Math 152 Class Notes
November 5, 2014
10.4 Other Convergence Tests
In this section, we study convergence tests for series whose terms are not necessarily
positive.
The Alternating Series Test
An alternating series is a series whose terms are alternately positive and negative.
Example 1. Determine whether the series
∞
X
n=1
is convergent or divergent.
(−1)n−1
1
1 1 1
= 1 − + − + ···
n
2 3 4
The Alternating Series Test. If the alternating series
∞
X
(−1)n−1 bn = b1 − b2 + b3 − b4 + · · · ,
(bn > 0)
n=1
satises
(a) If bn+1 ≤ bn for all n
(b) lim bn = 0
n→∞
Then the series is convergent.
The following gure illustrates the idea of the Alternating Series Test.
Example 2. For p > 0,
∞
P
(−1)n−1
n=1
1
is convergent by the Alternating Series Test.
np
∞ (−1)n
P
Example 3.
n=2 ln n
∞ (−1)n n
P
Example 4.
2
n=1 n + 1
∞ (−1)n n2
P
Example 5.
2
n=1 n + 1
Estimating the Sum of an Alternating Series
A partial sum sn of any convergent series can be used as an approximation to the exact
total sum s. The error involved is the remainder Rn = s − sn .
Alternating Series Estimation. If s =
series that satises
(a) 0 < bn+1 ≤ bn
∞
P
(−1)n−1 bn is the sum of an alternating
n=1
and
(b) lim bn = 0
n→∞
then |Rn | = |s − sn | < bn+1 .
Example 6. (a) Approximate the sum of the series
∞
P
(−1)n−1
n=1
1
by using the partial
n3
sum s4 of the rst 4 terms. Estimate the error involved in this approximation.
(b) How many terms are required to ensure that the sum is accurate to within 0.001?
Absolute Convergence
Given any series
P
an , we can consider the corresponding series
X
|an | = |a1 | + |a2 | + |a3 | + · · ·
whose terms are the absolute values of the terms of the original series.
Denition.
A series
P
values
P
an is called absolutely convergent if the series of absolute
|an | is convergent.
Example 7. The series
∞
P
(−1)n−1
n=1
Example 8. The series
∞
P
(−1)n−1
n=1
1
is absolutely convergent.
n2
1
is convergent but not absolutely convergent.
n
Thus it is possible for a series to be convergent but not absolutely convergent. However,
the next theorem shows that absolute convergence implies convergence.
Theorem. If a series
P
an is absolutely convergent, then it is convergent.
Example 9. Determine whether the series
∞ cos n
P
is convergent or divergent.
2
n=1 n
Example 10. From the fact that the series
∞ 1
P
is convergent, we can have new
2
n=1 n
convergent series by switching the signs of the terms back and forth arbitrarily (not
necessary alternately). For example, the following series are all convergent.
1
1
1
1
1
1
1
+
−
+
−
+
−
+ ···
22 32 42 52 62 72 82
1
1
1
1
1
1
1
1 − 2 − 2 + 2 − 2 + 2 + 2 − 2 + ···
2
3
4
5
6
7
8
1
1
1
1
1
1
1
−1 + 2 + 2 − 2 − 2 − 2 + 2 + 2 + · · ·
2
3
4
5
6
7
8
1−
The following test is very useful in determining whether a given series is absolutely
convergent.
The Ratio Test.
∞
an+1 = L < 1, then the series P an is absolutely convergent (and there(a) If lim n→∞ an n=1
fore convergent).
∞
an+1 an+1 = L > 1 or lim = ∞, then the series P an is divergent.
(b) If lim n→∞ an n→∞ an n=1
a (c) If lim n+1 = 1, the Ratio Test is inconclusive; that is, no conclusion can be
n→∞ an
∞
P
drawn about the convergence or divergence of
an .
n=1
Example 11. Determine whether the series
∞
P
(−1)n
n=1
n2 + 1
is convergent or divergent.
2n
∞ (−2)n
P
is convergent or divergent.
Example 12. Determine whether the series
n
n=1 n5
∞ (2n + 1)!
P
Example 13. Determine whether the series
is convergent or divergent.
n
n=1 n!10
∞ (−1)n
P
Example 14. Determine whether the series
is convergent or divergent.
n
n=1
Guidelines to Applying Convergence Tests
The main strategy to apply an appropriate test is to classify the series
to its form.
P
an according
1. If lim an 6= 0
n→∞
∞ (−1)n n2
P
Examples:
,
2
n=1 n + 1
2. If
P
an =
P
an =
P
P
∞
P
12
√ ,
n=1 n n
arn−1 or
∞ 1
P
,
Examples:
n
n=1 2
4. If
1
cos
,
n
n=1
∞
P
cos (nπ) . . .
n=1
P 1
(p-series)
np
∞ 1
P
Examples:
,
2
n=1 n
3. If
∞
P
P
∞ 5
P
√ , ...
n
n=1
arn (geometric series)
∞ 22n
P
,
1−n
n=1 3
∞ 22n+3
P
, ...
n
n=0 5
an is similar to a p-series or a geometric series
Examples:
∞ 2n − 1
P
,
n+1
3
n=1
∞
P
n
√
,
(n
+
2)
n
+
3
n=1
∞ sin2 n
P
,
2
n
n=1
∞ cos2 n + 5
P
√ , ...
3+
n
n
n=1
5. If
P
an =
Examples:
P
P
(−1)n−1 bn or (−1)n bn (alternating series)
∞
P
(−1)
n−1 1
n
n=1
,
∞
P
1
(−1)
,
ln n
n=2
n
∞ (−1)n n
P
, ...
2
n=1 n + 1
6. If an has factorials and/or exponentials
Examples:
∞
P
(−1)n
n=1
7. If an = f (n) and
Examples:
∞
P
n=1
n2 + 1
,
2n
´∞
1
ne−n ,
2
∞ (2n + 1)!
P
,
n
n=1 n!10
∞ 1
P
, ...
n=1 n!
f (x)dx is easily evaluated
∞ ln n
P
,
n=1 n
∞
P
1
, ...
n=1 n ln n
Choosing a Convergence Test for Infinite Series
Courtesy David J. Manuel
Do
the individual
terms approach 0?
Series Diverges by
No
the Divergence Test.
Yes
Does
the series
alternate
signs?
Do
individual terms
have factorials or
exponentials?
Yes
No
No
Is
individual term
easy to integrate?
Yes
Use Integral Test
No
Do
individual terms
involve fractions with
powers of n?
No
Yes
Use Comparison or
Limit Comp. Test
(Look at Dominating
Terms)
Use Alternating
Series Test
(do absolute value of
terms go to 0?)
Use Ratio Test
Yes
(Ratio of Consecutive
Terms)