Write the terms of a sequence Identify sequences as convergent or divergent Use summation notation

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.
 
n
a 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
n
a
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 24
S

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
, , ..., n
a a a 1 2
, , ..., n
b 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

 
    

 2
2
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
 11.1 Assignment (College Algebra)
 Page 1004: 24-46 (even), page 985: 26-36 (even);
page 978: 38-46 (even)
 11.1 Classwork Check
 Quiz 10.4