Infinite Series Tests: Convergent or Divergent?
First, it is important to recall some key definitions and notations before discussing the individual series
tests.
Sequence
: a function that takes on real number values
a
and is defined on the set of positive integers
= 1, 2, 3,…
n
n
Listed Notation:
{ }
123
,,,,,
nn
aa
a
a
a
=
……
For example:
{}
1234
,,,
12345
n
n
a
n
==
+
…
for
= 1, 2, 3, 4, .
..
n
Series
: a summation of a sequence of numbers
Notation:
1
n
a
∞
=
=
+++++
∑
Note that a sequence is a list of values, each value occurring for different value of
n
However a series is the sum of these values.
Convergence
- whenever a sequence or a series has a limit
Divergence
- whenever a sequence or a series does not have a limit
As a matter of notation, in the following pages, let
n
nc
a
∞
=
=
n
a
∑
∑
for some integer
0
c
≥
The first type of series test is used only for a series with a specific form:
TEST FOR CONVERGENCE OF A GEOMETRIC SERIES
:
A
geometric series
is a series which follows the pattern:
2
0
n
ar
a
ar
ar
ar
∞
=
=++ ++ +
∑
revised 12/07
1
where
a
is the initial term and
r
is a constant ratio (i.e.,
1
2
,
3
4
,
5
3
, etc. ).
If
1
r
<
(that is,
−
1
<
r
<
1
), then the series converges.
If
1
r
≥
, the series diverges.
Sum of the series:
Further, when convergent, this series converges to:
a
1
−
r
Example
:
Determine whether the series
2
+
1
+
1
2
+
1
4
+
…
converges.
Solution
:
To find
r
divide any term by the term preceeding it.
The series follows the pattern of a geometric series with
a
=
2
and
r
=
1
2
.