Slide 11.1- 1 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

OBJECTIVES

Sequences and Series

Learn sequence notation and how to find specific and general terms in a sequence.Learn to use factorial notation.Learn to use summation notation to write partial sums of a series.

SECTION 11.1

1

2

3

Slide 11.1- 3 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

DEFINITION OF A SEQUENCE

An infinite sequence is a function whose domain is the set of positive integers. The function values, written as

a1, a2, a3, a4, … , an, …

are called the terms of the sequence. The nth term, an, is called the general term of the sequence.

Slide 11.1- 4 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

EXAMPLE 2 Writing the First Several Terms of a Sequence

Write the first six terms of the sequence defined by:

bn 1 n1 1

n

b1 1 11 1

1

1 2

1 1

b2 1 21 1

2

1 3 1

2

1

2

Solution

Replace n with each integer from 1 to 6.

Slide 11.1- 5 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

EXAMPLE 2 Writing the First Several Terms of a Sequence

Solution continued

b3 1 31 1

3

1 4 1

3

1

3

b4 1 41 1

4

1 5 1

4

1

4

b6 1 61 1

6

1 7 1

6

1

6

b5 1 51 1

5

1 6 1

5

1

5

Slide 11.1- 6 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

DEFINITION OF FACTORIAL

For any positive integer n, n factorial (written n!) is defined as

As a special case, zero factorial (written 0!) is defined as

n!n n 1 4 321.

0!1.

Slide 11.1- 7 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

EXAMPLE 7Writing Terms of a Sequence Involving Factorials

Write the first five terms of the sequence whose general term is:

Solution

Replace n with each integer from 1 through 5.

an 1 n1

n!

a1 1 11

1!

1 2

11

a2 1 21

2!

1 3

21

1

2

Slide 11.1- 8 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

EXAMPLE 7Writing Terms of a Sequence Involving Factorials

Solution continued

a5 1 51

5!

1 6

54 321

1

120

a3 1 31

3!

1 4

321

1

6

a4 1 41

4!

1 5

4 321

1

24

Slide 11.1- 9 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

SUMMATION NOTATION

The sum of the first n term of a sequence a1, a2, a3, …, an, … is denoted by

The letter i in the summation notation is called the index of summation, n is called the upper limit, and 1 is called the lower limit, of the summation.

ai

i1

n

a1 a2 a3 L an .

Slide 11.1- 10 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

EXAMPLE 8Evaluating Sums Given in Summation Notation

Find each sum.

a. ii1

9

Solution

a. Replace i with integers 1 through 9, inclusive, and then add.

b. 2 j2 1 j4

7

c. 2k

k!k0

4

ii1

9

1 2 3 4 5 6 7 8 9 45

Slide 11.1- 11 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

EXAMPLE 8Evaluating Sums Given in Summation Notation

Solution continued

b. Replace j with integers 4 through 7, inclusive, and then add.

2 j2 1 j4

7

2 4 2 1 2 5 2 1

2 6 2 1 2 7 2 1 31 49 71 97 248

Slide 11.1- 12 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

EXAMPLE 8Evaluating Sums Given in Summation Notation

Solution continued

2k

k!k0

4

20

0!

21

1!

22

2!

23

3!

24

4!

1

1

2

1

4

2

8

6

16

24

1 2 2 4

3

2

37

c. Replace k with integers 0 through 4, inclusive, and then add.

Slide 11.1- 13 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

SUMMATION PROPERTIESLet ak and bk, represent the general terms of two sequences, and let c represent any real number. Then

1. ck1

n

cn

2. cakk1

n

c akk1

n

Slide 11.1- 14 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

SUMMATION PROPERTIES

3. ak bk k1

n

ak k1

n

bkk1

n

4. ak bk k1

n

ak k1

n

bkk1

n

5. akk1

n

ak k1

j

akkj1

n

, for 1 j n

Slide 11.1- 15 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

DEFINITION OF A SERIES

Let a1, a2, a3, … , ak, … be an infinite sequence. Then

1. The sum of the first n terms of the sequence is called the nth partial sum of the sequence and is denoted by

a1 a2 a3 L an ai

i1

n

.

Slide 11.1- 16 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

DEFINITION OF A SERIES

2. The sum of all terms of the infinite sequence is called an infinite series and is denoted by

a1 a2 a3 L an L ai

i1

.

Slide 11.1- 17 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

EXAMPLE 9 Writing a Partial Sum in Summation Notation

Write each sum in summation notation.

a. 3 5 7 L 21

3 5 7 L 21 2k 1

k1

10

Solution

a. This is the sum of consecutive odd integers from 3 to 21. Each can be expressed as 2k + 1, starting with k = 1 to 10.

b.

1

4

1

9L

1

49

Slide 11.1- 18 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

EXAMPLE 9 Writing a Partial Sum in Summation Notation

Solution continued

b. This finite series is the sum of fractions, each of which has numerator 1 and denominator k2, starting with k = 2 and ending with k = 7.

1

4

1

9L

1

49

1

k2k2

7