Transcript 12.1
Sequences and Series Copyright © Cengage Learning. All rights reserved. 12.1 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 a1, a2, a3, . . . . 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 = 5n 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 100th term of the sequence defined by each formula. (a) an = 2n – 1 (b) cn = n2 – 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 100th 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 2n, and odd numbers are of the form 2n – 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)n2n 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 F1 = 1, F2 = 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 F1 and F2, we proceed as follows. F3 = F2 + F1 = 1 + 1 = 2 F4 = F3 + F2 = 2 + 1 = 3 F5 = F4 + F3 = 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 13th 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 = 1/2n. 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 Sigma Notation 21 Sigma Notation Given a sequence a1, a2, a3, a4, . . . 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: 22 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 side of the equation. 23 Example 7 – Sigma Notation Find each sum. Solution: = 12 + 22 + 32 + 42 + 52 = 55 24 Example 7 – Solution cont’d = 5 + 6 + 7 + 8 + 9 + 10 = 45 =2+2+2+2+2+2 = 12 25 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 26 Sigma Notation The following properties of sums are natural consequences of properties of the real numbers. 27