Series
Section 14.3
Intro to Series
A person who conscientiously saves money by saving first $100 and
then saving $10 more each month than he saved the preceding month
is saving money according to the arithmetic sequence
𝑎𝑛 = 100 + 10 𝑛 − 1
Following this sequence, he can predict how much money he should
save any particular month. But if he also wants to know how much
money in total he has saved, say, by the fifth month, he must find the
sum of the first five terms of the sequence.
100 + 100 + 10 + 100 + 20 + 100 + 30 + 100 + 40
𝑎1
𝑎2
𝑎3
𝑎4
𝑎5
Series
A sum of the terms of a sequence is called a
series.
A series is a finite series if it is the sum of a
finite number of terms.
A series is an infinite series if it is the sum
of all the terms of an infinite sequence.
Sequence
Series
Type of
series
5, 9, 13
5 + 9 + 13
Finite
5, 9, 13, …
5 + 9 + 13 + ⋯
Infinite
1 1
4, −2, 1, − ,
2 4
4 + −2 + 1 + −
1
1
+
2
4
Finite
4, −2, 1, …
4 + −2 + 1 + ⋯
Infinite
3, 6, … , 99
3 + 6 + ⋯ + 99
Finite
Summation & Sigma
Shorthand notation for denoting a series when the general
term of the sequence is known is called summation
notation. The Greek uppercase letter sigma, ∑, is used to
mean “sum.”
5
For example: n = 1(3𝑛 + 1) is read “the sum of 3n + 1 as n
goes from 1 to 5”. Sometimes, i will be used instead of n.
This means: 3 ∙ 1 + 1 + 3 ∙ 2 + 1 + 3 ∙ 3 + 1 + 3 ∙ 4 + 1 + (3 ∙ 5 + 1)
4 + 7 + 10 + 13 + 16 = 50
Example 1
Evaluate:
a.
6
i=0
=
𝑖−2
2
0−2 1−2 2−2 3−2 4−2 5−2 6−2
+
+
+
+
+
+
2
2
2
2
2
2
2
= −1 + −
=
1
1
3
+0+ +1+ +2
2
2
2
7
1
𝑜𝑟 3
2
2
Example 1
Evaluate:
b.
5
2𝑖
i=3
= 23 + 24 + 25
= 8 + 16 + 32
= 56
OYO:
Evaluate:
a.
6
i=0
0
𝑖−3
4
b.
5
i=2
360
3𝑖
Example 2
Write each series with summation notation.
a. 3 + 6 + 9 + 12 + 15
What type of sequence would this be?
3, 6, 9, 12, 15
ARITHMETIC!!!
So 𝑎𝑛 = 𝑎1 + 𝑛 − 1 𝑑.
Example 2
Write each series with summation notation.
a. 3 + 6 + 9 + 12 + 15
𝑎𝑛 = 𝑎1 + 𝑛 − 1 𝑑
𝑎𝑛 = 3 + 𝑛 − 1 (3)
𝑎𝑛 = 3 + 3𝑛 − 3
𝑎1 = 3, 𝑑 = 3
5
3𝑖
i=1
𝑎𝑛 = 3𝑛
Example 2
Write each series with summation notation.
b.
1
2
1
4
+ +
1
8
1
+
16
What type of sequence would this be?
1 1 1 1
, , ,
2 4 8 16
GEOMETRIC!!!
So 𝑎𝑛 = 𝑎1 𝑟 𝑛−1 .
Example 2
Write each series with summation notation.
b.
1
2
1
4
1
8
+ + +
1
16
𝑎𝑛 = 𝑎1 𝑟 𝑛−1
𝑎𝑛 = 𝑎1 𝑟 𝑛−1
1
1
𝑎1 = 2, 𝑟 = 2
4
i=1
1
2
𝑖
1 1
𝑎𝑛 =
2 2
1
𝑎𝑛 =
2
𝑛
𝑛−1
OYO:
Write each series with summation notation.
a. 5 + 10 + 15 + 20 + 25 + 30
6
5𝑖
i=1
b.
1
5
+
1
10
4
i=1
+
1 𝑖
5
1
125
+
1
625
Partial Sums
The sum of the first n terms of a sequence is
a finite series known as a partial sum.
Denoted by 𝑆𝑛 .
n
𝑆𝑛 =
𝑎𝑖
i=1
Example 3:
Find the sum of the first three terms of the sequence whose
𝑛+3
general term is 𝑎𝑛 =
.
2𝑛
3
𝑖+3
𝑆3 =
2𝑖
i=1
=
1+3
2+3
3+3
+
+
2∙1
2∙2
2∙3
5
=2+ +1
4
1
4
4
𝑜𝑟
17
4
OYO:
Find the sum of the first three terms of the sequence whose
3𝑛+1
general term is 𝑎𝑛 =
.
𝑛
5
65
10
𝑜𝑟
6
6
Example 4: Application
The number of baby gorillas born at the San Diego Zoo is
a sequence defined by 𝑎𝑛 = 𝑛(𝑛 − 1), where n is the
number of years the zoo has owned gorillas. Find the total
number of baby gorillas born in the first 4 years.
We’ll find 𝑆4 =
i (i −1)
= 1 1 − 1 + 2 2 − 1 + 3 3 − 1 + 4(4 − 1)
= 0 + 2 + 6 + 12
= 20
There were 20 gorillas born in the first 4 years.
OYO:
The number of strawberry plants growing in a garden is a
sequence defined by 𝑎𝑛 = 𝑛(2𝑛 − 1), where n is the
number of years after planting a strawberry plant. Find the
total number of strawberry plants after 5 years.
95 strawberry plants
Homework

Unit 19 Homework page #14 – 20