23 Sequences and Series
Definition 23.1. A sequence is a function whose domain is a set of
integers
usually either the natural numbers 1, 2, 3,... or the non negative integers
0, 1, 2, 3, .... The output of the sequence when the input is n is called the
nth term of the sequence.
23.2. Usually the input is denoted by a letter like n near the middle of
the
alphabet and the output for input n is denoted by making n a subscript
rather than by surrounding n by parentheses, i.e. we write
rather than
a(n). Here are some examples.
Definition 23.3. An arithmetic sequence is one of form
where a and d are constants. An arithmetic sequence has the property that
the difference of successive terms is a constant:
Definition 23.4. A geometric sequence is one of form
where a and r are constants. A geometric sequence has the property that
the ratio of successive terms is a constant:
Example 23.5. The sequence 
is neither
arithmetic nor geometric.
The sequence is not arithmetic because the difference
is not constant (i.e. it depends on n) and the sequence is not geometric
because the ratio
is not constant (i.e. it also depends on n).
Definition 23.6. Every sequence determines another
sequence called the
corresponding series. The nth term of the series is the sum of the first n
terms of the sequence. Thus if the nth term of the sequence is
, then the
nth term of the series is
Definition 23.7. A series is often written using Sigma notation as
in
More generally
Theorem 23.8. The sum of the first n terms of a
geometric sequence is
given by
Proof.
23.9. Mortgages. A family takes out a 30 year $100,000 mortgage to buy
a house. The interest rate is 6% per year and they will repay the loan in 360
equal monthly payments. To compute the monthly payment imagine that
they family has taken 360 loans
and they will repay the kth loan at the end of the kth month with the
monthly payment of a dollars. The monthly interest rate is 6%/12 = 0.005
so the amount owed on the kth loan after k months is
. This is
the amount of the kth payment so 
so
so
Using Sigma notation and Theorem 23.8 with r = (1.005)-1 this may be
written
To evaluate the expression on the right multiply top and
bottom by 1.005 to
get
so 100000 = 166.79a so the monthly payment is
a = 100000/166.79 = 599.56.
24 Infinite Series
Definition 24.1. For a sequence
the notation
means that that the numbers
are arbitrarily close to
the number b when
n is sufficiently large. The expression on the left is called the the limit
of
as n becomes infinite. For a series we also use the notation
Theorem 24.2. If the ratio r of successive terms in
a geometric series is
less than one in absolute value , then the sum of the infinite geometric series
is
Proof. By Theorem 23.8
where c = ar/(1 - r). In Section 17 we saw that for l r l
< 1 the graph of
the exponential function y = rx decays exponentially as x becomes large
positive. This implies that
so
Example 24.3. (Zeno's Paradox) To travel one mile I must
first travel the
first half mile, then half of the remaining distance, then half of the remaining
distance, and so on. How can I ever go whole distance? The answer is that
the sum of all the distances is one:
The numerator is only one less than the denominator:
The finite sums are getting closer to one and the infinite
sum is
Example 24.4. We use Theorem 24.2 to prove that the
infinite repeating
decimal 0.1 36 36 36 36 36 36.... is equal to 3/22. The first step is to make
the decimal look like a geometric series.
Now by Theorem 23.8
so
Now we do the arithmetic:
24.5. The concept of an infinite sum makes the Definition
of decimal expan-
sion more precise. If x is a real number between 0 and 1 it has a decimal
expansion
where each 
is an integer between 0 and 9. For most real
numbers the
digits won't follow any pattern, but
Theorem 24.6. A real number x is rational if and only if has a repeating
decimal expansion like the one in Example 24.4.
Proof. A rational number is a ratio p/q of two integers p and q. To see why
the decimal expansion eventually repeats periodically imagine computing
the decimal expansion by long division. At each step in the long division
algorithm we compute the next digit in the quotient, multiply that digit
by q, subtract the product to get the next remainder, and bring down the
next digit from the dividend. The remainder is smaller than q, otherwise
we would have used a larger digit in the quotient we are computing. Once
we are computing digits of the right of the decimal point the digit we bring
down from the dividend is always zero and since the remainder is always less
than q we will eventually find ourselves redoing what we have already done.
The proof that a real number with a repeating decimal expansion is ratio-
nal is just like the computation in Example 24.4. We first write the number
as the sum of a finite decimal and a negative power of ten times a geometric
series. We then use Theorem 24.2 and do the arithmetic.
A Where to Look in the Textbook
This appendix tells you where to look in the course textbook
David Cohen: College Algebra Fifth Edition, Thomson Brooks/Cole
2003.
for additional reading .
1 (Laws of algebra) The material in this section is mostly review but some
of it is discussed in Appendices A.2 and B.1 of the textbook. The ma-
terial in Paragraph 1.1 Paragraph 1.2 Paragraph 1.3 Paragraph 1.4
Page A-5 in Appendix A.2 Paragraph 1.5 Paragraph 1.6 Page A-5
in Appendix A.2 Paragraph 1.7 Pages 17 and 77, Definition 1.8 Ap-
pendix B.2 Page A-8, Paragraph 1.9
2 (Kinds of Numbers) Most of the material in this section is covered in Sec-
tion 1.1 of the textbook. Page 2. Paragraph 2.2 Remark 2.3 Page 729.
Remark 2.5 See Page A-7 Appendix A.3 of the textbook for the proof
that
is irrational.
