Sections 10.1 and 10.2 Reading Assignment: Sequences and Series

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 careful with this assignment.

References: Thomas' Calculus: Early Transcendentals | Calculus | Calculus | Mathematics | Store | Pearson+

Sections 10.1 and 10.2 Reading Assignment: Sequences and Series Instructions. Read through this assignment and complete the three exercises below by reading the appropriate passages of the textbook. The subsection "Representing Sequences" (p. 578 -579) goes over the notation for sequences. Notice how sequences are denoted using a subscript (called the index) to indicate the place in the sequence. Also notice the sequence {d} near the bottom of p. 578 bounces back and forth between +1 and -1. This is a useful sequence that we will see more of later on. We should consider how to compute limits of sequences. The limits of sequences work just like limits of functions at infinity. Check out the subsections "Calculating Limits of Sequences" (p. 581 -582), "Using L'Hopital's Rule" (p. 582 -583) and "Commonly Occurring Limits" (p. 584). Moving on to Section 10.2, series are formed by adding up all the numbers in a sequence. It's important to keep these ideas clearly separated: a sequence is the list of numbers, while a series is the addition of those numbers. One of the big things is the use of Sigma notation to indicate a sum. The capital Greek letter E (Sigma) is used to denote a sum of numbers. The lower variable and value indicate the indexing variable (which tells you what numbers to add) along with what number to begin with the count. We typically begin with n = 0 or n = 1, though others are used too. The letter n too can be replaced depending on the situation, but the letter there tells you what the indexing variable is. The top number indicates when we stop, but typically we will deal with infinite sums, so that value will usually be infinity for us.Partial sums, where we start at the beginning and start adding but stop at some finite point, will be a natural choice of numerical approximation. We will discuss this more in later sections. We consider a group ofexamples called geometric series. which are nicely behaved. Exercise 1. Read the subsection \"Geometric Series\" ip. 592 593}. State what a geometric series is and when it converges. Geometric series are special and should not be treated as the norm. We typically can't find the value of a series. and it's generally even harder to find the value of the series by looking at their partial sums. Dur emphasis for the next few sections will be on determining whether a given series converges or diverges. Exercise 2. Read "The nthTerm Test for Divergence\" (boxed near the bottom of p. 594} and Example T (p. 59:1 595}. Based on the different parts of this example, explain the different ways the nth-Term Test can apply. The indirect nature of these tests can make these difcult to tell if they're used correctly. The biggest issue with this test in particular is that it's a oneway street. This causes some confusion. Exercise 3. Read the "Caution\" note in the margin of p. 594. Explain what it is that the nth term test doesn't say.2 1I will be strict on this distinction during grading. So be sure to keep that distinction in mind when writing. 2 This is a huge issue. Claiming that the nth term test applies when it doesn't isn't just a common mistake; but also a big one. Considering this a small mistake is an even worse mistake; it's missing the whole point. Chapter 10 Infinite Sequences and Series 578 Chapter 10 Infinite Sequences and Series Representing Sequences A sequence is a list of numbers in a given order. Each of a, dy, d, and so on represents a number. These are the terms of the sequence. For example, the sequence 2, 4, 6, 8, 10, 12, . .., 2n, ... has first term a, = 2, second term a, = 4, and ath term a,, = 2n. The integer n is called the index of a,, and indicates where a, occurs in the list. Order is important. The sequence 2. 4, 6, 8 . . . is not the same as the sequence 4, 2. 6, 8 . . .. We can think of the sequence as a function that sends 1 to a, 2 to az, 3 to a,, and in general sends the positive integer n to the nth term a,. More precisely, an infinite sequence of numbers is a function whose domain is the set of positive integers. For example, the function associated with the sequence 2, 4. 6, 8, 10, 12, . .. . 2n, . . . sends 1 to a, = 2,2 to a, = 4, and so on. The general behavior of this sequence is described by the formula a, = 2n. We can change the index to start at any given number n. For example, the sequence 12, 14. 16. 18. 