The series convergence tests we have used require that the underlying sequence be a positive sequence. (We can relax this with Theorem 9.2.5 and state that there must be an such that for all ; that is, is positive for all but a finite number of values of .)
In this section we explore series whose summation includes negative terms. We start with a very specific form of series, where the terms of the summation alternate between being positive and negative.
Let be a positive sequence. An alternating series is a series of either the form
We want to think that an alternating sequence is related to a positive sequence by .
Recall that the terms of Harmonic Series come from the Harmonic Sequence . An important alternating series is the Alternating Harmonic Series:
Geometric Series can also be alternating series when . For instance, if , the geometric series is
Theorem 9.2.1 states that geometric series converge when and gives the sum: . When as above, we find
A powerful convergence theorem exists for other alternating series that meet a few conditions.
Let be a positive, decreasing sequence where . Then
The basic idea behind Theorem 9.5.1 is illustrated in Figure 9.5.1. A positive, decreasing sequence is shown along with the partial sums
Because is decreasing, the amount by which bounces up and down decreases. Moreover, the odd terms of form a decreasing, bounded sequence, while the even terms of form an increasing, bounded sequence. Since bounded, monotonic sequences converge (see Theorem 9.1.6) and the terms of approach 0, we will show below that the odd and even terms of converge to the same common limit , the sum of the series.
Because is a decreasing sequence, we have . We will consider the even and odd partial sums separately. First consider the even partial sums.
We now have
so is an increasing sequence. But we can also write
Each term in parentheses is positive and is positive so we have for all . We now have the sequence of even partial sums, , is increasing and bounded above so by Theorem 9.1.6 converges. Since we know it converges, we will assume it’s limit is or
Next we determine the limit of the sequence of odd partial sums.
Both the even and odd partial sums converge to so we have , which means the series is convergent. ∎
Determine if the Alternating Series Test applies to each of the following series.
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 the Alternating Series Test and conclude this series converges. While the test does not state what the series converges to, we will see later that
The underlying sequence is . This is positive for and (use L’Hôpital’s Rule). However, the sequence is not decreasing for all . It is straightforward to compute , , and : the sequence is increasing for at least the first 2 terms. We do not immediately conclude that we cannot apply the Alternating Series Test. Rather, consider the long-term behavior of . Treating as a continuous function of defined on , we can take its derivative:
The derivative is negative for all (actually, for all ), meaning is decreasing on . We can apply the Alternating Series Test to the series when we start with and conclude that converges; adding the terms with does not change the convergence (i.e., we apply Theorem 9.2.5). The important lesson here is that as before, if a series fails to meet the criteria of the Alternating Series Test on only a finite number of terms, we can still apply the test.
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 cannot remove a finite number of terms to make decreasing, therefore we cannot apply the Alternating Series Test. Keep in mind that this does not mean we conclude the series diverges; in fact, it does converge. We are just unable to conclude this based on Theorem 9.5.1.
One of the famous results of mathematics is that the Harmonic Series, diverges, yet the Alternating Harmonic Series, , converges. The notion that alternating the signs of the terms in a series can make a series converge leads us to the following definitions.
(a) A series converges absolutely if converges. (b) A series converges conditionally if converges but diverges.
Thus we say the Alternating Harmonic Series converges conditionally.
Determine if the following series converge absolutely, conditionally, or diverge.
We can show the series
diverges using the Limit Comparison Test, comparing with . The sequence is monotonically decreasing, so that the series converges using the Alternating Series Test; we conclude it converges conditionally.
diverges using the Test for Divergence, so it does not converge absolutely. The series fails the conditions of the Alternating Series Test as does not approach as . We can state further that this series diverges; as , the series effectively adds and subtracts over and over. This causes the sequence of partial sums to oscillate and not converge. Therefore the series diverges.
Knowing that a series converges absolutely allows us to make two important statements, given in the following theorem. The first is that absolute convergence is “stronger” than regular convergence. That is, just because converges, we cannot conclude that will converge, but knowing a series converges absolutely tells us that will converge.
Let be a series that converges absolutely.
