Download Sequences 9.1

Calculus Section 9.1 Sequences
-List the terms of a sequence and write a sequence.
-Determine whether a sequence converges or diverges
Homework: page 602 #’s 1, 7, 13, 15, 19, 25, 29,
31, 37, 38, 41, 47, 49, 53, 67, 69, 78, 80, 83, 87, 90
Mathematically, 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, not standard notation. For
example: 1, 2, 3, 4, 5, … n, …
is a representation of a sequence.
a1 a2 a3 a4 a5 … a n …
1 is mapped onto a1, 2 is mapped onto a2, and so on. The numbers a1, a2, a3, …, an, … are the terms of the
sequence. The number an is called the nth term of the sequence, and the entire sequence is called {an}.
Example) Listing Terms List the terms of each sequence:
 n 
1) {an} = {3 + (-1)n}
2) {an} = 
3) {dn} is dn+1 = dn – 5, d1 = 25

1  2 n 
The primary focus of this chapter concerns sequences whose terms approach limiting values. These
1
sequences are said to converge. For instance, the sequence  n  =
converges to 0.
2 
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 exists if for each ε > 0, there
x 
exists M > 0 such that |an – L| < ε whenever n > M. If the limit L of a sequence exists, then the sequence
converges to L. If the limit of a sequence does not exist, then the sequence diverges.
This sequence converges to 1
because it approaches the
value y = 1 as the x values
approach infinity. If the gap
between the dots does not
decrease as you approach
infinity, then the sequence
will diverge.
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(anbn )  LK
4) lim(
n 
n 
n 
n 
an
L
)  , bn  0 and K  0
bn
K
Example) Finding a Limit, Does it Converge or Diverge
Find the limit of each of the following functions.
 n 
2) {an} = 

1  2 n 
1) {an} = {3 + (-1)n}
3) {an} =
n2
2n  1
Squeeze Theorem for Sequences
If lim an  L  lim bn and there exists an integer N such that
n 
n 
an ≤ cn ≤ bn for all n > N, then lim cn  L .
n 
1

Example) Show that the sequence {cn} = (1) n  converges, and find its limit.
n !

Find the nth Term of a Sequence
Find a sequence {an} whose first five terms are
2 8 26 80 242
, ,
, ,
… then determine if it converges.
1 2 6 24 120
A sequence {an} is called monotonic if its terms are nondecreasing a1 ≤ a2 ≤ a3 ≤ … ≤ an ≤ an+1 ≤ … OR if its terms
are nonincreasing a1 ≥ a2 ≥ a3 ≥ … ≥ an ≥ an+1 ≥ … If any term does not follow the pattern, it isn’t monotonic.
A sequence {an} is bounded above if there is a real number M such that an ≤M for all n. M is the upper bound.
A sequence {an} is bounded below if there is a real number N such that an ≥N for all n. N is the lower bound.
A sequence {an} is bounded if it is both bounded above and bounded below.
If a sequence {an} is bounded and monotonic, then it converges.