All of the series convergence tests we have used require that the underlying sequence \(\{a_n\}\) be a positive sequence. (We can relax this with Theorem 8.2.21 and state that there must be an \(N>0\) such that \(a_n>0\) for all \(n>N\text{;}\) that is, \(\{a_n\}\) is positive for all but a finite number of values of \(n\text{.}\))

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.

Recall the terms of Harmonic Series come from the Harmonic Sequence \(\{a_n\} = \{1/n\}\text{.}\) An important alternating series is the Alternating Harmonic Series: \begin{equation*} \infser (-1)^{n+1}\frac1n = 1-\frac12+\frac13-\frac14+\frac15-\frac16+\cdots \end{equation*}

Geometric Series can also be alternating series when \(r\lt 0\text{.}\) For instance, if \(r=-1/2\text{,}\) the geometric series is \begin{equation*} \infser[0] \left(\frac{-1}{2}\right)^n = 1-\frac12+\frac14-\frac18+\frac1{16}-\frac1{32}+\cdots \end{equation*}

Theorem 8.2.5 states that geometric series converge when \(\abs{r}\lt 1\) and gives the sum: \(\ds \infser[0] r^n = \frac1{1-r}\text{.}\) When \(r=-1/2\) as above, we find \begin{equation*} \infser[0] \left(\frac{-1}{2}\right)^n = \frac1{1-(-1/2)} = \frac 1{3/2} = \frac23. \end{equation*}

A powerful convergence theorem exists for other alternating series that meet a few conditions.

The basic idea behind Theorem 8.5.2 is illustrated in Figure 8.5.3. A positive, decreasing sequence \(\{a_n\}\) is shown along with the partial sums \begin{equation*} S_n = \sum_{i=1}^n(-1)^{i+1}a_i =a_1-a_2+a_3-a_4+\cdots+(-1)^{n+1}a_n. \end{equation*}

Because \(\{a_n\}\) is decreasing, the amount by which \(S_n\) bounces up/down decreases. Moreover, the odd terms of \(S_n\) form a decreasing, bounded sequence, while the even terms of \(S_n\) form an increasing, bounded sequence. Since bounded, monotonic sequences converge (see Theorem 8.1.26) and the terms of \(\{a_n\}\) approach 0, one can show the odd and even terms of \(S_n\) converge to the same common limit \(L\text{,}\) the sum of the series.

Key Idea 8.2.17 gives the sum of some important series. Two of these are \begin{equation*} \infser \frac1{n^2} =\frac{\pi^2}6 \approx 1.64493 \text{ and } \infser \frac{(-1)^{n+1}}{n^2} = \frac{\pi^2}{12}\approx 0.82247. \end{equation*}

These two series converge to their sums at different rates. To be accurate to two places after the decimal, we need 202 terms of the first series though only 13 of the second. To get 3 places of accuracy, we need 1069 terms of the first series though only 33 of the second. Why is it that the second series converges so much faster than the first?

While there are many factors involved when studying rates of convergence, the alternating structure of an alternating series gives us a powerful tool when approximating the sum of a convergent series.

Part 1 of Theorem 8.5.5 states that the \(n^\text{ th }\) partial sum of a convergent alternating series will be within \(a_{n+1}\) of its total sum. You can see this visually in Figure 8.5.3.(b). Look at the distance between \(S_6\) and \(L\text{.}\) Clearly this distance is less than the length of the arrow corresponding to \(a_7\text{.}\)

Also consider the alternating series we looked at before the statement of the theorem, \(\ds \infser \frac{(-1)^{n+1}}{n^2}\text{.}\) Since \(a_{14} = 1/14^2 \approx 0.0051\text{,}\) we know that \(S_{13}\) is within \(0.0051\) of the total sum.

Moreover, Part 2 of the theorem states that since \(S_{13} \approx 0.8252\) and \(S_{14}\approx 0.8201\text{,}\) we know the sum \(L\) lies between \(0.8201\) and \(0.8252\text{.}\) One use of this is the knowledge that \(S_{14}\) is accurate to two places after the decimal.

Some alternating series converge slowly. In Example 8.5.4 we determined the series \(\ds\infser (-1)^{n+1}\frac{\ln(n) }{n}\) converged. With \(n=1001\text{,}\) we find \(\ln(n) /n \approx 0.0069\text{,}\) meaning that \(S_{1000} \approx 0.1633\) is accurate to one, maybe two, places after the decimal. Since \(S_{1001} \approx 0.1564\text{,}\) we know the sum \(L\) is \(0.1564\leq L\leq0.1633\text{.}\)

One of the famous results of mathematics is that the Harmonic Series, \(\ds \infser \frac1n\) diverges, yet the Alternating Harmonic Series, \(\ds \infser (-1)^{n+1}\frac1n\text{,}\) converges. The notion that alternating the signs of the terms in a series can make a series converge leads us to the following definitions.

Thus we say the Alternating Harmonic Series converges conditionally.

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 {\(\infser a_n\)} converges, we cannot conclude that {\(\infser \abs{a_n}\)} will converge, but knowing a series converges absolutely tells us that {\(\infser a_n\)} will converge.

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 \(1+2+3 = 3+1+2\text{.}\)) 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.

In Example 8.5.8, we determined the series in Part 2 converges absolutely. Theorem 8.5.9 tells us the series converges (which we could also determine using the Alternating Series Test).

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, including \(\infty\text{!}\)

As an example, consider the Alternating Harmonic Series once more. We have stated that \begin{equation*} \infser (-1)^{n+1}\frac1n = 1-\frac12+\frac13-\frac14+\frac15-\frac16+\frac17\cdots = \ln(2) , \end{equation*}

(see Key Idea 8.2.17 or Example 8.5.4).

Consider the rearrangement where every positive term is followed by two negative terms: \begin{equation*} 1-\frac12-\frac14+\frac13-\frac16-\frac18+\frac15-\frac1{10}-\frac1{12}\cdots \end{equation*}

(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: \begin{align*} \left(1-\frac12\right)-\frac14+\left(\frac13-\frac16\right)-\frac18+\left(\frac15-\frac1{10}\right)-\frac1{12}+\cdots \amp =\\ \frac12-\frac14+\frac16-\frac18+\frac1{10}-\frac{1}{12}+\cdots \amp =\\ \frac12\left(1-\frac12+\frac13-\frac14+\frac15-\frac16+\cdots\right) \amp = \frac12\ln(2) . \end{align*}

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 \(\ln(2)\text{,}\) because rearranging the terms of the series shouldn't change the sum. However, the Alternating Series Test proves this series converges to \(L\text{,}\) for some number \(L\text{,}\) and if the rearrangement does not change the sum, then \(L = L/2\text{,}\) implying \(L=0\text{.}\) But the Alternating Series Approximation Theorem quickly shows that \(L>0\text{.}\) The only conclusion is that the rearrangement did change the sum.) This is an incredible result.

We end here our study of tests to determine convergence. The back cover of this text contains a table summarizing the tests that one may find useful.

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. We will use power series to create functions where the output is the result of an infinite summation.