Let be any rearrangement of the sequence . Then
One reason this is important is that our convergence tests all require that the underlying sequence of terms be positive. By taking the absolute value of the terms of a series where not all terms are positive, we are often able to apply an appropriate test and determine absolute convergence. This, in turn, determines that the series we are given also converges.
The second statement relates to rearrangements of series. When dealing with a finite set of numbers, the sum of the numbers does not depend on the order which they are added. (So .) One may be surprised to find out that when dealing with an infinite set of numbers, the same statement does not always hold true: some infinite lists of numbers may be rearranged in different orders to achieve different sums. The theorem states that the terms of an absolutely convergent series can be rearranged in any way without affecting the sum.
The theorem states that rearranging the terms of an absolutely convergent series does not affect its sum. This implies that perhaps the sum of a conditionally convergent series can change based on the arrangement of terms. Indeed, it can. The Riemann Rearrangement Theorem (named after Bernhard Riemann) states that any conditionally convergent series can have its terms rearranged so that the sum is any desired value or infinity.
Before we consider an example, we state the following theorem that illustrates how the alternating structure of an alternating series is a powerful tool when approximating the sum of a convergent series.
Let be a sequence that satisfies the hypotheses of the Alternating Series Test, and let and be the partial sum and sum, respectively, of either or . Then
is between and .
Part 1 of Theorem 9.5.3 states that the partial sum of a convergent alternating series will be within of its total sum. Consider the alternating series we looked at before the statement of the theorem, . Since , we know that is within of the total sum.
Moreover, Part 2 of the theorem states that since and , we know the sum lies between and . One use of this is the knowledge that is accurate to two places after the decimal.
Some alternating series converge slowly. In Example 9.5.1 we determined the series converged. With , we find , meaning that is accurate to one, maybe two, places after the decimal. Since , we know the sum is .
Approximate the sum of the following series, accurate to within .
Using Theorem 9.5.3, we want to find where :
Let be the sum of this series. By Part 1 of the theorem, . We can compute , which our theorem states is within of the total sum. We can use Part 2 of the theorem to obtain an even more accurate result. As we know the term of the series is , we can easily compute . Part 2 of the theorem states that is between and , so .
We want to find where . We start by solving for . This cannot be solved algebraically, so we will use Newton’s Method to approximate a solution. Let ; we want to know where . We make a guess that must be “large,” so our initial guess will be . Recall how Newton’s Method works: given an approximate solution , our next approximation is given by
We find . This gives
Using a computer, we find that Newton’s Method seems to converge to a solution after 8 iterations. Taking the next integer higher, we have , where . Again using a computer, we find . Part 1 of the theorem states that this is within of the actual sum . Already knowing the 9,119 term, we can compute , meaning
Notice how the first series converged quite quickly, where we needed only 10 terms to reach the desired accuracy, whereas the second series took over 9,000 terms.
We now consider the Alternating Harmonic Series once more. We have stated that
(see Example 9.5.1).
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 Alternating Harmonic 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 , because rearranging the terms of the series shouldn’t change the sum. However, the Alternating Series Test proves this series converges to , for some number , and if the rearrangement does not change the sum, then , implying . But the Alternating Series Approximation Theorem quickly shows that . The only conclusion is that the rearrangement did change the sum.) This is an incredible result.
We mentioned earlier that the Integral Test did not work well with series containing factorial terms. The next section introduces the Ratio Test, which does handle such series well. We also introduce the Root Test, which is good for series where each term is raised to a power.
Why is not an alternating series?
A series converges when is , and .
Give an example of a series where converges but does not.
The sum of a convergent series can be changed by rearranging the order of its terms.
In Exercises 5–18., an alternating series is given.
Determine if the series converges or diverges.
Determine if converges or diverges.
If converges, determine if the convergence is conditional or absolute.
Let be the partial sum of a series. In Exercises 19–22., a convergent alternating series is given and a value of . Compute and and use these values to find bounds on the sum of the series.
In Exercises 23–26., a convergent alternating series is given along with its sum and a value of . Use Theorem 9.5.3 to find such that the partial sum of the series is within of the sum of the series.