Root Test to each series, noting that each series has positive terms. (a) converges because 1- 2