The convergence tests I've discussed (such as the Ratio Test and Limit Comparsion) apply to positive term series. What can you say about convergence if a series has negative terms?
If there are only finitely many negative terms, you can "chop them off" and consider the series that remains, which will have only positive terms. What about the case where there are infinitely many negative terms?
If the positive and negative terms alternate, the Alternating Series Test may tell you that the series converges. But there are series to which it does not apply.
One approach you might take to series with negative terms is to force all the negative terms to be positive by taking absolute values.
Definition. A series converges absolutely if the absolute value series converges.
Forcing all the terms to be positive should make it more difficult for a series to converge, since you lose the benefit of having negative terms cancelling with positive terms (which might keep the partial sums from blowing up). You would think that if you can do a more difficult thing (converge absolutely) then you ought to be able to do the easier thing (converge), and this turns out to be true.
Theorem. If converges absolutely, then it converges.
Proof. Suppose that converges absolutely, so converges.
Step 1. is a series with nonnegative terms.
If , then , and . If , then (because a positive number must be greater than a negative number), and so .
Step 2. converges.
By taking cases as in Step 1, I have . Adding to both sides, I get . The series converges by assumption, so converges as well. Therefore, the inequality shows that converges by comparison.
Step 3. converges.
Since converges, its negative converges. I have
Since the two series on the right converge, it follows that converges as well.
The theorem says something which is reasonable, and it's also useful: Sometimes the easiest way to show a series converges is to show that the absolute value series converges.
On the other hand, perhaps taking absolute values results in a series which no longer converges, even if the original series does.
Definition. A series converges conditionally if the absolute value series diverges, but the original series converges.
Note that to conclude that a series converges conditionally, you need to know two things:
Example. The alternating harmonic series
converges by the Alternating Series Rule.
If I replace each term with its absolute value (removing the ), I get the harmonic series
which diverges.
Therefore, the series converges conditionally.
Example. Consider the series .
If I replace each term with its absolute value (removing the ), I get . This is a p-series with so it converges.
Therefore, the series converges absolutely.
Example. The series
does not alternate. In fact,
Thus, you can't use the Alternating Series Test. On the other hand, since the series has negative terms, many convergence tests --- the Integral Test, the Ratio Test, the Root Test --- don't apply.
The trick is to consider the absolute value series, which is . Since for all k,
The series is a convergent geometric series. Therefore, the series converges by comparison.
Thus, the original series converges absolutely. Hence, the series converges.
In problems which ask you to check for absolute or conditional convergence, you should be careful to do things in the correct order. Here's how to approach the question: "Does the series converge absolutely, converge conditionally, or diverge?"
Note: There is no point in taking absolute values if the series has positive terms. For a series with positive terms, you simply check for convergence or divergence.
Example. Does the series
converge absolutely, converge conditionally, or diverge?
The series diverges by the Zero Limit Test.
Example. Does the series
converge absolutely, converge conditionally, or diverge?
so the Zero Limit Test fails.
Consider the absolute value series . (Taking absolute values removes the .) For large n,
Apply Limit Comparison:
The limit is finite and positive.
The series diverges, because it's a p-series with . Therefore, the absolute value series diverges by Limit Comparison, and the original series does not converge absolutely.
Return to the original series. The terms alternate. If , then
for , so the terms decrease in magnitude. Finally,
The hypotheses of the Alternating Series Test are satisfied, so the original series converges.
Since the original series converges, but does not converge absolutely, it converges conditionally.
Example. Does the series
converge absolutely, converge conditionally, or diverge?
Consider the absolute value series . Apply Limit Comparison:
The limit is a finite positive number. The series converges, because it is a p-series with . Therefore, converges by Limit Comparison.
Hence, converges absolutely.
Example. Does the series
converge absolutely, converge conditionally, or diverge?
The table below shows the sign of for to .
This is not an alternating series.
Consider the absolute value series .
converges, since it's a multiple of a p-series with .
Therefore, the absolute value series converges by comparison.
Hence, the original series converges absolutely.
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Copyright 2005 by Bruce Ikenaga