F
STRATEGIES FOR
TESTING SERIES
W
e have considered many basic convergence tests for inFnite series in Chapter 10.
The Basic Tests for InFnite Series
(A)
Divergence Test
(B)
Integral Test
(C)
Comparison Test
(D)
Limit Comparison Test
(E)
Ratio and Root Tests
(±)
Leibniz Test for Alternating Series
So given a particular inFnite series, which test should you use? There is no single
answer, but there are some facts and general guidelines you should keep in mind. ±irst of
all, to develop your intuition, you should be familiar with the following key examples:
•
Geometric series:
∞
X
n
=
0
r
n
converges if

r

<
1 and diverges otherwise.
•
p
series:
∞
X
n
=
1
1
n
p
converges if
p
>
1 and diverges otherwise.
•
∞
X
n
=
0
1
n
!
converges by the Ratio Test.
•
∞
X
n
=
1
(
−
1
)
n
n
converges conditionally but not absolutely (Example 4, Section 10.4).
A Frst step, when testing an inFnite series
∞
X
n
=
1
a
n
is to check that the general term
a
n
approaches zero. By the Divergence Test, if lim
n
→∞
a
n
does not exist, or if lim
n
→∞
a
n
exists but is not equal to zero, then
X
a
n
diverges.
EXAMPLE 1
First Check Whether the General Term Approaches Zero
Determine the
convergence of
(a)
∞
X
n
=
0
n
2
n
2
+
1
and
(b)
∞
X
n
=
0
(
−
1
)
n
n
2
n
2
+
1
.
Solution
Both series diverge:
REMINDER
The Leibniz Test states that
∞
X
n
=
0
(
−
1
)
n
a
n
converges if the sequence
{
a
n
}
is positive, decreasing, and
lim
n
→∞
a
n
=
0
.
This test
does not
apply to the series in
Example 1 (b) because the terms
a
n
=
n
2
n
2
+
1
do not satisfy either of the two hypotheses:
{
a
n
}
is not decreasing and
lim
n
→∞
a
n
is
nonzero.
(a)
lim
n
→∞
n
2
n
2
+
1
=
lim
n
→∞
1
1
+
n
−
2
=
1
(nonzero)
⇒
∞
X
n
=
0
n
2
n
2
+
1
diverges
(b)
lim
n
→∞
(
−
1
)
n
n
2
n
2
+
1
=
lim
n
→∞
(
−
1
)
n
1
+
n
−
2
(does not exist)
⇒
∞
X
n
=
0
(
−
1
)
n
n
2
n
2
+
1
diverges
Next, ask yourself if the series resembles a series whose behavior is known. If so,
try using the Comparison or Limit Comparison Tests. The following guidelines may be
useful:
F1
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APPENDIX F
STRATEGIES FOR TESTING SERIES
If the series contains:
Try:
n
k
or other powers of
n
Comparison or Limit Comparison with
p
series
b
n
(
b
constant)
Ratio Test
factorials such as
n
!
, (2n)!
Ratio Test
n
n
Root Test
(
−
1
)
n
Check for absolute convergence or use Leibniz Test
EXAMPLE 2
Check for a Simple Comparison
Determine the convergence of
∞
X
n
=
1
1
1
.
6
n
+
n
−
1
Solution
The series converges by the Comparison Test because
1
1
.
6
n
+
n
−
1
≤
1
1
.
6
n
and
∞
X
n
=
1
1
1
.
6
n
is a convergent geometric series
µ
with
r
=
1
1
.
6
¶
. Although the inequal
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 Spring '08
 ningzhong
 Calculus, Mathematical Series, lim, limit comparison

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