Download 9.1 Sequences and Series

9.1 Sequences and Series
Recursive Sequence
Factorial Notation
Summation Notation
Definition of a Sequence
The values of a function whose domain is
positive set of integers.
f(1), f(2), f(3), f(4),…f(n)....called terms
written as
a1, a2, a3, a4, …, an, …
This is an infinite sequence since it does not
end. If the sequence has a final term, it is
called a finite sequence.
Find the first 4 terms of the
sequence an = 2n + 1
n=1
n=2
n=3
n=4
a1 = 2(1) + 1 = 3
a2= 2(2) + 1 = 5
a3 = 2(3) + 1 = 7
a4 = 2(4) + 1 = 9
3, 5, 7, 9
Find the first 4 terms of the
sequence a   1n1
n
n=1
n=2
n=3
n=4
2n  1
11

 1
1
a1 
 1
21  1 1
2 1

 1
1
a2 

22  1 3
31

 1
1
a3 

23  1 5
4 1

 1
1
a4 

24  1 7
1 1 1
1,
, ,
3 5 7
Recursive Sequence
A sequence where each term of the
sequence is defined as a function of the
preceding terms.
Example: an = an -2 + an-1 : a1 = 1, a2 = 1
a3 = 1 + 1 = 2
a4 = 1 + 2 = 3
a5 = 2 + 3 = 5
1,1, 2, 3, 5, …….
The Fibonacci Sequence is a
Recursive Sequence
In mathematics, the Fibonacci numbers
are the numbers in the following integer
sequence:
By definition, the first two numbers in the
Fibonacci sequence are 0 and 1, and each
subsequent number is the sum of the
previous two.
Why are they called Fibonacci
Leonardo Pisano Bigollo (c. 1170 – c. 1250) also known
as Leonardo of Pisa, Leonardo Pisano, Leonardo
Bonacci, Leonardo Fibonacci, or, most commonly,
simply Fibonacci, was an Italian mathematician,
considered by some "the most talented western
mathematician of the Middle Ages."
Fibonacci is best known to the
modern world for the spreading of
the Hindu-Arabic numeral system
in Europe,
Definition of Factorial Notation
If n is a positive integer, n factorial is
n! 1 2  3  4 n  1  n
0! = 1
5! 1 2  3  4  5  120
Find the first 5 terms of the sequence
Start where n = 1
n
3
an 
n!
Find the first 5 terms of the sequence
31
a1 
3
1!
Start where n = 1
n
3
an 
n!
32 9
a2 

2! 2
33 27 9
a3 


3!
6 2
3 4 81 27
a4 


4! 24 8
35 243 81
a5 


5! 120 40
Evaluate
Expand
6!
1 2  3  4  5  6

8!3! 1  2  3  4  5  6  7  8  1  2  3
Evaluate
Expand
6!
1 2  3  4  5  6

8!3! 1  2  3  4  5  6  7  8  1  2  3
6!
1
1


8!3! 7  8  1  2  3 336
Summation Notation
or Sigma Notation
If you add the terms of a sequence, the
sequence is called a Series.
b
Upper
limit
 f n  f a   .......  f b 
na
Index
variable
Lower
limit
Series
Find each Sum
•
3
 4n 
n 1
Find each Sum
•
3
 4n  41  42  43
n 1
4  8  12  24
Find each Sum
•
 n
6
n 3
2

2 
Find each Sum
•
 n
6
n 3
2
 
 
 
 
 2  3  2  4  2  5  2  6  2
2
7  14  23  34  78
2
2
2

Properties of Sums
If c is a constant
n
 c  cn
i 1
n
 ca
k 1
n
k
 c ak
k 1
Properties of Sums
n
 a
k 1
k
n
 a
k 1
k
n
n
k 1
k 1
n
n
k 1
k 1
 bk    a k   bk
 bk    a k   bk
Find the Sum
4


2
n

6

n 1
Find the Sum
4
4
4
n 1
n 1
n 1
 2n  6   2n   6
Find the Sum
4
4
4
n 1
n 1
n 1
 2n  6   2n   6
21  22  23  24  64
2  4  6  8  24  44
The Infinite Series
Infinite Series go on forever, some converge
on one number, other decrease or
increase forever.

 2n  21  22  23  24  
n 1
Does this Series ever end in one number?
The Infinite Series
Infinite Series go on forever, some converge
on one number, another decrease or
increase forever.

1 1 1
1
1
1
  


 

n
4 16 64 256 1024
n 1 4
Does this Series ever end in one number?
Homework
Page 621 – 622
# 1, 10, 17, 28,
40, 55, 58, 66,
68, 74, 77, 84,
87, 92