Sequences, Series, and the Binomial Theorem - 2012 Book Archive

114 Pages·2012·1.45 MB·English

This is “Sequences, Series, and the Binomial Theorem”, chapter 9 from the book Advanced Algebra(index.html)
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i
Chapter 9
Sequences, Series, and the Binomial Theorem
1977
Chapter 9 Sequences, Series, and the Binomial Theorem
9.1 Introduction to Sequences and Series
LEARNING OBJECTIVES
1. Find any element of a sequence given a formula for its general term.
2. Use sigma notation and expand corresponding series.
3. Distinguish between a sequence and a series.
4. Calculate thenth partial sum of sequence.
Sequences
Asequence1is a function whose domain is a set of consecutive natural numbers
beginning with 1. For example, the following equation with domain{1,2,3,…}
defines aninfinite sequence2:
a(n) = 5n − 3 or an = 5n − 3
The elements in the range of this function are called terms of the sequence. It is
common to define thenth term, or thegeneral term of a sequence3, using the
subscritped notationa , which reads “asubn.” Terms can be found using
n
substitution as follows:
Generalterm : an=5n − 3
1.A function whose domain is a Firstterm(n = 1) : a =5(1) − 3 = 2
1
set of consecutive natural
Secondterm(n = 2) :a =5(2) − 3 = 7
numbers starting with 1. 2
Thirdterm(n = 3) : a =5(3) − 3 = 12
2.A sequence whose domain is 3
the set of natural numbers Fourthterm(n = 4) :a =5(4) − 3 = 17
3
{1,2,3,…}.
Fifthterm(n = 5) : a3=5(5) − 3 = 22
3.An equation that defines the
nth term of a sequence ⋮
commonly denoted using
subscriptsan.
1978
Chapter 9 Sequences, Series, and the Binomial Theorem
This produces an ordered list,
2,7,12,17,22,…
The ellipsis (…) indicates that this sequence continues forever. Unlike a set, order
matters. If the domain of a sequence consists of natural numbers that end, such as
{1,2,3,…,k}, then it is called afinite sequence4.
4.A sequence whose domain is
{1,2,3,…,k}wherekis a
natural number.
9.1 Introduction to Sequences and Series 1979
Chapter 9 Sequences, Series, and the Binomial Theorem
Example 1
th
Given the general term of a sequence, find the first 5 terms as well as the 100
term:an = n(n−1) .
2
Solution:
To find the first 5 terms, substitute 1, 2, 3, 4, and 5 fornand then simplify.
1(1 − 1) 1(0) 0
a = = = = 0
1
2 2 2
2(2 − 1) 2(1) 2
a = = = = 1
2
2 2 2
3(3 − 1) 3(2) 6
a = = = = 3
3
2 2 2
4(4 − 1) 4(3) 12
a = = = = 6
4
2 2 2
5(5 − 1) 5(4) 20
a = = = = 10
5
2 2 2
Usen = 100to determine the 100thterm in the sequence.
100(100 − 1) 100(99) 9,900
a = = = = 4,950
100
2 2 2
Answer: First five terms: 0, 1, 3, 6, 10;a = 4,950
100
Sometimes the general term of a sequence will alternate in sign and have a variable
other thann.
9.1 Introduction to Sequences and Series 1980
Chapter 9 Sequences, Series, and the Binomial Theorem
Example 2
Find the first 5 terms of the sequence:an = (−1)nxn+1.
Solution:
Here we take care to replacenwith the first 5 natural numbers and notx.
a =(−1)1x1+1 = −x2
1
a =(−1)2x2+1 = x3
2
a =(−1)3x3+1 = −x4
3
a =(−1)4x4+1 = x5
4
a =(−1)5x5+1 = −x6
5
Answer:−x2,x3,−x4,x5,−x6
Try this!Find the first 5 terms of the sequence:an = (−1)n+12n.
Answer: 2, −4, 8, −16, 32.
(click to see video)
One interesting example is the Fibonacci sequence. The first two numbers in the
Fibonacci sequence are 1, and each successive term is the sum of the previous two.
