13 Sequences and Series Copyright Cengage Learning All

13 Sequences and Series Copyright © Cengage Learning. All rights reserved.

13. 1 Sequences and Summation Notation Copyright © Cengage Learning. All rights reserved.

Objectives ■ Sequences ■ Recursively Defined Sequences ■ The Partial Sums of a Sequence ■ Sigma Notation 3

Sequences and Summation Notation Roughly speaking, a sequence is an infinite list of numbers. The numbers in the sequence are often written as a 1, a 2, a 3, . . The dots mean that the list continues forever. A simple example is the sequence We can describe the pattern of the sequence displayed above by the following formula: an = 5 n 4

Sequences 5

Sequences Any ordered list of numbers can be viewed as a function whose input values are 1, 2, 3, . . . and whose output values are the numbers in the list. So we define a sequence as follows. 6

Example 1 – Finding the Terms of a Sequence Find the first five terms and the 100 th term of the sequence defined by each formula. (a) an = 2 n – 1 (b) cn = n 2 – 1 (c) (d) 7

Example 1 – Solution To find the first five terms, we substitute n = 1, 2, 3, 4, and 5 in the formula for the nth term. To find the 100 th term, we substitute n = 100. This gives the following. 8

Example 2 – Finding the nth Term of a Sequence Find the nth term of a sequence whose first several terms are given. (a) (b) – 2, 4, – 8, 16, – 32, . . . Solution: (a) We notice that the numerators of these fractions are the odd numbers and the denominators are the even numbers. Even numbers are of the form 2 n, and odd numbers are of the form 2 n – 1 (an odd number differs from an even number by 1). 9

Example 2 – Solution cont’d So a sequence that has these numbers for its first four terms is given by (b) These numbers are powers of 2, and they alternate in sign, so a sequence that agrees with these terms is given by an = (– 1)n 2 n You should check that these formulas do indeed generate the given terms. 10

Recursively Defined Sequences 11

Recursively Defined Sequences Some sequences do not have simple defining formulas like those of the preceding example. The nth term of a sequence may depend on some or all of the terms preceding it. A sequence defined in this way is called recursive. Here is an example. 12

Example 4 – The Fibonacci Sequence Find the first 11 terms of the sequence defined recursively by F 1 = 1, F 2 = 1 and Fn = Fn – 1 + Fn – 2 Solution: To find Fn, we need to find the two preceding terms, Fn – 1 and Fn – 2. Since we are given F 1 and F 2, we proceed as follows. F 3 = F 2 + F 1 = 1 + 1 = 2 F 4 = F 3 + F 2 = 2 + 1 = 3 F 5 = F 4 + F 3 = 3 + 2 = 5 13

Example 4 – Solution cont’d It’s clear what is happening here. Each term is simply the sum of the two terms that precede it, so we can easily write down as many terms as we please. Here are the first 11 terms. 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, . . . 14

Recursively Defined Sequences The sequence in Example 4 is called the Fibonacci sequence, named after the 13 th century Italian mathematician who used it to solve a problem about the breeding of rabbits. 15

The Partial Sums of a Sequence 16

The Partial Sums of a Sequence In calculus we are often interested in adding the terms of a sequence. This leads to the following definition. 17

Example 5 – Finding the Partial Sums of a Sequence Find the first four partial sums and the nth partial sum of the sequence given by an = 1/2 n. Solution: The terms of the sequence are The first four partial sums are 18

Example 5 – Solution cont’d Notice that in the value of each partial sum, the denominator is a power of 2 and the numerator is one less than the denominator. 19

Example 5 – Solution cont’d In general, the nth partial sum is The first five terms of an and Sn are graphed in Figure 7. Graph of the sequence an and the sequence of partial sums Sn Figure 7 20

11. 1 Sequences A finite sequence has domain the finite set {1, 2, 3, …, n} for some natural number n. Example 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 An infinite sequence has domain {1, 2, 3, …}, the set of all natural numbers. Example 1, 2, 4, 8, 16, 32, … 21

11. 1 Convergent and Divergent Sequences A convergent sequence is one whose terms get closer and closer to a some real number. The sequence is said to converge to that number. A sequence that is not convergent is said to be divergent. 22

11. 1 Convergent and Divergent Sequences Example The sequence converges to 0. The terms of the sequence 1, 0. 5, 0. 33. . , 0. 25, … grow smaller and smaller approaching 0. This can be seen graphically. 23

11. 1 Convergent and Divergent Sequences Example The sequence is divergent. The terms grow large without bound 1, 4, 9, 16, 25, 36, 49, 64, … and do not approach any one number. 24

Sigma Notation 25

Sigma Notation Given a sequence a 1, a 2, a 3, a 4, . . . we can write the sum of the first n terms using summation notation, or sigma notation. This notation derives its name from the Greek letter (capital sigma, corresponding to our S for “sum”). Sigma notation is used as follows: 26

Sigma Notation The left side of this expression is read, “The sum of ak from k = 1 to k = n. ” The letter k is called the index of summation, or the summation variable, and the idea is to replace k in the expression after the sigma by the integers 1, 2, 3, . . . , n, and add the resulting expressions, arriving at the right-hand side of the equation. 27

Example 7 – Sigma Notation Find each sum. Solution: = 12 + 2 2 + 3 2 + 4 2 + 5 2 = 55 28

Example 7 – Solution cont’d = 5 + 6 + 7 + 8 + 9 + 10 = 45 =2+2+2+2 = 12 29

Example 7 – Solution cont’d We can use a graphing calculator to evaluate sums. For instance, Figure 8 shows how the TI-83 can be used to evaluate the sums in parts (a) and (b) of Example 7. Figure 8 30

Sigma Notation The following properties of sums are natural consequences of properties of the real numbers. 31

11. 1 Series and Summation Notation Example Use the summation properties to evaluate (a) (b) (c) Solution (a) 32

11. 1 Series and Summation Notation Summation Rules 33

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