Section 10.4: Other Convergence Tests
Def: An alternating series is a series whose terms alternate signs. For example,
1 1 1
1 − + − + ... is an alternating series. We would like to know under what conditions
2 3 4
does an alternating series converge?
The Alternating Series Test: The alternating series
∞
P
(−1)n an , where an > 0, con-
n=1
verges if it satisfies both conditions given below:
• an+1 ≤ an for all n (ie the sequence {an } is decreasing).
• lim an = 0
n→∞
Illustration as to why this is true.
Consider
∞
P
(−1)n−1 an , where an > 0 and an decreases to zero.
n=1
s 1 = a1
s 2 = a1 − a2
s3 = a1 − a2 + a3
s4 = a1 − a2 + a3 − a4
s5 = a1 − a2 + a3 − a4 + a5
.
.
.
.
Plot the sequence of partial sums below:
EXAMPLE 1: Determine whether the following series are convergent.
∞ (−1)n
P
√
a.)
n+3
n=1
b.)
c.)
1
1
1
1
−
+
−
+ ..
ln 2 ln 3 ln 4 ln 5
∞ (−1)n n2
P
n=1
1 + n2
Def: A series is absolutely convergent if
∞
P
n=1
∞
P
n=1
|an | converges. If
∞
P
an converges, but
n=1
|an | diverges, then the series is conditionally convergent. Note: If
of positive terms, and
∞
P
∞
P
an is a series
n=1
an converges, then by default it is absolutely convergent.
n=1
EXAMPLE 2: Determine whether the following series are convergent, absolutely convergent, or divergent.
∞ 1
P
a.)
4
n=1 n
b.)
∞ (−1)n
P
n=1
√
n n
c.)
∞ (−1)n
P
n=2
d.)
n ln n
∞ (−1)n (n + 1)
P
n=1
π4 n
Remainder Estimate and The Alternating Series Theorem
If
∞
P
(−1)n an , an > 0, is a convergent alternating series, and a partial sum
n=1
sn =
n
P
(−1)i ai is used to approximate the sum of the series with remainder Rn , then
i=1
|Rn | ≤ an+1
EXAMPLE 3: Consider
∞ (−1)n
P
n=1
n3
a.) Prove the series is absolutely convergent.
b.) Use s6 to approximate the sum of the series and estimate the error (remainder).
EXAMPLE 4: How many terms of the series
∞ (−1)n
P
n=1
approximate the sum to within
EXAMPLE 5: Approximate
n!4n
do we need to add together to
1
?
100
∞ (−1)n
P
n=1
√
n
with error less than 10−4 .
The Ratio Test:
∞
an+1 P
= L < 1, then the series
• If lim an is absolutely convergent (and therefore
n→∞ an
n=1
convergent).
∞
an+1 an+1 P
= ∞, then the series
• If lim = L > 1 or lim an is divergent.
n→∞ an
n→∞ an
n=1
an+1 = 1, then the test fails.
• If lim n→∞ a
n
EXAMPLE 6: Determine whether the following series converge or diverge.
∞ (−2)n
P
a.)
n+1
n=1 n5
b.)
∞ (2n + 1)!
P
n=1
n!10n
∞ (−2)n
P
converges absolutely
n5n+1
by the ratio test. Use s2 to approximate the sum of the series and estimate the error.
Next, find the sum of the series to within 5 decimal places.
EXAMPLE 7: We showed in example 6a that the series
n=1
EXAMPLE 8: For which of the following series is the ratio test inconclusive?
∞ (−1)n
P
a.)
n
n=1
b.)
∞ (−1)n
P
n!
n=0
c.)
∞ (−1)n n!
P
n=1
d.)
(n + 1)!
∞ 2n
P
n=1
n