8.4. Series With Odd or Even Negative Terms http://www.ck12.org
Determine if the series∑k∞= 1 (− 2 k^1 +)k+ 11 converges absolutely.
Solution
The series made up of the absolute values of the terms is∑∞k= 1
∣∣
∣(− 21 k+)k 1 +^1∣∣
∣=^13 +^15 +^17 +.... This series behaves like∑∞k= (^121) k=^12 ∑∞k= (^11) k, which diverges. The series∑∞k= 1 (−^1 )k+^12 k^1 + 1 does not converge absolutely.
It is possible to have a series that is convergent, but not absolutely convergent. We know that∑k∞= 1 (− 2 k^1 +)k 1 +^1 is an
alternating series. By the Alternating Series Tes we know that this series will converge if both of two subtests are
satisfied, that is, if 1.μ≥μk+ 1 for allKand 2. limk→∞μk=0. In this case,μ= 2 k^1 + 1 andμk+ 1 = 2 (K+^11 )+ 1 = 2 k^1 + 3 ,
so we know, since the second denominator is always larger than the first, that the first subtest(μ≥μk+ 1 )of the
Alternating Series Test is satisfied. Then we can also see that limk→∞ 2 k^1 + 1 =0, and so the second subtest is also
satisfied, and so the series converges.
Conditional Convergence
An infinite series that converges, but does not converge absolutely, is called aconditionally convergentseries.
Example 5
Determine if∑∞k= 1 (−^1 k)k+^1 converges absolutely, converges conditionally, or diverges.
Solution
The series of absolute values is∑∞k= 11 k. This is the harmonic series, which does not converge. So, the series
∑∞k= 1 (−^1 k)k+^1 does not converge absolutely. The next step is to check the convergence∑∞k= 1 (−^1 k)k+^1. This will tell us
if the series converges conditionally. Applying the Alternating Series Test:
The sequence^11 >^12 >^13 >...is nonincreasing and limk→∞^1 k=0.
The series∑∞k= 1 (−^1 k)k+^1 converges. Hence, the series converges conditionally, but not absolutely.
Rearrangement
Making arearrangementof terms of a series means writing all of the terms of a series in a different order. The
following theorem explains how rearrangement affects convergence.
Theorem
If∑∞k= 1 ukis an absolutely convergent series, then the new series formed by a rearrangement of the terms of the series
also converges absolutely.
This tells us that rearrangement does not affect absolute convergence.
Review Questions
Determine if the series converges or diverges.
1.∑∞k= 1 (− 1 )k+^13 k 2 k+k
2.∑∞k= 1 (− 1 )k+^132 kk
- Computes 3 for∑∞k= 1 (− 1 )k+^1 k^43.
- The series∑∞k= 1 (− 1 )kk^52 converges according to the Alternating Series Test. Let∑∞k= 1 (− 1 )kk^52 =S. Compute
s 3 for∑∞k= 1 (− 1 )kk^52 and determine the bound on|s 3 −S|.