CONVERGENCE TESTS FOR INFINITE SERIES
The
geometric series
∑
∞
n
=0
ar
n
=
a
+
ar
+
ar
2
+
ar
3
+
· · ·
is convergent if

r

<
1 and
in that case its sum is
∑
∞
n
=0
ar
n
=
a
1

r
. The geometric series is divergent if

r
 ≥
1.
Test for Divergence
If lim
n
→∞
a
n
does not exist or if lim
n
→∞
a
n
= 0 then the series
∑
∞
n
=1
a
n
is
divergent
. If lim
n
→∞
a
n
= 0, then the test is
inconclusive.
The Integral Test
Suppose
f
(
x
) is a positive, continuous and
decreasing
function
on an interval [
a,
∞
) such that
f
(
n
) =
a
n
. Then
(i)
if
∞
a
f
(
x
)
dx
is convergent, then
∑
∞
n
=1
a
n
is convergent, and
(ii)
if
∞
a
f
(
x
)
dx
is divergent, then
∑
∞
n
=1
a
n
is divergent.
The
pseries
∑
∞
n
=1
1
n
p
is convergent if
p >
1 and divergent if
p
≤
1.
The Comparison Test
Suppose that
∑
∞
n
=1
a
n
and
∑
∞
n
=1
b
n
are series with positive
terms and
a
n
≤
b
n
eventually.
Then
(i)
if
∑
∞
n
=1
b
n
is convergent, then
∑
∞
n
=1
a
n
is convergent also, and
(ii)
if
∑
∞
n
=1
a
n
is divergent, then
∑
∞
n
=1
b
n
is divergent also.
The Limit Comparison Test
Suppose that
∑
∞
n
=1
a
n
and
∑
∞
n
=1
b
n
are series with
positive terms.
If lim
n
→∞
a
n
b
n
=
c
where
c
is a finite positive number, then either both
series converge or both diverge.