Chapter 11
Infinite Sequences and Series
Math 1B
§ 11.1 Sequences
A sequence is a list of numbers written in a specific order: a1 , a2 , a3 , … , an , an +1 , …
Notation: All of the following mean the same thing:
{a1,a2,a3 ,...}
€
€ € €
∞
{an }
{an }n=1
€ €
Note: We deal only with infinite sequences.
€
€
Sequence
Defining Formula
Several Terms
Note: n does not have to start at 1.
We were treating the formulas as functions that can only have integers plugged into them:
n +1
f ( n) = 2
n
g( n )
(−1)
=
n +1
2n
Treating sequences as functions will allow us to do many things with sequences that we couldn’t do
otherwise.
€
€
Example: Find a formula for the general term an of the sequence.
# 1 2 3 4 &
$− , ,− , ,...'
% 4 9 16 25 (
€
€
Some sequences don’t have a simple defining formula:
∞
{bn }n=1 where bn = n th digit of π
€
€
€
To graph the sequence {an } , plot the points ( n,an ) i.e. (1,a1 ) , (2,a2 ) , ( 3,a3 ) , …
∞
" n + 1%
Example: Graph the sequence # 2 &
$ n€ ' n=1
€
€
€
€
€
Definition: A sequence {an } has the limit L and we write
lim an = L
n →∞
or
an → L as n → ∞
€
if we can make an as close to L as we want for all sufficiently large n.
€
€
€
If lim an exists, we say the sequence converges. Otherwise, the sequence diverges.
n →∞
€
€
How do we find the limits of sequences?
Most limits of sequences can be found using one of the following theorems.
Theorem 1:
If lim f ( x ) = L and f ( n ) = an when n is an integer, then lim an = L .
n →∞
x →∞
(This tells us we take limits of sequences like we take limits of functions.)
€
€
€
The properties
of limits for functions also hold for limits of sequences.
Limit Laws for Sequences: If {an } and {bn } are convergent sequences and c is a constant,
1. lim( an ± bn ) = lim an ± lim bn
n →∞
n →∞
2. lim can = c lim an
n →∞
€
€
€
€
3. lim( an bn ) = lim an ⋅ lim bn
n →∞
n →∞
an
an lim
= n →∞
n →∞ b
lim bn
n
4. lim
€
€
n →∞
n →∞
€
n →∞
lim bn ≠ 0
n →∞
n →∞
[
5. lim anp = lim an
€n →∞
n →∞
]
p
p > 0,an ≥ 0
Squeeze Theorem for Sequences: If an ≤ bn ≤ c n and n ≥ N and lim an = lim c n = L , then lim bn = L .
n →∞
n →∞
n →∞
€
Theorem 2: If lim an = 0 , then lim an = 0 .
n →∞
n →∞
€
€
€
€
(This is useful for sequences that can’t be written as functions, especially sequences that alternate in
sign.)
€
€
Example: Determine whether the sequence converges or diverges. If it converges, find the limit.
3n 2 −1
a) an =
10n + 5n 2
€
∞
" n %
b) # 2n &
$ e ' n=1
€
Note: Usually we use l’Hospital’s rule on the sequence terms without converting to x’s since the work
is identical. But remember that we technically can’t differentiate when dealing with sequence terms.
#% (−1) n '%∞
c) $ 2 (
%& n %)
n=1
€
d)
n ∞
{(−1) }
n=1
€
n
n
Note: Theorem 2 does not apply since lim (−1) = 1 ≠ 0 , so we can’t say lim(−1) = 1.
n →∞
n →∞
€
€
Theorem:
If lim an = L and the function f is continuous at L, then lim f ( an ) = f ( L) .
n →∞
n →∞
(We can bring the limit inside the function.)
€
€ Find lim ln n + 1 − ln n .
Example:
( ( )
)
n →∞
€
3n
Example: Discuss the convergence of the sequence an = , where n!= 1⋅ 2 ⋅ 3⋅ ⋅ ⋅ n .
n!
€
€
Useful Fact: The sequence {r n } is convergent if −1 < r ≤ 1 and divergent for all other values of r.
$0
lim r n = % if
n →∞
&1
€
€
−1 < r < 1
r€= 1
€
Definition: A sequence {an } is increasing if an < an +1 for all n ≥ 1, a1 < a2 < a3 < ...
It is decreasing if an > an +1 for all n ≥ 1, a1 > a2 > a3 > ...
It is called monotonic if it is either increasing or decreasing.
€
€
€
€
Example: Determine
whether the sequence is increasing, decreasing, or not monotonic.
€
€
€
" n %∞
#
&
$ n + 1' n=1
€
Definition: 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 m ≤ an for all n ≥ 1.
If it is bounded above and below, then {an } is a bounded sequence.
€
€
€
€
Example: Determine if the following sequences are monotonic and/or bounded.
" n %∞
a) #
&
$ n + 1' n=1
€
€
€
b)
n ∞
{(−1) }
n=1
€
c) an =
2
n2
€
Theorem: If {an } is bounded and monotonic then {an } is convergent.
If {an } is bounded above and increasing then it converges and likewise if {an } is bounded below and
decreasing then it converges.
€
€
€
€