Active Calculus

Subsection 8.4.1 The Alternating Series Test

Preview Activity 8.4.1 gives us several approximations to \(\ln(2)\text{.}\) The linear approximation is \(1\text{,}\) and the quadratic approximation is \(1 - \frac{1}{2} = \frac{1}{2}\text{.}\) If we continue this process, cubic, quartic (degree \(4\)), quintic (degree \(5\)), and higher degree polynomials give us the approximations to \(\ln(2)\) in Table 8.4.1.

The pattern here shows that \(\ln(2)\) can be approximated by the partial sums of the infinite series

where the alternating signs are indicated by the factor \((-1)^{k+1}\text{.}\) We call such a series an alternating series.

Using computational technology, we find that the sum of the first 100 terms in this series is 0.6881721793. As a comparison, \(\ln(2) \approx 0.6931471806\text{.}\) This shows that even though the series (8.4.1) converges to \(\ln(2)\text{,}\) it must do so quite slowly, since the sum of the first 100 terms isn't particularly close to \(\ln(2)\text{.}\) We will investigate the issue of how quickly an alternating series converges later in this section.

We have some flexibility in how we write an alternating series; for example, the series

whose index starts at \(k = 1\text{,}\) is also alternating. As we will soon see, there are several very nice results that hold for alternating series, while alternating series can also demonstrate some unusual behaivior.

It is important to remember that most of the series tests we have seen in previous sections apply only to series with nonnegative terms. Alternating series require a different test.

Activity 8.4.2 illustrates the general behavior of any convergent alternating series. We see that the partial sums of the alternating harmonic series oscillate around a fixed number that turns out to be the sum of the series.

Recall that if \(\lim_{k \to \infty} a_k \neq 0\text{,}\) then the series \(\sum a_k\) diverges by the Divergence Test. From this point forward, we will thus only consider alternating series

in which the sequence \(a_k\) consists of positive numbers that decrease to \(0\text{.}\) The \(n\)th partial sum \(S_n\) is

Notice that

• \(S_2 = a_1 - a_2\text{,}\) and since \(a_1 \gt a_2\) we have \(0 \lt S_2 \lt S_1 \text{.}\)

• \(S_3 = S_2+a_3\) and so \(S_2 \lt S_3\text{.}\) But \(a_3 \lt a_2\text{,}\) so \(S_3 \lt S_1\text{.}\) Thus, \(0 \lt S_2 \lt S_3 \lt S_1 \text{.}\)

• \(S_4 = S_3-a_4\) and so \(S_4 \lt S_3\text{.}\) But \(a_4 \lt a_3\text{,}\) so \(S_2 \lt S_4\text{.}\) Thus, \(0 \lt S_2 \lt S_4 \lt S_3 \lt S_1 \text{.}\)

• \(S_5 = S_4+a_5\) and so \(S_4 \lt S_5\text{.}\) But \(a_5 \lt a_4\text{,}\) so \(S_5 \lt S_3\text{.}\) Thus, \(0 \lt S_2 \lt S_4 \lt S_5 \lt S_3 \lt S_1 \text{.}\)

This pattern continues as illustrated in Figure 8.4.4 (with \(n\) odd) so that each partial sum lies between the previous two partial sums.

Note further that the absolute value of the difference between the \((n-1)\)st partial sum \(S_{n-1}\) and the \(n\)th partial sum \(S_n\) is

Because the sequence \(\{a_n\}\) converges to \(0\text{,}\) the distance between successive partial sums becomes as close to zero as we'd like, and thus the sequence of partial sums converges (even though we don't know the exact value to which it converges).

The preceding discussion has demonstrated the truth of the Alternating Series Test.

Note that if the limit of the sequence \(\{a_k\}\) is not 0, then the alternating series diverges.

Subsection 8.4.2 Estimating Alternating Sums

If the series converges, the argument for the Alternating Series Test also provides us with a method to determine how close the \(n\)th partial sum \(S_n\) is to the actual sum of the series. To see how this works, let \(S\) be the sum of a convergent alternating series, so

Recall that the sequence of partial sums oscillates around the sum \(S\) so that

Therefore, the value of the term \(a_{n+1}\) provides an error estimate for how well the partial sum \(S_n\) approximates the actual sum \(S\text{.}\) We summarize this fact in the statement of the Alternating Series Estimation Theorem.

Subsection 8.4.3 Absolute and Conditional Convergence

A series such as

whose terms are neither all nonnegative nor alternating is different from any series that we have considered so far. The behavior of such a series can be rather complicated, but there is an important connection between a series with some negative terms and series with all positive terms.

As the example in Activity 8.4.5 suggests, if a series \(\sum a_k\) has some negative terms but \(\sum |a_k|\) converges, then the original series, \(\sum a_k\text{,}\) must also converge. That is, if \(\sum | a_k |\) converges, then so must \(\sum a_k\text{.}\)

As we just observed, this is the case for the series (8.4.2), because the corresponding series of the absolute values of its terms is the convergent \(p\)-series \(\sum \frac{1}{k^2}\text{.}\) But there are series, such as the alternating harmonic series \(\sum (-1)^{k+1} \frac{1}{k}\text{,}\) that converge while the corresponding series of absolute values, \(\sum \frac{1}{k}\text{,}\) diverges. We distinguish between these behaviors by introducing the following language.

In this terminology, the series (8.4.2) converges absolutely while the alternating harmonic series is conditionally convergent.

Conditionally convergent series turn out to be very interesting. If the sequence \(\{a_n\}\) decreases to 0, but the series \(\sum a_k\) diverges, the conditionally convergent series \(\sum (-1)^k a_k\) is right on the borderline of being a divergent series. As a result, any conditionally convergent series converges very slowly. Furthermore, some very strange things can happen with conditionally convergent series, as illustrated in some of the exercises.

