SECT. 9-1 SEQUENCES
Sequence
An ordered collection of numbers in a prescribed order defined by a function f(n)
The values, an are called terms
a1, a2, a3, a4 ,..., an ,...
notation {a1, a2, a3, a4, ...} {an} {an}n1
1) Write the first five terms of the sequence
a)
b)
an nn 1
an 1 n (n 1)
3n
Defining Sequences
Determine the pattern in the sequenceAnd use pattern to determine the nth term using inductive reasoning
1, 24
, 79
, 1416
, 2325
...
12 212 ,
22 222 ,
32 232 ,
42 242 ,
52 252 ...
an n2 2n2
2) Write the next two terms for the given sequence
72
, 4, 92
, 5 ....
rewrite : 3.5, 4, 4.5, 5, 5.5, 6, ...
A recursively defined sequence: given the first term, all other terms are defined using that term. d1 = 3.5 and {dn}= dn+1
3) Find next three terms: pattern?
5, 10, 20, 40, ...
4) Find the general term an
35
, - 425
, 5125
, - 6625
, 73125
...
Numerator: start with 1 for first term, add 2
Denominator: powers of 5
Notice terms alternate signs n1 )1(or )1( n
5) Write and expression for the nth term
2, 1, 45
, 57
, 69
,...
rewrite21
, 33
, 45
, 57
, 69
,...
an n 1
2n 1 there may be several ways to write the nth term
Convergence and Divergence of a sequence
We say that a sequence converges to a limit L if
If no limit exists then an diverges.If the terms increase without bound, {an} diverges to infinity
limn
an L
Convergence or Divergence?
Convergence and Divergence ?
limn
an L
Properties of Sequences
if limn
an L and limn bn K then
1. limn
an bn LK
2. limn can cL
3. limn
anbn Lk
4. limn
anbn
LK
, bn 0 and K 0
6) Evaluate
limn
1 n
n
limn
1n
1
Squeeze Theorem for Sequences:If {an}, {bn}, and {cn} are sequences and an ≤ bn ≤ cn for every n andif , then lim limn nn n
a L c
lim nnb L
7) Determine whether the sequence converges or diverges. If it converges, find the limit.
a.
b.
c.
2
5n
ne
1( 1)n
232 n
diverges
to 0converges
to 2converges
8) Determine whether the sequence converges or diverges. If it converges, find the limit.
a.
b.
c.
2cos3nn
1 1( 1)nn
to 0converges
to 0converges
!
2lim
21
!1
nnn
n
n
to 0converges
9) Determine whether the sequence converges or diverges. If it converges, find the limit
2
2
2
2 53n
lim 53
nnn
nnnan
Factorial (!)
5! = 54 321
10) simplify
25!23!
11) simplify
n 2 !n!
Increasing and Decreasing Sequences
•A sequence {an} is increasing if
an an1 for all n 1 a1 a2 a3 ...
•A sequence {an} is decreasing if
an an1 for all n 1 a1 a2 a3 ...
•A sequence {an} is monotonic if it is either always increasing or always decreasing
Monotonic? Sequences
Bounded Sequences
•A sequence {an} is bounded from above if
there is a number M such that an M for all n
•A sequence {an} is bounded from below if
there is a number M such that an M for all n
Bounded Sequences
•A sequence {an} is called Bounded if it is bounded either from above or below. •If a sequence {an} is bounded
and monotonic,then it converges.
12) Is the sequence {an} bounded?
HOME WORKPage 604 # 1,2,3,5, 15-18, 19-22, 25,27,29, 35,39,45, 47, 51, 59, 61,73, 77, 87, 89 and 91