3 (Coordinates on the Line and Order ) Interval notation is explained in Sec-
tion 1.1 (Page 3). Paragraph 3.1 Page 1. Paragraph 3.2 Paragraph
3.3
Paragraph 3.4 Page 4 Paragraph 3.5 Section 1.2 Page 6. Paragraph
3.6
Section 1.2 Page 9. The properties of inequalities are reviewed at the
beginning of Section 2.5 of the textbook.
4 (Exponents) See Appendices B.2 and B.3 in the textbook. Theorem 4.1
Paragraph 4.2
5 (Coordinates in the Plane and Graphs) Coordinates in the plane are in-
troduced in Sections 1.2 of the textbook Paragraph 5.1 Definition 5.2
Page 36. The Distance Formula 5.4 first appears on Page 23-24. Para-
graph 5.5 Pages 68-70. The Midpoint Formula 5.6 is on Page 27. Def-
inition 7.1 Page 66, Page 636. Theorem 7.2 Page 638. Example 7.3
Page 639. Remark 7.4 Page 644, Page 652 (Exercise 19). Definition 7.5
Page 653. Theorem 7.6 Page 655. Example 7.7 Page 656.
6 (Lines) This material is covered in section 1.6 of the textbook. Para-
graph 6.1 Page 55. Definition 6.2 Page 48. Paragraph 6.3 Page 52.
Paragraph 6.4 Page 55. Theorem 6.5 Page 56, Theorem 6.6 Page 56.
8 ( Solving Equations ) Definition 8.1 appears on Page 13 of
the textbook.
Extraneous solutions are explained on pages 15-16. Paragraph 8.2
Paragraph 8.3 Paragraph 8.4 Page 104 Remark 8.5 Paragraph
8.6 This
material is reviewed in Section 1.3 of the textbook.
9 ( Systems of Equations ) Systems of two linear equations in two unknowns
are reviewed in Section 6.1 of the textbook. Paragraph 9.1 Example 9.2
Paragraph 9.3 Paragraph 9.4
10 (Symmetry) Definition 10.1 Section 1.7 Page 63. Paragraph 10.2 Page 64.
11 (Completing the Square) Paragraph 11.1 Theorem 11.2 Pages 18 and 87.
Theorem 11.3 Page 70. Theorem 11.4 Page 273. Paragraph 11.5
Pages 198-209.
12 (Functions) Definition 12.1 Section 3.1 Page 160. Paragraph 12.2 Sec-
tion 3.1 Page 167. Paragraph 12.3 12.3 Section 3.1 Pages 162-163.
Definition 12.4 Section 3.2 Page 173. The Vertical Line Test 12.4 Sec-
tion 3.2 Page 174. Example 12.5 Example 2 Page 174. Remark 12.6
Section 3.2 Pages 174-175. Paragraph 12.7 Section 3.5 Page 209. De -
nition 12.8 Section 3.5 Page 211. Remark 12.9 Section 3.5 Example 2
Page 211. Paragraph 12.10
13 (Inverse Functions) Definition 13.1 Section 3.6 Page 222. Example
13.2
Section 3.6 Example 2 Page 223. Remark 13.3 Paragraph 13.4 Sec-
tion 3.6 Example 2 Page 229. The Horizontal Line Test 13.4 Section 3.6
Page 229. Paragraph 13.5 Section 3.6 Page 229. Definition 13.6 186,
244 Theorem 13.7 Remark 13.8 Paragraph 13.9 Paragraph 13.10
Sec-
tion 3.6 Page 227.
14 (Average Rate of Change ) Definition 14.1 Section 3.3 Page 189. Exam-
ple 14.2 Paragraph 14.3 Paragraph 14.4
15 (Polynomials) Section 4.6 Page 326. Paragraph 15.1 Definition
15.2 Para-
graph 15.3 Paragraph 15.4 15.4 Remark 15.5 Paragraph 15.6 Para-
graph 15.7
16 (Rational Functions) Definition 16.1 Section 4.7 Theorem 21.1 345, 569-
571 Paragraph 16.2 Section 4.7 Example 8 Page 350. Example 16.3
17 (Exponentials and Logarithms) Paragraph 17.1 Section
5.1 Paragraph 17.2
Remark 17.3 Definition 17.4 Section 5.3 Paragraph 17.5 Box
17.5 Re-
mark 17.6 Paragraph 17.7
18 (Exponential Growth and Decay) Section 5.7. Definition 18.1 Para-
graph 18.2 Paragraph 18.3
19 (The Natural Logarithm) Paragraph 19.1 Section 5.2 Paragraph 19.2
Paragraph 19.3 Remark 19.4 Paragraph 19.5
20 (Complex Numbers) Definition 20.1 Section 7.1 Paragraph 20.2 Para-
graph 20.3
21 (Division of Polynomials) Theorem Division Algorithm 21.1 Remark
21.2
An analogous statement holds for integers.
22 (The Fundamental Theorem of Algebra) Theorem Fundamental Theorem
of Algebra] 22.1 Theorem Remainder Theorem] 22.3 Corollary Factor
Theorem] 22.4 Theorem Complete Factorization] 22.5 Corollary 22.6
23 (Sequences and Series) Section 9.3. Definition 23.1 Paragraph
23.2 Def-
inition 23.3 Section 9.4. Definition 23.4 Section 9.5. Example 23.5
Definition 23.6 Definition 23.7 Section 9.3 Theorem 23.8 Section 9.5.
Paragraph 23.9 (Extra)
24 (infinite Series) Definition 24.1 Theorem 24.2 Example
24.3 Example 24.4
Paragraph 24.5 Theorem 24.6 Page 729.
