Alternating series have nice properties.
All of the series convergence tests we have used require that the underlying sequence be a positive sequence. We can actually relax this and state that there must be an such that for all ; that is, is positive for all but a finite number of values of . We’vealso stated this by saying that the tail of the sequence must have positiveterms. In this section we explore series whose summation includes negativeterms.
Alternating series test
We start with a very specific form of series, where the terms of the summationalternate between being positive and negative.
Let be a positive sequence. An alternating series is a series of either the form
In essence, the signs of the terms of alternate between positive and negative.
Recall that the terms of the harmonic series come from the harmonic sequence. An important alternating series is the alternating harmonic series:
Geometric series are also alternating series when . For instance, if , the geometricseries is
We know that geometric series converge when and have the sum: When as above,we find
A powerful convergence theorem exists for other alternating series that meet a fewconditions.
Alternating Series Test Let be a positive, nonincreasing sequence where . Then converge.
Does the alternating series test apply to the series
yes no
This is the alternating harmonic series as seen previously. The underlying sequence is, which is positive, decreasing, and approaches 0 as . Therefore we can apply thealternating series test and conclude this series converges.
Does the alternating series test apply to the series
yes no
The underlying sequence is . This is positive and approaches as (use L’Hôpital’s Rule). However, the sequence is not decreasing for all . It is straightforward tocompute , , , and : the sequence is increasing for at least the first terms.
However, we do not “give-up” and immediately conclude that we cannot apply thealternating series test. Rather, consider the long term behavior of . Treating as acontinuous function of defined on , we can take its derivative: The derivativeis negative for all (actually, for all ), meaning is decreasing on . We cannow apply the alternating series test to the series when we start with andconclude that converges; adding the terms with and do not change theconvergence.
The important lesson here is that as before, if a series fails to meet the criteria of thealternating series test on only a finite number of terms, we can still apply the test.
Does the alternating series test apply to the series
yes no
The underlying sequence is . This sequence is positive and approaches as . However,it is not a decreasing sequence; the value of oscillates between and as . We cannotremove a finite number of terms to make decreasing, therefore we cannot apply thealternating series test.
Keep in mind that this does not mean we conclude the series diverges; in fact, it doesconverge. We are just unable to conclude this based on the alternating series test.
Approximating alternating series
While there are many factors involved when studying rates of convergence, thealternating structure of an alternating series gives us a powerful tool whenapproximating the sum of a convergent series.
Alternating Series Approximation Let be a sequence that satisfies the hypotheses ofthe alternating series test, let be the th partial sum, and let Then
- , and
- is between and .
In this case, is called the th remainder of the series.
Here is the basic idea behind this theorem. Say we have an alternating sequence, .Let’s assume the first term is positive, so the second is negative, and so on. We addthe first two numbers and get some number . Now is smaller than , because wesubtracted something from . Next, we add on the third term, , to get the partial sum .This is bigger than , because the sequence is alternating, but is smaller than ,because the sequence is decreasing.
If we know the series converges to some , we can see that we must be bouncing backand forth around as we add and subtract terms. At one point, is larger than , andthen subtracting off the next term makes the partial sum smaller than . In otherwords, the true limit must be between and . Imagine plotting , , and on a numberline. (Or, try it yourself with the alternating harmonic series!) can be nofurther from than whatever the next term in the sequence is. How do weget from to ? By adding (or subtracting) , which takes us back “across” again. In other words, the distance between and can be no more than.
See if you can use these same ideas to prove the alternating series test!
Let’s see an example of approximating an alternating series.
Approximate the sum of the alternating harmonic series with an error less than .
Look at the th term. Using a computer, we can see that We will see later that the true value of this seriesis . Comparing we have our desired accuracy.
Absolute convergence versus conditional convergence
It is an interesting result that the harmonic series, diverges, yet the alternatingharmonic series, converges. The notion that simply alternating the signs of the termsin a series can change a series from divergent to convergent leads us to the followingdefinitions.
- A series converges absolutely if converges.
- A series converges conditionally if converges but diverges.
Note, in the definition above, is not necessarily an alternating series; it may justhave some negative terms.
Does the series converge absolutely, converge conditionally, or diverge?
The seriesconverges conditionally. The series converges absolutely. The series diverges.
We can show the series diverges using the limit comparison test, comparing with .The series converges using the alternating series test; we conclude the seriesconverges conditionally.
Does the series converge absolutely, converge conditionally, or diverge?
The seriesconverges conditionally. The series converges absolutely. The series diverges.
We can show the series converges using the ratio test. Therefore we conclude converges absolutely.
Does the series converge absolutely, converge conditionally, or diverge?
The seriesconverges conditionally. The series converges absolutely. The series diverges.
The series diverges using the divergence test, so it does not converge absolutely.The series fails the conditions of the alternating series test as does notapproach as . We can state further that this series diverges; as , the serieseffectively adds and subtracts over and over. This causes the sequence ofpartial sums to oscillate and not converge. Therefore the series diverges.
Knowing that a series converges absolutely allows us to make two importantstatements. The first, given in the following theorem, is that absolute convergence is“stronger” than regular convergence. That is, just because converges, we cannotconclude that will converge, but knowing a series converges absolutely tells us that will converge.
One reason this is important is that our convergence tests all require that theunderlying sequence of terms be positive. By taking the absolute value of the termsof a series where not all terms are positive, we are often able to apply an appropriatetest and determine absolute convergence. This, in turn, determines that the series weare given also converges.
The second statement relates to rearrangements of series. When dealing witha finite set of numbers, the sum of the numbers does not depend on theorder in which they are added. (So .) One may be surprised to find out thatwhen dealing with an infinite set of numbers, the same statement does notalways hold true: some infinite lists of numbers may be rearranged in differentorders to achieve different sums. The theorem states that the terms of anabsolutely convergent series can be rearranged in any way without affecting thesum.
Absolute Convergence Let be a series that converges absolutely. Let be anyrearrangement of the sequence . Then
This theorem states that rearranging the terms of an absolutely convergent seriesdoes not affect its sum. Making such a statement implies that perhaps the sum of aconditionally convergent series can change based on the arrangement of terms.Indeed, it can. The Riemann rearrangement theorem (named after BernhardRiemann) states that any conditionally convergent series can have its termsrearranged so that the sum is any desired value, including !
As an example, consider the alternating harmonic series once more. We have stated that Consider the rearrangement where every positive term is followed by two negative terms: (Convince yourself that these are exactly the same numbers as appear in the alternatingharmonic series, just in a different order.) Now group some terms and simplify:
By rearranging the terms of the series, we have arrived at a different sum!
One could try to argue that the alternating harmonic series does not actually converge to , and here is an example of such an argument. According to the alternating series test, we know that this series converges to some number . If, as our intuition tells us should be true, the rearrangement does not change the sum, then we have just seen that . The only possibility for is then . But the alternating series approximation theorem quickly shows that . The only conclusion is that the rearrangement did, contrary to our intuition, change the sum.
The fact that conditionally convergent series can be rearranged to equal any numberis really an incredible result.
While series are worthy of study in and of themselves, our ultimate goal within calculus is the study of power series, which we will consider in the next section. Wewill use power series to create functions where the output is the result of an infinitesummation.
