AP CALCULUS BC
Section Number:
9.1
LECTURE NOTES
MR. RECORD
Day: 1 of 2
Topics: Sequences
- Limits
- Pattern Recognition
Sequences
Example:
1, 4, 7, 10, ,3n  2
a1 , a2 , a3 , a4 , , an
Recursive Example:
an   3n  2
b1  25
25, 20, 15, 10,
bn 1  bn  5
Example 1: the Terms of a Sequence.

n
Write out the terms of the sequence an  
.
1  2n
y
x




Limit of a Sequence
DEFINITION OF THE LIMIT OF A SEQUENCE
Let L be a real number. The limit of a sequence an  is L, written as
lim an  L
n 
if for each   0 , there exists M  0 such that an  L   whenever
nM .
If the limit L of a sequence exists, then the sequence converges to L. If
the limit to a sequence does not exist, then the sequence diverges.
THEOREM 9.1: LIMIT OF A SEQUENCE
Let L be a real number. Let f be a function of a real variable such that
lim f ( x)  L
x 
If an  is a sequence such that f (n)  an for every positive integer n,
then
lim an  L
x 
Example 2: Finding the Limit of a Sequence
n
 1
Find the limit of the sequence whose nth term is an   1   .
 n
(If you don’t recall the answer, use your TI-Nspire.)







THEOREM 9.2: PROPERTIES OF LIMITS OF SEQUENCES
Let lim an  L and lim bn  K .
n 
n 
1. lim  an  bn   L  K
2. lim can  cL, c is any real number
3. lim  a nbn   LK
a  L
4. lim  n   , bn  0 and K  0
n  b
 n  K
n 
n
n
Example 3: Determining Convergence or Divergence.
Determine the convergence or divergence of the sequence with the given nth term. If the sequence
converges, find its limit.
n
n2
a. an  3  (1) n
b. bn 
c. cn  n
1  2n
2 1
THEOREM 9.3: SQUEEZE THEOREM FOR SEQUENCES
If
lim an  L  lim bn
n 
n 
and there exists an integer N such that an  cn  bn for all n  N , then
lim cn  L
n 
Example 4/ PROOF: Using the Squeeze Theorem.
1

Show the sequence cn   (1) n  converges and find its limit.
n !

n
Note: lim k  0, for any constant k
n 
THEOREM 9.4: ABSOLUTE VALUE THEOREM
For the sequence an  ,
if lim an  0
n 
then
lim an  0
n 
n!
Pattern Recognition for Sequences
Example 5: Finding the nth term of a Sequence.
Find a sequence an  whose first five terms are given and then determine whether the particular
sequence you have chosen converges or diverges.
2 4 8 16 32
, , , , ,
a.
1 3 5 7 9
====================================================================================
Note that merely the listing of the first three terms of
a sequence may produce many different results.
(See the box to the right.)
We hope to be given five (or more) terms and
rely on our deductive reasoning skills to generate what
we hope is a unique sequence.
1 1 1 1
1
, , , , , n,
2 4 8 16
2
1 1 1 1
6
,
bn  : , , , , ,
2 4 8 15
(n  1)(n 2  n  6)
an  :
1 1 1 7
n 2  3n  3
, , , , ,
,
2 4 8 62
(n  1)(n 2  n  6)
1 1 1
n(n  1)(n  4)
,
d n  : , , , 0, , 2
2 4 8
6(n  3n  2)
cn  :
====================================================================================
2 8 26 80 242
,
b.  , ,  , , 
1 2
6 24 120
AP CALCULUS BC
Section Number:
9.1
LECTURE NOTES
Topics: Sequences
- Monotonic Sequences
- Bounded Sequences
Monotonic Sequences and Bounded Sequences
DEFINITION OF A MONOTONIC SEQUENCE
A sequence an  is monotonic if its terms are nondecreasing
a1  a2  a3 
or if its terms are nonincreasing
a1  a2  a3 
 an 
 an 
Example 6: Determining Whether a Sequence is Monotonic.
Determine whether each sequence having the given nth term
is monotonic
a. an  3  (1) n
b. bn 
2n
1 n
c. cn 
n2
2n  1
MR. RECORD
Day: 2 of 2
DEFINITION OF A BOUNDED SEQUENCE
1. A sequence an  is bounded above if there is a real number M such that an  M for
all n. The number M is called an upper bound of the sequence.
2. A sequence an  is bounded below if there is a real number N such that N  an for
all n. The number N is called a lower bound of the sequence.
3. A sequence an  is bounded if it is bounded above AND below.
or if its terms
are nonincreasing
THEOREM
9.5: BOUNDED MONOTONIC SEQUENCES
If a sequence an  is bounded and monotonic, then it converges.
Example 7: Bounded and Monotonic Sequences.
Determine whether each sequence is bounded and/or monotonic.
1 
a. an    
n
b.
 n2 
bn    
 n  1
c.
cn   (1)n 