Sequences
Chapter 9(1)
Sequence Definition and Notation
A Sequence is a function whose domain is the
positive integers. We use subscript notation to
denote terms of a sequence.
a1 , a2 , a3 , a4 , ...
Terms of a Sequence
There are several ways to find terms of a sequence
In terms of n
an  n  3
Recursively
based on past terms
an  3an1  7
a1  1  3  4
a1  6 (must be given)
a2  2  3  5
a3  3  3  6
a2  3(6)  7  25
a3  3(25)  7  82
a20  20  3  23 a4  3(82)  7  253
Limit of a Sequence
A sequence that approaches a
value is said to Converge.
1
2
, 14 , 18 , 161 , ...
1 1
2 8
1
32
1
64
, , , ,...
Converges to 0
Converges to 0
1, , , , ... Convergesto
2
3
1
2
3
5
4
7
, 14 , 18 , 0,...
Diverges
You can’t tell if a sequence converges
just by looking at it.
1
2
Limit of a Sequence
Let L be a real number and let f be a function where
Lim f ( x)  L
x 
If fn = an for each positive integer n, then
Lim an  L
x 
This ties sequences to functions and we can
then use what we have learned about limits of
functions to find limits of sequences.
Absolute Value Theorem
If Lim an  0, then Lim an  0
n 
n 
This allows you to find limits when
the signs change in a sequence.
1
2
,
1
4
, 81 ,
1
16
,...
Converges to 0
Show if the sequence has
a limit and find the limit.
The sequence is:
n 2
an  2
n 1
2
11
a1  32 , a2  65 , a3  10
, ...
x 2
Lim 2
x  x  1
2
1
So  Lim an  1
x 
Show if the sequence has
a limit and find the limit.
Lim1
 Lim e

a1  2, a2  2.25, a3  2.37
The sequence is:
x 
a n  1
1 n
n

1 x
x
 
Ln 1 1
x
x 1
 Lim e
xLn 1 1x 
x 
L’Hopital
x 
 Lim e
x 
So  Lim an  e
x 
1
1 x1
e
Show if the sequence has
a limit and find the limit.
The sequence is:
a1  4, a2  4 34 , a3  4 89
Lim 5  n12
x 
a n  5  n12
5n 2  1
 Lim

5
2
x 
n
So  Lim an  5
x 
Show if the sequence has
a limit and find the limit.
an 
( n1)!
( n2)!
The sequence is:
a1  2!3!  13 , a2  4!3!  14 , a3  5!4!  15
Lim
( n 1)!
( n  2)!
Lim
1
n2
x 
x 


( n 1)( n )( n 1) ...
( n  2)( n 1) n ( n 1)...
1

0
Monotonic Sequence
A sequence with terms that are
non-decreasing
a1  a2  a3  . . .  an
Or non-increasing
a1  a2  a3  . . .  an
Bounded Sequence
A sequence is bounded above if
there is a real number M such that
an  M for all n
A sequence is bounded below if
there is a real number N such that
N  an for all n
Is the sequence monotonic
and/or bounded?
Monotonic
No Upper Bound
Lower Bound:
an 
n2
n 1
a1  12 , a2  34 , a3  94 , ...
Lim an  
n 
1
2
(Since it is monotonic it starts
at it’s lowest value)
Graphing a sequence
an 
n2
n 1
Select MODE, SEQ, and DOT
Type in the function
Under Window set the n values from 1 to 10
and the x & y max & min