Therefore, the general term is expressed in terms of the previous two as follows:
Fn = Fn−2 + Fn−1
9.1 Introduction to Sequences and Series 1981
Chapter 9 Sequences, Series, and the Binomial Theorem
HereF = 1,F = 1, andn > 2.A formula that describes a sequence in terms of
1 2
its previous terms is called arecurrence relation5.
Example 3
Find the first 7 Fibonacci numbers.
Solution:
Given thatF1 = 1andF2 = 1, use the recurrence relationFn = Fn−2 + Fn−1
wherenis an integer starting withn = 3to find the next 5 terms:
F =F + F = F + F = 1 + 1 = 2
3 3−2 3−1 1 2
F =F + F = F + F = 1 + 2 = 3
4 4−2 4−1 2 3
F =F + F = F + F = 2 + 3 = 5
5 5−2 5−1 3 4
F =F + F = F + F = 3 + 5 = 8
6 6−2 6−1 4 5
F =F + F = F + F = 5 + 8 = 13
7 7−2 7−1 5 6
Answer: 1, 1, 2, 3, 5, 8, 13
Figure 9.1
5.A formula that uses previous
terms of a sequence to describe
subsequent terms.
9.1 Introduction to Sequences and Series 1982
Chapter 9 Sequences, Series, and the Binomial Theorem
Leonardo Fibonacci (1170–1250) Wikipedia
Fibonacci numbers appear in applications ranging from art to computer science and
biology. The beauty of this sequence can be visualized by constructing a Fibonacci
spiral. Consider a tiling of squares where each side has a length that matches each
Fibonacci number:
Connecting the opposite corners of the squares with an arc produces a special spiral
shape.
This shape is called the Fibonacci spiral and approximates many spiral shapes found
in nature.
Series
6.The sum of the terms of a
sequence.
Aseries6is the sum of the terms of a sequence. The sum of the terms of an infinite
7.The sum of the terms of an
infinite sequence denotedS∞. sequence results in aninfinite series7, denotedS∞.The sum of the firstnterms in
a sequence is called apartial sum8, denotedSn.For example, given the sequence of
8.The sum of the firstnterms in
positive odd integers 1, 3, 5,… we can write:
a sequence denotedSn.
9.1 Introduction to Sequences and Series 1983
Chapter 9 Sequences, Series, and the Binomial Theorem
S =1 + 3 + 5 + 7 + 9 + ⋯ Infiniteseries
∞
S =1 + 3 + 5 + 7 + 9 = 25 5thpartialsum
5
Example 4
rd th
Determine the 3 and 5 partial sums of the sequence: 3,−6, 12,−24, 48,…
Solution:
S3=3 + (−6) + 12 = 9
S5=3 + (−6) + 12 + (−24) + 48 = 33
Answer:S = 9;S = 33
3 5
If the general term is known, then we can express a series usingsigma9(or
summation10)notation:
∞
S = Σ n2 = 12 + 22 + 32 + … Infiniteseries
∞
n=1
3
9.A sum denoted using the S = Σ n2 = 12 + 22 + 32 3rdpartialsum
3
symbolΣ (upper case Greek n=1
letter sigma).
10.Used when referring to sigma
notation.
The symbolΣ (upper case Greek letter sigma) is used to indicate a series. The
11.The variable used in sigma expressions above and below indicate the range of theindex of summation11, in
notation to indicate the lower this case represented byn. The lower number indicates the starting integer and the
and upper bounds of the
summation.
9.1 Introduction to Sequences and Series 1984
Chapter 9 Sequences, Series, and the Binomial Theorem
upper value indicates the ending integer. Thenth partial sumS can be expressed
n
using sigma notation as follows:
n
Sn = Σ ak = a1 + a2 + ⋯ + an
k=1
This is read, “the sum ofa askgoes from 1 ton.” Replacenwith ∞ to indicate an
k
infinite sum.
Example 5
5
Evaluate: Σ (−3)n−1.
k=1
5
k−1 1−1 2−1 3−1 4−1 5−1
Σ (−3) =(−3) + (−3) + (−3) + (−3) + (−3)
k=1
0 1 2 3 4
=(−3) + (−3) + (−3) + (−3) + (−3)
=1 − 3 + 9 − 27 + 81
=61
Answer: 61
When working with sigma notation, the index does not always start at 1.
9.1 Introduction to Sequences and Series 1985