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10.1 Sequences Contemporary Calculus 1 10.1 SEQUENCES Sequences play important roles in several areas of theoretical and applied mathematics. As you study additional mathematics you will encounter them again. In this course, however, their role is primarily as a foundation for our study of series ("big polynomials"). In order to understand how and where it is valid to represent a function such as sine as a series, we need to examine what it means to add together an infinite number of values. And in order to understand this infinite addition we need to analyze lists of numbers (called sequences) and determine whether or not the numbers in the list are converging to a single value. This section examines sequences, how to represent sequences graphically, what it means for a sequence to converge, and several techniques to determine if a sequence converges. Example 1: A person places $100 in an account that pays 8% interest at the end of each year. How much will be in the account at the end of 1 year, 2 years, 3 years, and n years? Solution: After one year, the total is the principal plus the interest: 100 + (.08)100 = (1.08).100 = $108. At the end of the second year, the amount is 108% of the amount at the start of the second year: (1.08) { (1.08)100 } = (1.08)2 .100 = $116.64 . At the end of the third year, the amount is 108% of the amount at the start of the third year: (1.08) { (1.08)2 100 } = (1.08)3 .100 = $125.97 . These results are shown in Fig. 1. In general, at the end of the nth year, the amount in the account is (1.08)n .100 dollars. Practice 1: A layer of protective film transmits two thirds of the light that reaches that layer. How much of the incoming light is transmitted through 1 layer, 2 layers, 3 layers, and n layers? (Fig. 2) The Example and Practice each asked for a list of numbers in a definite order: a first number, then a second number, and so on. Such a list of numbers in a definite order is called a sequence. An infinite sequence is one that just keeps going and has no last number. Often the pattern of a sequence is clear from the first few numbers, but in order to precisely specify a sequence, a rule for finding the value of the nth term , an ("a sub n") , in the sequence is usually given. Money in account (dollars)42 31 5 6Years160140120100Fig. 1 Fig. 242 31n0.51.0Number of layersLight TransmittedIncoming lightProtective layers1 2 3 4. . . .10.1 Sequences Contemporary Calculus 2 Example 2: List the next two numbers in each sequence and give a rule for calculating the nth number, an: (a) 1, 4, 9, 16, . . . (b) –1, 1, –1, 1, . . . (c) 12 , 14 , 18 , 116 , . . . . Solution: (a) a5 = 25, a6 = 36, and an = n2 . (b) a5 = –1, a6 = 1, and an = (–1)n . (c) a5 = 132 , a6 = 164 , and an = ( 12 )n = 12n . Practice 2: List the next two numbers in each sequence and give a rule for calculating the nth number, an: (a) 1, 12 , 13 , 14 , . . . (b) –12 , 14 , –18 , 116 , . . . (c) 2, 2, 2, 2, . . . Definition and Notation Since a sequence gives a single value for each integer n, a sequence is a function, but a function whose domain is restricted to the integers. Definition A sequence is a function whose domain is all integers greater than or equal to a starting integer. Most of our sequences will have a starting integer of 1, but sometimes it is convenient to start with 0 or another integer value. Notation: The symbol an represents a single number called the nth term . The symbol { an } represents the entire sequence of numbers, the set of all terms. The symbol { rule } represents the sequence generated by the rule. The symbol { an }n = 3 represents the sequence that starts with n = 3. Because sequences are functions, we can add, subtract, multiply, and divide them, and we can combine them with other functions to form new sequences. We can also graph sequences, and their graphs can sometimes help us describe and understand their behavior. Example 3: For the sequences given by an = 3 – 1n and bn = 12n , graph the points (n, an) and (n, bn) for n = 1 to 5. Calculate the first 5 terms of cn = an + bn and graph the points (n, cn). Solution: c1 = ( 3 – 11 ) + ( 121 ) = 2.5, c2 = 2.75, c3 ≈ 2.792 , c4 = 2.8125, c5 = 2.83125. The graphs of (n, an), (n, bn), and (n, cn) are shown in Fig. 3. Practice 3: For an and bn in the previous example, calculate the first 5 terms of cn = an – bn and dn = ( –1 )n bn and graph the points (n, cn) and (n, dn) . 42 31 5 6231ancnbnnFig. 310.1 Sequences Contemporary Calculus 3 Recursive Sequences A recursive sequence is a sequence defined by a rule that gives each new term in the sequence as a combination of some of the previous terms. We already encountered a recursive sequence when we studied Newton's Method for approximating roots of a function (Section 2.7). Newton's method for finding the roots of a function generates a recursive sequence { x1, x2, x3, x4, ...} , as do successive iterations of a function and other operations. Example 4: Let f(x) = x2 – 4 . Take x1 = 3 and apply Newton's method (Section 2.7) to calculate x2 and x3. Give a rule for xn . Solution: f(x) = x2 – 4 so f '(x) = 2x , and, by Newton's method, x2 = x1 – f(x1)f '(x1) = 3 – f(3)f '(3) = 3 – 56 = 136 ≈ 2.1667. x3 = x2 – f(x2)f '(x2) = 136 – f(13/6)f '(13/6) = 136 – 25156 = 313156 ≈ 2.0064 . In general, xn = xn–1 – f(xn–1)f '(xn–1) = xn–1 – (xn–1)2 – 42xn–1 . The terms x1, x2, . . . approach the value 2, one solution of x2 – 4 = 0. The sequence { xn } is a recursive sequence since each term xn is defined as a function of the previous term xn–1 . Practice 4: Let f(x) = 2x – 1, and define an = f( f( f( . . . f( a0 ) . . . ) ) ) where the function is applied n times. Put a0 = 3 and calculate a1, a2, and a3. Note that an can be defined recursively as an = f(an–1) . Example 5: Let an = 1/2n , and define a second sequence { sn } by the rule that sn is the sum of the first n terms …


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BC MATH 153 - Sequences

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