AP CALCULUS BC
Section 9.1: SEQUENCES, pg. 594
DO NOW:
List the first five terms of the following sequences:
1.
an   3   1 
n
2.
bn   
n 

1  2n 
AP CALCULUS BC
3.
 n2 
c

 n  n 
 2  1
Section 9.1: SEQUENCES, pg. 594
4.
The recursively defined sequence d n  ,
where d1  25 and d n 1  d n  5
AP CALCULUS BC
Section 9.1: SEQUENCES, pg. 594
SEQUENCES
A sequence is defined as a function whose domain is the set of positive integers. Although a
sequence is a function, it is common to represent sequences by subscript notation rather than
standard function notation.
As you could see in the four above Do Now examples, the sub-index indicates the position of the term
in the sequence.
A sequence is said to converge when its terms approach a limiting value as n   .
If the sequence an  agrees with the function f ( x) at every positive integer, an if f ( x ) approaches
a limit L as n   , then the sequence must converge to the same limit L.
AP CALCULUS BC
Sample Problem #1: FINDING THE LIMIT OF A SEQUENCE
Find the limit of the sequence whose nth term is:
a)
5n 2
an  2
n 2
Section 9.1: SEQUENCES, pg. 594
AP CALCULUS BC
b)
an 
Section 9.1: SEQUENCES, pg. 594
2n
n2  1
AP CALCULUS BC
c)
1
an  sin  
n
Section 9.1: SEQUENCES, pg. 594
AP CALCULUS BC
d)
 2n 2 
an  cos  3 
 n 
Section 9.1: SEQUENCES, pg. 594
AP CALCULUS BC
Section 9.1: SEQUENCES, pg. 594
Sample Problem #2: DETERMINING CONVERGENCE OR DIVERGENCE OF A SEQUENCE
Determine whether the sequence converge or diverge, if the sequence converge find its limit.
a)
n
an   1
n2
n
b)
 5
an  1  
 n
n
AP CALCULUS BC
Section 9.1: SEQUENCES, pg. 594
Sample Problem #3: USING THE SQUEEZE THEOREM


Show that the sequence cn    1
n
1
 converges, and find its limit.
n!
AP CALCULUS BC
Section 9.1: SEQUENCES, pg. 594
AP CALCULUS BC
Section 9.1: SEQUENCES, pg. 594
PATTERN RECOGNITION FOR SEQUENCES
Sample Problem #4:
Write the nth term of the sequence. (There is more than one correct answer.)
a)
1, 
1
1
1
,
,
,...
13 135 1357
b)
1,6,120,5040,362880,...
AP CALCULUS BC
Section 9.1: SEQUENCES, pg. 594
MONOTONIC SEQUENCES AND BOUNDED SEQUENCES
Sample Problem #5:
Determine if the sequence is monotonic. bn 
2n
1 n
AP CALCULUS BC
Section 9.1: SEQUENCES, pg. 594
Sample Problem #6:
Determine if the sequence converges. You do not need to find its limit.
an   
2n 

2
 n  1