Overview of Sequences and Series
MAT 104 – Frank Swenton, Summer 2000
Sequences
are ordered lists of real numbers, such as “
a
1
, a
2
, a
3
, . . .
”, sometimes written
{
a
n
}
•
Just as with limits of functions on the real line, we can talk about limits of sequences; if the numbers
in the sequence approach some real number
L
, then the sequence has a limit. Otherwise, it may
approach
∞
or
∞
, or it may not approach anything at all in particular.
•
Note that
if
lim
x
→∞
f
(
x
) either exists or is infinite, then the limit
lim
n
→∞
f
(
n
) is the same (the graph of
the sequence lies on the graph of the function, so if the function approaches a limit, the sequence
is stuck to the same limit). This fact can be useful in computing limits of some sequences, since
limits of functions can often be more easily evaluated by l’Hˆ
opital’s rule. For example:
Using l’Hˆ
opital’s rule,
lim
x
→∞
x
2
e
x
= lim
x
→∞
2
x
e
x
= lim
x
→∞
2
e
x
= 0, so
lim
n
→∞
n
2
e
n
= 0 as well.
•
Some useful limits to know:
If

r

<
1, then
lim
n
→∞
r
n
= 0
lim
n
→∞
n
√
n
= 1
(and the same for
n
√
n
2
,
n
√
n
3
, etc.)
lim
n
→∞
n
√
n
! =
∞
Some limits that equal
e
:
lim
n
→∞
1 +
1
n
n
,
lim
n
→∞
n
+ 1
n
n
Some limits that equal
1
e
:
lim
n
→∞
1

1
n
n
,
lim
n
→∞
n

1
n
n
,
lim
n
→∞
n
n
+ 1
n
•
Remember the hierarchy of functions (on the “limit comparison test” page);
If
f
(
n
)
g
(
n
), then
lim
n
→∞
f
(
n
)
g
(
n
)
= 0, and
lim
n
→∞
g
(
n
)
f
(
n
)
=
∞
.
For example,
lim
n
→∞
100
n
n
!
= 0,
lim
n
→∞
n
100
2
n
= 0, etc.
•
Remember the rules of exponentiation:
a
m
+
n
=
a
m
·
a
n
,
a

n
=
1
a
n
, and
a
mn
= (
a
m
)
n
Series
(sums of infinitely many terms)
•
Given a sequence
a
1
, a
2
, a
3
, . . .
, we may want to add up all of the values, i.e.
a
1
+
a
2
+
a
3
+
· · ·
.
This is called a
series
, and it is denoted
∞
X
n
=1
a
n
. The individual numbers being added together are
called the
terms
of the series. To add up an infinite number of terms, we first define the
partial
sums
of the series,
S
n
=
a
1
+
a
2
+
· · ·
+
a
n
, i.e.
S
n
is the sum of the first
n
terms of the series.
We then define the meaning of the infinite sum by
∞
X
n
=1
a
n
=
lim
n
→∞
S
n
. If this limit exists as a real
number, we call the series
convergent
; if the limit doesn’t exist, we call the series
divergent
.