This document defines sequences and series and provides examples of how to write terms of sequences and evaluate partial sums of series. It discusses writing sequences as functions with the natural numbers as the domain and the term values as the range. Examples are provided of finding the next term in a sequence and using Desmos to list terms. The document also defines convergent and divergent sequences, introduces summation notation for writing series, and provides properties and rules for manipulating summations including evaluating finite series.

1. 11.1 Sequences and Series
Chapter 11 Further Topics in Algebra

2. Concepts and Objectives
⚫ Sequences and Series
⚫ Write the terms of a sequence
⚫ Identify sequences as convergent or divergent
⚫ Use summation notation to write the terms of a series
and evaluate its sum

3. Sequences
⚫ A sequence is a function whose domain is the set of
natural numbers () (the term numbers), and whose
range is the set of term values.
Examples: Find a next term for the following:
A. 3, 10, 17, 24, 31, …
B. 3, 6, 12, 24, 48, …

4. Sequences
⚫ A sequence is a function whose domain is the set of
natural numbers () (the term numbers), and whose
range is the set of term values.
Examples: Find a next term for the following:
A. 3, 10, 17, 24, 31, …
38 (add 7)
B. 3, 6, 12, 24, 48, …
96 (multiply by 2)

5. Sequences (cont.)
⚫ Instead of using f(x) notation to indicate a sequence, it is
customary to use an, where . The letter n is
used instead of x as a reminder that n represents a
natural (counting) number.
⚫ The elements in the range of a sequence, called the
terms of the sequence, are . The first term
is found by letting n = 1, the second term is found by
letting n = 2, and so on. The general term, or the nth
term, of the sequence is an.
( )na f n=
1 2 3, , , ...a a a

6. Sequences (cont.)
⚫ You can use Desmos to list the term in a sequence:
⚫ Type the sequence function into Desmos as a
function, f(n).
⚫ Add a table.
⚫ Change the x1 to n1 and y1 to f(n1). (To put in a
subscript, put an underline in front.
⚫ Enter 1 for n1. When you hit the Enter key, it will fill
in the value for f(n1). Enter 2, and press the Enter key
again, and it will start to populate the list for you.

7. Sequences (cont.)
⚫ Example: Write the first five terms of the sequence.
Step 1: Enter the sequence into Desmos as a function.
1
2
n
n
a
n
+
=
+
(Notice that I
used parentheses
so that Desmos
would divide the
right expression.)

8. Sequences (cont.)
⚫ Example: Write the first five terms of the sequence.
Step 2: Add a table by clicking on the “+” button.
1
2
n
n
a
n
+
=
+

9. Sequences (cont.)
⚫ Example: Write the first five terms of the sequence.
Step 3: Change the x and y.
1
2
n
n
a
n
+
=
+

10. Sequences (cont.)
⚫ Example: Write the first five terms of the sequence.
Step 4: Enter 1-5 for n.
1
2
n
n
a
n
+
=
+
There’re our answers:
a1 = 0.67
a2 = 0.75
a3 = 0.8
a4 = 0.83
a5 = 0.86

11. Infinite Sequences
⚫ If the terms of an infinite sequence get closer and closer
to some real number, the sequence is said to be
convergent and to converge to that real number.
⚫ For example, the sequence defined by
approaches 0 as n becomes large.
⚫ A sequence that does not converge to any number is
divergent.
⚫ An example would be the sequence an = n2 because
the numbers just keep getting bigger.
1
na
n
=

12. Series
⚫ A series is the indicated sum of the terms of a sequence.
⚫ The sum of part of a series is called a partial sum.
⚫ The sum of the first n terms of a series is called the nth
partial sum of that series. It is usually represented by Sn.
⚫ Example: For the sequence 3, 5, 7, 9, …, find S4.
= + + + =4 3 5 7 9 24S

13. Series (cont.)
⚫ Series are usually written using summation notation.
We use the Greek letter  (sigma) to represent this.
A finite series is an expression of the form
An infinite series is an expression of the form
The letter i is called the index of summation.
1 2
1
...
n
n n i
i
S a a a a
=
= + + + = 
1 2
1
... ...n i
i
S a a a a


=
= + + + + = 

14. Summation Properties
⚫ If and are two sequences and
c is a constant, then for every positive integer n,
1 2, , ..., na a a 1 2, , ..., nb b b
1
(a)
n
i
c nc
=
= 1 1
(b)
n n
i i
i i
ca c a
= =
= 
( )
1 1 1
(c)
n n n
i i i i
i i i
a b a b
= = =
+ = +   ( )
1 1 1
(d)
n n n
i i i i
i i i
a b a b
= = =
− = −  

15. Summation Properties (cont.)
⚫ Summation Rules:
( )
1
1
1 2 ...
2
n
i
n n
i n
=
+
= + + + =
( )( )2 2 2 2
1
1 2 1
1 2 ...
6
n
i
n n n
i n
=
+ +
= + + + =
( )
22
3 3 3 3
1
1
1 2 ...
4
n
i
n n
i n
=
+
= + + + =

16. Summation Properties (cont.)
⚫ Example: Evaluate ( )
7
2
1
3 5
i
i i
=
+ +
( )
7 7 7 7
2 2
1 1 1 1
3 5 3 5
i i i i
i i i i
= = = =
+ + = + +   
7 7 7
2
1 1 1
3 5
i i i
i i
= = =
= + +  
( )( ) ( )
( )
7 7 1 2 7 1 7 7 1
3 7 5
6 2
 + + +
= + + 
 
140 84 35 259= + + =

17. Classwork
⚫ College Algebra
⚫ Page 1004: 24-46, page 978: 28-36, page 969: 44-48