Subsection 8.4.4 Summary of Tests for Convergence of Series

We have discussed several tests for convergence/divergence of series in our sections and in exercises. We close this section of the text with a summary of all the tests we have encountered, followed by an activity that challenges you to decide which convergence test to apply to several different series.

Geometric Series

The geometric series \(\sum ar^k\) with ratio \(r\) converges for \(-1 \lt r \lt 1\) and diverges for \(|r| \geq 1\text{.}\)

The sum of the convergent geometric series \(\displaystyle \sum_{k=0}^{\infty} ar^k\) is \(\frac{a}{1-r}\text{.}\)

Divergence Test

If the sequence \(a_n\) does not converge to 0, then the series \(\sum a_k\) diverges.

This is the first test to apply because the conclusion is simple. However, if \(\lim_{n \to \infty} a_n = 0\text{,}\) no conclusion can be drawn.

Integral Test

Let \(f\) be a positive, decreasing function on an interval \([c,\infty)\) and let \(a_k = f(k)\) for each positive integer \(k \geq c\text{.}\)

• If \(\int_c^{\infty} f(t) \ dt\) converges, then \(\sum a_k\) converges.

• If \(\int_c^{\infty} f(t) \ dt\) diverges, then \(\sum a_k\) diverges.

Use this test when \(f(x)\) is easy to integrate.

Direct Comp. Test

(see Ex 4 in Section 8.3)

Let \(0 \leq a_k \leq b_k\) for each positive integer \(k\text{.}\)

• If \(\sum b_k\) converges, then \(\sum a_k\) converges.

• If \(\sum a_k\) diverges, then \(\sum b_k\) diverges.

Use this test when you have a series with known behavior that you can compare to — this test can be difficult to apply.

Limit Comp. Test

Let \(a_n\) and \(b_n\) be sequences of positive terms. If

\begin{equation*} \displaystyle \lim_{k \to \infty} \frac{a_k}{b_k} = L \end{equation*}

for some positive finite number \(L\text{,}\) then the two series \(\sum a_k\) and \(\sum b_k\) either both converge or both diverge.

Easier to apply in general than the comparison test, but you must have a series with known behavior to compare. Useful to apply to series of rational functions.

Ratio Test

Let \(a_k \neq 0\) for each \(k\) and suppose

\begin{equation*} \displaystyle \lim_{k \to \infty} \frac{|a_{k+1}|}{|a_k|} = r\text{.} \end{equation*}

• If \(r \lt 1\text{,}\) then the series \(\sum a_k\) converges absolutely.

• If \(r \gt 1\text{,}\) then the series \(\sum a_k\) diverges.

• If \(r=1\text{,}\) then test is inconclusive.

This test is useful when a series involves factorials and powers.

Root Test

(see Exercise 2 in Section 8.3)

Let \(a_k \geq 0\) for each \(k\) and suppose

\begin{equation*} \displaystyle \lim_{k \to \infty} \sqrt[k]{a_k} = r\text{.} \end{equation*}

• If \(r \lt 1\text{,}\) then the series \(\sum a_k\) converges.

• If \(r \gt 1\text{,}\) then the series \(\sum a_k\) diverges.

• If \(r=1\text{,}\) then test is inconclusive.

In general, the Ratio Test can usually be used in place of the Root Test. However, the Root Test can be quick to use when \(a_k\) involves \(k\)th powers.

Alt. Series Test

If \(a_n\) is a positive, decreasing sequence so that \(\displaystyle \lim_{n \to \infty} a_n = 0\text{,}\) then the alternating series \(\sum (-1)^{k+1} a_k\) converges.

This test applies only to alternating series — we assume that the terms \(a_n\) are all positive and that the sequence \(\{a_n\}\) is decreasing.

Alt. Series Est.

Let \(S_n = \displaystyle \sum_{k=1}^n (-1)^{k+1} a_k\) be the \(n\)th partial sum of the alternating series \(\displaystyle \sum_{k=1}^{\infty} (-1)^{k+1} a_k\text{.}\) Assume \(a_n \gt 0\) for each positive integer \(n\text{,}\) the sequence \(a_n\) decreases to 0 and \(\displaystyle \lim_{n \to \infty} S_n = S\text{.}\) Then it follows that \(|S - S_n| \lt a_{n+1}\text{.}\)

This bound can be used to determine the accuracy of the partial sum \(S_n\) as an approximation of the sum of a convergent alternating series.

Subsection 8.4.5 Summary

• An alternating series is a series whose terms alternate in sign. It has the form

  \begin{equation*} \sum (-1)^ka_k \end{equation*}

  where \(a_k\) is a positive real number for each \(k\text{.}\)

• The sequence of partial sums of a convergent alternating series oscillates around the sum of the series if the sequence of \(n\)th terms converges to 0. That is why the Alternating Series Test shows that the alternating series \(\sum_{k=1}^{\infty} (-1)^ka_k\) converges whenever the sequence \(\{a_n\}\) of \(n\)th terms decreases to 0.

• The difference between the \(n-1\)st partial sum \(S_{n-1}\) and the \(n\)th partial sum \(S_n\) of a convergent alternating series \(\sum_{k=1}^{\infty} (-1)^ka_k\) is \(|S_n - S_{n-1}| = a_n\text{.}\) Since the partial sums oscillate around the sum \(S\) of the series, it follows that

  \begin{equation*} |S - S_n| \lt a_n\text{.} \end{equation*}

  So the \(n\)th partial sum of a convergent alternating series \(\sum_{k=1}^{\infty} (-1)^ka_k\) approximates the actual sum of the series to within \(a_n\text{.}\)