20, 22 . . . is described by the formula a, - 10 + 20, if we start with a = 1. It can also be described by the simpler formula b, = 2n. where the index n starts at 6 and increases. To allow such simpler formulas, we let the first index of the sequence be any appropriate integer, In the sequence above, {o, } starts with a while { b, } starts with be. Sequences can be described by writing rules that specify their terms, such as an = Vn. b, = (-1)+1! do = (-1 )+1or by listing terms: (a,) = {Vi. Vz V3 .... VA... } 1234 1 - (d,) = {1,-1, 1, -1, 1, -1,....(-1)" + ,... ). We also sometimes write a sequence using its rule, as with and Figure 10.1 shows two ways to represent sequences graphically. The first marks the first few points from aj. dy, age . . . . Ope . . . on the real axis. The second method shows the graph of the function defining the sequence. The function is defined only on integer inputs, and the graph consists of some points in the xy-plane located at (1, a ), (2, az). . ... (n, a,), . . . .Chapter 10 Infinite Sequences and Series 10.1 Sequences 579 C2 03 8405 2345 2 3 4 5 a = (-1)"! FIGURE 10.1 Sequences can be represented as points on the real line or as points in the plane where the horizontal axis a is the index number of the term and the vertical axis o, is its value. L-B L L+e Convergence and Divergence Sometimes the numbers in a sequence approach a single value as the index n increases. This happens in the sequence1 1 1 - -(n. a,)-4---- whose terms approach 0 as a gets large, and in the sequence L-E "(N. ay) 123 N whose terms approach 1. On the other hand. sequences like FIGURE 10.2 In the representation of a ( Vi. V2, V3. .... Vn. ... } sequence as points in the plane, a -> L if have terms that get larger than any number as n increases, and sequences like y = L is a horizontal asymptote of the se- {1, -1, 1, -1, 1, -1, ... . (-1"+, . .. } quence of points {(x, a ) ) . In this figure, all the a,'s after ay lie within s of L. bounce back and forth between 1 and - 1. never converging to a single value. The follow- ing definition captures the meaning of having a sequence converge to a limiting value. It says that if we go far enough out in the sequence, by taking the index a to be larger than some value N, the difference between a,, and the limit of the sequence becomes less than any preselected number s > 0. DEFINITIONS The sequence {o, } converges to the number & if for every positive number & there corresponds an integer A such that whenever R > N. If no such number _ exists, we say that {a, } diverges. If { a, } converges to L. we write lim, . (, = L, or simply a, - L. and call L the limit of the sequence (Figure 10.2). HISTORICAL BIOGRAPHY Nicole Oresme The definition is very similar to the definition of the limit of a function f(x) as x tends (ca. 1320-1382) to co (lim, f(x) in Section 2.6). We will exploit this connection to calculate limits of www. goo.gl/r7154z sequences.Chapter 10 Infinite Sequences and Series 10.1 Sequences 581 A sequence may diverge without diverging to infinity or negative infinity, as we saw in Example 2. The sequences { 1. -2, 3, -4.5, -6, 7, -8. . . . } and { 1. 0, 2, 0,3, 0.. . . } are also examples of such divergence. The convergence or divergence of a sequence is not affected by the values of any number of its initial terms (whether we omit or change the first 10, 1000, or even the first million terms does not matter). From Figure 10.2, we can see that only the part of the sequence that remains after discarding some initial number of terms determines whether the sequence has a limit and the value of that limit when it does exist. Calculating Limits of Sequences Since sequences are functions with domain restricted to the positive integers, it is not sur- prising that the theorems on limits of functions given in Chapter 2 have versions for sequences. THEOREM 1 Let {a,, } and {b,, } be sequences of real numbers, and let A and B be real numbers. The following rules hold if lim, .a,, = A and lim, b. = B. 1. Sum Rule: lim,(a, + b, ) = A+ B 2. Difference Rule: lim, (a, - b,) = A - B 3. Constant Multiple Rule: lim, (k . b) = k . B (any number k) A. Product Rule: lim,-(a, ' b, ) = A . B 5. Quotient Rule: On _ A lim, b. B if B # 0 The proof is similar to that of Theorem 1 of Section 2.2 and is omitted.EXAMPLE 3 By combining Theorem I with the limits of Example 1, we have; (a) lim = -1 . lim -=-1.0 =0 Constant Multiple Rule and Example la Difference Rule (b) lim = lim = lim 1 - lim 631-0=1 1-+30 1-+00 11-+00 and Example la (e) lim = 5 . lim * him 1 1 =5.0.0=0 Product Rule Divide numerator and denominator (d) lim 4 - 7m lim (4/m) - 7 0-7 -7. 1-+30 27" + 3 8-+301 + (3/m') 1+0 by at and use the Sum and Quotient Rules, Be cautious in applying Theorem 1. It does not say. for example, that each of the sequences {a, } and {b, } have limits if their sum {a, + b, } has a limit. For instance. {al = {1. 