S E R I E S

If you make a sequence of bank deposits, then you might be interested in the total
value of the terms of the sequence. Of course, if the sequence has only a few terms,
you can simply add them. In Sections 13.3 and 13.4 we will develop formulas that
give the sum of the terms for certain nite and innite sequences. In this section you
will rst learn a notation for expressing the sum of the terms of a sequence.
Summation Notation
To describe the sum of the terms of a sequence, we use summation notation. The
Greek letter (sigma) is used to indicate sums. For example, the sum of the rst
ve terms of the sequence a
n
n
2
is written as

5
n1
n
2
.
You can read this notation as the sum of n
2
for n between 1 and 5, inclusive. To
nd the sum, we let n take the values 1 through 5 in the expression n
2
:

5
n1
n
2
1
2
2
2
3
2
4
2
5
2
1 4 9 16 25
55
In this context the letter n is the index of summation. Other letters may also be
used. For example, the expressions

5
n1
n
2
,

5
j1
j
2
, and

5
i1
i
2
all have the same value. Note that i is used as a variable here and not as an imagi-
nary number.
E X A M P L E 1 Evaluating a sum in summation notation
Find the value of the expression

3
i1
(1)
i
(2i 1).
Solution
Replace i by 1, 2, and 3, and then add the results:

3
i1
(1)
i
(2i 1) (1)
1
[2(1) 1] (1)
2
[2(2) 1] (1)
3
[2(3) 1]
3 5 7
5 I
Series
The sum of the terms of the sequence 1, 4, 9, 16, 25 is written as
1 4 9 16 25.
This expression is called a series. It indicates that we are to add the terms of the
given sequence. The sum, 55, is the sum of the series.
684 (13-8) Chapter 13 Sequences and Series
13.2
I n t h i s
s e c t i o n
G
Summation Notation
G
Series
G
Changing the Index
Series
The indicated sum of the terms of a sequence is called a series.
Just as a sequence may be nite or innite, a series may be nite or innite. In
this section we discuss nite series only. In Section 13.4 we will discuss one type of
innite series.
Summation notation is a convenient notation for writing a series.
E X A M P L E 2 Converting to summation notation
Write the series in summation notation:
2 4 6 8 10 12 14
Solution
The general term for the sequence of positive even integers is 2n. If we let n take the
values from 1 through 7, then 2n ranges from 2 through 14. So
2 4 6 8 10 12 14

7
n1
2n.
I
E X A M P L E 3 Converting to summation notation
Write the series

1
2

1
3

1
4

1
5

1
6

1
7

5
1
0

in summation notation.
Solution
For this series we let n be 2 through 50. The expression (1)
n
produces alternating
signs. The series is written as

50
n2

(
n
1)
n
.
I
Changing the Index
In Example 3 we saw the index go from 2 through 50, but this is arbitrary. Aseries
can be written with the index starting at any given number.
E X A M P L E 4 Changing the index
Rewrite the series

6
i1

(
i
2
1)
i

with an index j, where j starts at 0.

Solution
Because i starts at 1 and j starts at 0, we have i j 1. Because i ranges from 1 through
6 and i j 1, j must range from 0 through 5. Now replace i by j 1 in the
summation notation:

5
j0

(
(
j
1)
1
j
)
1
2

Check that these two series have exactly the same six terms. I
13.2 Series (13-9) 685
h e l p f u l h i n t
A series is called an indicated
sumbecause the addition is in-
dicated by not actually being
performed. The sum of a series
is the real number obtained
by actually performing the in-
dicated addition.
True or false? Explain your answer.
1. Aseries is the indicated sum of the terms of a sequence. True
2. The sum of a series can never be negative. False
3. There are eight terms in the series

10
i2
i
3
. False
4. The series

9
i1
(1)
i
i
2
and

8
j0
(1)
j
( j 1)
2
have the same sum. False
5. The ninth term of the series

100
i1

(i
(
1

)(
1
i
)
i
2)
is
1
1
10
. False
6.

2
i1
(1)
i
2
i
2 True 7.

5
i1
3i 3

5
i1
i

True
8.

5
i1
4 20 True 9.

5
i1
2i

5
i1
7i

5
i1
9i True
10.

3
i1
(2i 1)

3
i1
2i

1 False
686 (13-10) Chapter 13 Sequences and Series
W A R M - U P S
Reading and Writing After reading this section, write out the
answers to these questions. Use complete sentences.
1. What is summation notation?
Summation notation provides a way to write a sum without
writing out all of the terms.
2. What is the index of summation?
The index of summation is the variable used in summation
notation.
3. What is a series?
A series is the indicated sum of the terms of a sequence.
4. What is a nite series?
A nite series is the indicated sum of the terms of a nite
sequence.
Find the sum of each series. See Example 1.
5.

