Download 8.1 Sequences Objectives: To determine whether a sequence

8.1 Sequences
Objectives:
To determine whether a sequence converges and find its
limit if any.
Defn: A sequence can be thought of as a list of numbers
w/a definite order: a1 , a2 , ..., an , ...
Note: For every positive integer n, there is a
corresponding an term, so a sequence can be defined as a
function w/ domain = set of positive integers.
Notation: f (n) = an
{a1 , a2 , ...}
Or
Ex: an =
n
2n + 1
1 2
3 4
Ex: {− , , − , ,...}, the general term an =?
4 9 16 25
{5, 1, 5, 1, 5, 1,...}
Defn: A sequence{an } has the limit L and we write
lima n = L or an → L as n → ∞ if we can make the terms an
n →∞
as close to L as we like by taking n sufficiently large. If
a n exists, we say the sequence converges (or is
lim
n →∞
convergent), otherwise, we say the sequence diverges (or
is divergent). See graphs:
Theorem: If lim
f ( x) = L and f (n) = an , where n is an
x →∞
integer, then lim
a =L
n →∞ n
Ex: a) Does an =
n +1
converge or diverge?
3n − 1
(−1) n n 3
b) an = 3
n + 2n 2 + 1
Defn: lim
a n = ∞ means that for every positive number M
n →∞
there is an integer N such that if n > N, then a n > M
Note: in this case the sequence still diverges but diverges
to ∞
n2 + 1
Ex: an =
3n − 3
Limit Laws for Sequences
If {an } and {bn } are convergent sequences and c is a
constant, then:
1) lim(a n ±bn ) =
n →∞
2) lim c a n =
n →∞
3) lim(a n bn ) =
n →∞
4) lim(
n →∞
an
)=
bn
5) lim(a n ) p =
n →∞
The Squeeze Theorem
If an ≤ bn ≤ cn for n ≥ n0 , and lim
a = lim cn = L , then
n →∞ n
n →∞
b =L
lim
n →∞ n
cos 2 (n)
Ex: an =
2n
Theorem: lim | an |= 0 , then lim an = 0
n →∞
n →∞
Why?
(−1) n
Ex: an = 3
n
[ln(n)]2
Ex: an =
n
Caution: we cannot use L’Hopital’s rule directly on {an },
we have to use its corresponding function
Ex: an =
(n + 2)!
n!
Defn: A sequence {an } is called increasing if an < an+1 for
all n ≥ 1, that is, a1 < a2 < a3 < .... It is called decreasing if
an > an+1 for all n ≥ 1. A sequence is ‘monotonic’ if it is
either increasing or decreasing.
Ex : Determine if the sequence is increasing, decreasing,
or neither.
1
a) an =
2n + 1
b) an =
2n − 3
3n + 4
Defn: A sequence {an } is bounded above if there is a
number M such that an ≤ M for all n ≥ 1. It is bounded
below if there is a number m such that an ≥ m for all
n ≥ 1. If it is bounded above and bounded below, then
{an } is a bounded sequence.
Ex: (cont)
1
a) an =
2n + 1
b) an =
2n − 3
3n + 4
Monotonic Sequence Theorem
Every bounded, monotonic sequence is convergent.