Section 10.1
Definitions:
A sequence ,
location of
Sequences
{a n }
an
or a n n  1 , is an ordered list of numbers. The subscript n tells the
in the list.

Example: The sequence
1 
 
 n  n 1
is
1
1,
,
2
1
3
,
1
, ...
4

Example: In the sequence,
n 1
,


 n  1  n 1
the 5th term is
51
51

4
6

2
.
3
A sequence need not have a formula for the nth term. For example, the nth term could be
the nth decimal of the number e.
A recursive sequence is a sequence for which the first term is given and for each n,
a n  1  f ( a n ) for some function f.
Example:
a 1  3,
a1  3
a2 
4
,
3
4
a n 1 
and
a 3  3,
a4 
a1  2,
a n 1  1 
a1  2,
a2  1 
Example:
3
We can write the first few terms;
an
4
and so the sequence alternates between 3 and 4/3.
, ...
3
3
an
 2 .5,
a3  1 
2
3
2 .5

11
5
and so on.
The limit of a sequence is L, that is,
| an  L |
lim
n 
an  L
, if
becomes arbitrarily close to 0 if n becomes arbitrarily large.
If the limit L exists, the sequence converges. If no such L exists, the sequence diverges.
1
 
n
Example: The limit of the sequence
is 0 since all terms become arbitrarily close to
0 as n becomes arbitrarily large.


Example: The sequence (  1 )
diverges since the terms alternate between -1 and 1.
No single number L exists so that all terms become arbitrarily close to L if n approaches
infinity.
Example 1: Find
lim
n 
n
n
2
4n
2
 2n  3
 3n  4
.

Example 2: Find the limit of the sequence

 1 
 n sin   
 n   n 1

.
Monotone Sequences: A sequence is monotone decreasing if, for each n,
A sequence is monotone increasing if, for each n, a n 1  a n .
a n 1  a n
.
Bounded Sequences: A sequence is bounded if there exists a number, M, for which
| a n | M for all n.


Example: The sequences,
1
,
 
 n  n 1

The sequences,
2

1  
n

 n 1
,
2 n n  0

 1 n
  
  
2 


n0
are monotone decreasing.
are monotone increasing. Only the last one of these
four sequences is not bounded.
Example: The sequence
n

 (  1)

n







 n 1
is bounded but is not monotone.
Example 3: Show that the sequence defined recursively as
monotone.
a 1  1,
a n 1  5 
6
an
is not
Theorem: If a sequence is both monotone and bounded, then it converges.
Induction: Let P(n) be a statement which depends on n. If P(1) is true and if, for all n,
P(n) is true implies P(n+1) is true, then P(n) is true for all n.
Example 4: Show that the sequence defined recursively by
converges and find the limit.
a 0  1,
a n 1 
1
2
an 
2
an