Section 5.5 Alternating Series (and Conditional vs. Absolute Convergence)
References.
- Calculus, Early Transcendentals by Stewart, Section 11.5.
An Alternating Series is one whose terms \(a_n\) are alternately positive and negative:
\begin{equation*}
b_1 - b_2 + b_3 + \cdots = \sum_{n=1}^\infty (-1)^{n-1}b_n
\end{equation*}
where the \(b_n\) are non-negative. That is, \(a_n = (-1)^{n-1}b_n\text{,}\) \(b_n = |a_n|\text{.}\)
As is often the case, indexing from zero can be more elegant:
\begin{equation*}
\sum_{n=0}^\infty (-1)^n b_n = b_0 - b_1 + b_2 - \cdots \qquad b_n \geq 0
\end{equation*}
Examples.
One example we have already seen is a geometric series with negative ratio \(r=-s\)
\begin{equation*}
\sum_{n=0}^\infty (-1)^n s^n = 1 - s + s^2 - s^3 + \cdots = \frac{1}{1+s}, \qquad 0 < s < 1
\end{equation*}
Another important example is the Alternating Harmonic Series:
\begin{equation*}
\sum_{n=1}^\infty \frac{(-1)^{n-1}}{n} = 1 - \frac{1}{2} + \frac{1}{3} - \frac{1}{4} + \cdots
\end{equation*}
which as we will soon see is convergent, unlike the Harmonic Series. (In fact the value is \(\ln(2)\text{,}\) as will be seen in Section 6.3)
We will also see in Section 6.3 that the “almost but not quite geometric” series
\begin{equation*}
\sum_{n=0}^\infty \frac{(-1)^n r^{2n}}{(2n)!}
= \sum_{n=0}^\infty \frac{(-r^2)^n}{(2n)!}
= 1 - \frac{r^2}{2!} + \frac{r^4}{4!} - \frac{r^6}{6!} + \cdots
\end{equation*}
converges for any number \(r\text{.}\) (In fact the value is \(\cos(r)\text{,}\) as will be seen in Section 6.3 )
Show that the series \(\ds \sum_{n=1}^\infty \frac{(-1)^n 3n}{4n - 1}\) is alternating, but diverges.
Convergence of alternating series with terms that decrease in size to zero.
Under two simple conditions, we can both show that an alternating series converges, and also rather easily get upper and lower bounds on the value of its sum, making such series very convenient for practical calculations:
Theorem 5.5.2. The Alternating Series Test for convergence.
If the alternating series
\begin{equation*}
\sum_{n=0}^\infty (-1)^n b_n = b_0 - b_1 + b_2 - \cdots
\end{equation*}
satisfies the two conditions that
- the terms are decreasing in magnitude: \(b_{n+1} \leq b_n\text{,}\) and
- the terms converge to zero: \(b_n \to 0\) (as is always needed for convergence.)
then the series converges.
Measuring the accuracy of partial sums.
This is much easier to do than with the comparison to improper integrals seen in Section 5.3
Theorem 5.5.3. The Alternating Series Estimation Theorem.
If an alternating series satisfies the above two conditions, its value \(S\) lies between any two consecutive partial sums, and the error in any partial sum \(S_N\) is no larger than the first term not used in that partial sum, and so smaller than \(b_N\text{,}\) the last term used:
\begin{equation*}
\left| R_N \right| = \left| S - S_N \right| < b_{N+1}, \leq b_N
\end{equation*}
That is:
- As you successively add terms, after each addition of a positive value the running total \(S_N\) is more than the exact value \(S\) of the series, and after each addition of a negative term it is less.
- The size of the last term added gives an upper limit on error in the partial sum.
Absolute Convergence and Conditional Convergence.
To move beyond the special cases of series with all positive terms or ones with alternating signs, it helps to start by looking at just the magnitudes of the terms:
Definition 5.5.4. Absolutely Convergent.
The series \(\sum a_n\) is called absolutely convergent if the positive-term series \(\sum |a_n|\) is convergent.
Checkpoint 5.5.5.
Verify that the series \(\ds \sum_{n=0}^\infty \frac{(-1)^n}{n!}\) is absolutely convergent.
Not surprisingly:
Theorem 5.5.6.
If a series is absolutely convergent, it is also convergent.
But the opposite is not true!
Checkpoint 5.5.7.
Verify that the Alternating Harmonic Series \(\ds\sum \frac{(-1)^{n-1}}{n}\) is convergent but not absolutely convergent.
Checkpoint 5.5.8.
Determine whether the series \(\ds \sum_{n=1}^\infty \frac{\cos n}{n^2}\) is convergent.
Hint.Use both the idea of absolute convergence and then a comparison test from Section 5.4.
Definition 5.5.9. Conditionally Convergent.
A series that is convergent but not absolutely convergent is called conditionally convergent.
As a rule of thumb, conditionally convergent series are far less useful and when we seek to express the solution to a problem as the sum of a series, we greatly prefer to find an absolutely convergent series that does the job.
Checkpoint 5.5.10.
Verify directly that the alternating \(p\)-series \(\ds \sum \frac{(-1)^{n-1}}{n^2}\) is both convergent and absolutely convergent.
Checkpoint 5.5.11.
Verify that the alternating \(p\)-series \(\ds \sum \frac{(-1)^{n-1}}{n^p}\) is
- absolutely convergent for \(p > 1\text{.}\)
- conditionally convergent for \(0 < p \leq 1\text{.}\)
- divergent for \(p \leq 0\text{.}\)
Checkpoint 5.5.12.
Verify that any convergent geometric series is also absolutely convergent.
(But note that this is not true for all series!)
Study Guide.
Study Calculus Volume 2, Section 5.5 2 ; in particular
- The definition of an Alternating Series
- The following proof that the Alternating Harmonic Series converges, even though the original Harmonic Series diverges
- The Alternating Series Test in Theorem 13, noting the extra requirement that the terms are decreasing in magnitude.
- Theorem 14 about how accurate a partial sum is at approximating the infinite sum, including whether it is an underestimate or an over-estimate.
- The definitions of Absolute Convergence and Conditional Convergence, and Theorem 15
- Examples 19, 20, 21
- Checkpoints 18, 19, 20
- and one or several exercises from each of the groups 250–257 and 280–283.
openstax.org/books/calculus-volume-2/pages/5-5-alternating-seriesopenstax.org/books/calculus-volume-2/pages/5-5-alternating-series