2.3. ...} and {b, } = {-1. -2. -3. .. . } both diverge, but their sum {a, + b ) = {0, 0, 0, . . . } clearly converges to (. One consequence of Theorem 1 is that every nonzero multiple of a divergent sequence {a,} diverges. Suppose, to the contrary, that {ca, } converges for some number c # 0. Then, by taking & = 1/c in the Constant Multiple Rule in Theorem I, we see that the sequence {tocan) = fant converges. Thus, { ca,, } cannot converge unless {a, } also converges. If {a,, } does not converge, then { ca, } does not converge.Chapter 10 Infinite Sequences and Series 582 Chapter 10 Infinite Sequences and Series The next theorem is the sequence version of the Sandwich Theorem in Section 2.2. You are asked to prove the theorem in Exercise 119. (See Figure 10,4.) THEOREM 2-The Sandwich Theorem for Sequences Let {a,). {6, }, and {c ) be sequences of real numbers. If a,, s b, = G. holds for all n beyond some index /, and if lim,-..d, = lim,_.G, = L, then FIGURE 10.4 The terms of sequence lim,-.b, = L also. {b, } are sandwiched between those of {a } and {c } , forcing them to the same common limit L. An immediate consequence of Theorem 2 is that, if b, | s c, and c, -+ 0, then b,, - 0 because -c,, 00, 1 -0 continuous at /. and defined at all a,, then f(a) - f(1.). and 21/" - 2" (Example 6). The terms of (1 ) are shown on the x-axis; the terms of {21/} are shown as the y-values on the EXAMPLE 5 Show that V(n + 1)- 1. graph of f(x) = 25.Solution We know that (n + 1) - 1. Taking ((x) = Vx and L = 1 in Theorem 3 gives V( + 1)/m - VI = 1. EXAMPLE 6 The sequence {1/w} converges to 0. By taking a, = 1, f(x) = 2', and L - 0 in Theorem 3. we see that 21/* = f(1) - f(L) - 2" - 1. The sequence { 21/} converges to 1 (Figure 10.5). Using L'Hopital's Rule The next theorem formalizes the connection between lim, ,a,, and lim, f(x). It enables us to use I'Hopital's Rule to find the limits of some sequences. THEOREM 4 Suppose that f(x) is a function defined for all x 2 n, and that {a, } is a sequence of real numbers such that a = f(n) for n 2 no. Then lim a,, = L whenever lim f(x) = L. X-100Chapter 10 Infinite Sequences and Series 10.1 Sequences 583 Proof Suppose that lim, .f(x) = L. Then for each positive number & there is a number M such that If(x) - 1 8 whenever x > M. Let / be an integer greater than M and greater than or equal to m- Since a, = f(n), it follows that for all n > N we have |an - L| = 15(m) - L| 0) 4. lim x" = 0 (| x | 1 define the sequence 1, 2, 3, . ... n, . . . of positive integers. With a, = 1. we have a, =a, + 1 = 2, dj = a +1 = 3, and so on. (b) The statements a, = 1 and a,, =n'a, , for a > 1 define the sequence 1, 2, 6, 24. ..., a!, ... of factorials. With a, = 1, we have a, = 2.a1 = 2, a; = 3+ 02 = 6, 04 = 4 0; = 24, and so on.Chapter 10 Infinite Sequences and Series 592 Chapter 10 Infinite Sequences and Series L'1 3 S 6 FIGURE 10.9 The sum of a series with positive terms can be interpreted as a total area of an infinite collection of rectangles. The series converges when the total area of the rectangles is finite (a) and diverges when the total area is unbounded (b). Note that the total area can be infinite even if the area of the rectangles is decreasing. Geometric Series Geometric series are series of the form at art art . .. + an-+ ... in which a and r are fixed real numbers and a > 0. The series can also be written as En-our". The ratio r can be positive, as in 1 + : + ... + + .... 7 = 1/2,0 = 1 or negative, as in 1 . + + r = -1/3,a = 1 If r = 1, the ath partial sum of the geometric series is $=at all) + a(ly+ . + all)" != na,and the series diverges because lim,-.5, = 10o, depending on the sign of a. If r = -1, the series diverges because the ath partial sums alternate between a and ( and never approach a single limit. If | | # 1, we can determine the convergence or divergence of the series in the following way: s =atartart..tam- Write the ath partial sum. Multiply & by r. Subtract rs, from s,- Most of the terms on the right cancel. s(1 - ) = all - r) Factor. 5. = 1- r (r # 1 ). We can solve for s, ifr # 1. If |r 1, then |" | - co and the series diverges. If r a,, converges, then a,, - 0. converges if a, -0. It is possible for a series to diverge when a, - 0. (See Example S.) Theorem 7 leads to a test for detecting the kind of divergence that occurred in Example 6. The rith-Term Test for Divergence 2 an diverges if lim a, fails to exist or is different from zero. EXAMPLE 7 The following are all examples of divergent series. (a) > n' diverges because n' - co.Chapter 10 Infinite Sequences and Series 10.2 Infinite Series 595 (b) n+ diverges because " + 1_ lima, * 0 (c) > (-1)" diverges because lim,,_(-1)"+ does not exist. (d) 4 2n + 5 diverges because lim,-.