slide 11.1- 1 copyright © 2007 pearson education, inc. publishing as pearson addison-wesley
TRANSCRIPT
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