Intermediate Algebra (11th edition)

OBJECTIVE 4 Use summation notation to evaluate a series.By adding the

terms of a sequence, we obtain a series.

680 CHAPTER 12 Sequences and Series

Series

The indicated sum of the terms of a sequence is called a series.

For example, if we consider the sum of the payments listed in Example 3,namely,

we have a series that represents the total payments for the first four months. Since a

sequence can be finite or infinite, there are both finite and infinite series.

We use a compact notation, called summation notation,to write a series from

the general term of the corresponding sequence. In mathematics, the Greek letter

(sigma)is used to denote summation. For example, the sum of the first six terms of

the sequence with general term is written as

The letter iis called the index of summation.We read this as “the sum from to

6 of ” To find this sum, we replace the letter iin with 1, 2, 3, 4, 5, and

6, and add the resulting terms.

3 i+2. 3 i+ 2

i= 1

a

6

i= 1

13 i+ 22.

an= 3 n+ 2

π

550 + 545 + 540 + 535,

CAUTION This use of ias the index of summation has no connection with the

complex number i.

Evaluating Series Written in Summation Notation

Write out the terms and evaluate each series.

(a)

Replace iwith

1, 2, 3, 4, 5, 6.

= 75 Add.

= 5 + 8 + 11 + 14 + 17 + 20

• 13 # 4 + 22 + 13 # 5 + 22 + 13 # 6 + 22

= 13 # 1 + 22 + 13 # 2 + 22 + 13 # 3 + 22

a

6

i= 1

13 i+ 22

EXAMPLE 4

Multiply and

then add.

(b)

Subtract.

=- 5 Simplify.

=- 3 - 2 - 1 + 0 + 1

= 11 - 42 + 12 - 42 + 13 - 42 + 14 - 42 + 15 - 42 i=1, 2, 3, 4, 5

a

5

i= 1

1 i- 42

Work inside the

parentheses.