4
i1
i
2
6.

3
j0
( j 1)
2
30 30
7.

5
j0
(2j 1) 8.

6
i1
(2i 3)
24 24
9.

5
i1
2
i
10.

5
i1
(2)
i

3
3
1
2

1
3
1
2

11.

10
i1
5i
0
12.

20
j1
3
50 60
13.

3
i1
(i 3)(i 1) 14.

5
i0
i(i 1)(i 2)(i 3)
7 144
15.

10
j1
(1)
j
16.

11
j1
(1)
j
0 1
Write each series in summation notation. Use the index i, and let
i begin at 1 in each summation. See Examples 2 and 3.
17. 1 2 3 4 5 6

6
i1
i
18. 2 4 6 8 10

5
i1
2i
19. 1 3 5 7 9 11

6
i1
(1)
i
(2i 1)
20. 1 3 5 7 9

5
i1
(1)
i1
(2i 1)
21. 1 4 9 16 25 36

6
i1
i
2
22. 1 8 27 64 125

5
i1
i
3
23.
1
3

1
4

1
5

1
6

4
i1

2
1
i

E X E R C I S E S 13. 2
24. 1
1
2

1
3

1
4

1
5

1
6

6
i1

(1
i
)
i1

25. ln(2) ln(3) ln(4)

3
i1
ln(i 1)
26. e
1
e
2
e
3
e
4

4
i1
e
i
27. a
1
a
2
a
3
a
4
4
i1
a
i
28. a
2
a
3
a
4
a
5

4
i1
a
i1
29. x
3
x
4
x
5
x
50
48
i1
x
i2
30. y
1
y
2
y
3
y
30
30
i1
y
i
31. w
1
w
2
w
3
w
n
n
i1
w
i
32. m
1
m
2
m
3
m
k
k
i1
m
i
Complete the rewriting of each series using the new index as
indicated. See Example 4.
33.

5
i1
i
2

j0
34.

6
i1
i
3

j0

4
j0
( j 1)
2

5
j0
( j 1)
3
35.

12
i0
(2i 1)

j1
36.

3
i1
(3i 2)

j0

13
j1
(2j 3)

2
j0
(3j 5)
37.

8
i4

1
i

j1
38.

10
i5
2
i

j1

5
j1

j
1
3

6
j1
2
j4
39.

4
i1
x
2i3

j0
40.

2
i0
x
32i

j1

3
j0
x
2j5

3
j1
x
52j
41.

n
i1
x
i

j0
42.

n
i0
x
i

j1

n1
j0
x
j1

n1
j1
x
j1
Write out the terms of each series.
43.

6
i1
x
i
x x
2
x
3
x
4
x
5
x
6
44.

5
i1
(1)
i
x
i1
1 x x
2
x
3
x
4
13.2 Series (13-11) 687
45.

3
j0
(1)
j
x
j
x
0
x
1
x
2
x
3
46.

5
j1

x
1
j

x
1
1

x
1
2

x
1
3

x
1
4

x
1
5

47.

3
i1
ix
i
x 2x
2
3x
3
48.

5
i1

x
i
x
2
x

3
x

4
x

5
x

A series can be used to model the situation in each of the

following problems.
49. Leap frog. A frog with a vision problem is 1 yard away
from a dead cricket. He spots the cricket and jumps halfway
to the cricket. After the frog realizes that he has not reached
the cricket, he again jumps halfway to the cricket. Write a
series in summation notation to describe how far the frog
has moved after nine such jumps.

9
i1
2
i
50. Compound interest. Cleo deposited $1,000 at the begin-
ning of each year for 5 years into an account paying 10%
interest compounded annually. Write a series using summa-
tion notation to describe how much she has in the account
at the end of the fth year. Note that the rst $1,000 will
receive interest for 5 years, the second $1,000 will receive
interest for 4 years, and so on.

5
i1
1000(1.1)
i
51. Total economic impact. In Exercise 43 of Section 13.1 we
described a factory that spends $1 million annually in a
community in which 80% of all money received in the
community is respent in the community. Use summation
notation to write the sum of the rst four terms of the
economic impact sequence for the factory.

4
i1
1,000,000(0.8)
i1
52. Total spending. Suppose you earn $1 on January 1, $2 on
January 2, $3 on January 3, and so on. Use summation no-
tation to write the sum of your earnings for the entire month
of January.

31
i1
i
GETTI NG MORE I NVOLVED
53. Discussion. What is the difference between a sequence
and a series?
A sequence is basically a list of numbers. A series is the
indicated sum of the terms of a sequence.
54. Discussion. For what values of n is

n
i1

1
i
4?
n 31