488
8.4. ALTERNATING SERIES
though the series (8.15) converges to ln(2), it must do so quite slowly, since the sum of
the first 100 terms isn’t particularly close to ln(2). We will investigate the issue of how
quickly an alternating series converges later in this section. Again, note particularly that
the series (8.15) is different from the series we have consider earlier in that some of the
terms are negative. We call such a series an alternating series.
Definition 8.6. An alternating series is a series of the form
∞
k=0
(−1)
k a k ,
where a k ≥ 0 for each k.
We have some flexibility in how we write an alternating series; for example, the series
∞
k=1
(−1)
k+1 a k ,
whose index starts at k = 1, is also alternating. As we will soon see, there are several very
nice results that hold about 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. Thus, alternating series require a
different test. To investigate this idea, we return to the example in Preview Activity 8.4.
Activity 8.16.
Remember that, by definition, a series converges if and only if its corresponding
sequence of partial sums converges.
(a) Complete Table 8.7 by calculating the first few partial sums (to 10 decimal
places) of the alternating series
∞
k=1
(−1)
k+1 1
k
.
(b) Plot the sequence of partial sums from part (a) in the plane. What do you
notice about this sequence?
⊳
Activity 8.16 exemplifies the general behavior that any convergent alternating series
will demonstrate. In this example, 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.
8.4. ALTERNATING SERIES
though the series (8.15) converges to ln(2), it must do so quite slowly, since the sum of
the first 100 terms isn’t particularly close to ln(2). We will investigate the issue of how
quickly an alternating series converges later in this section. Again, note particularly that
the series (8.15) is different from the series we have consider earlier in that some of the
terms are negative. We call such a series an alternating series.
Definition 8.6. An alternating series is a series of the form
∞
k=0
(−1)
k a k ,
where a k ≥ 0 for each k.
We have some flexibility in how we write an alternating series; for example, the series
∞
k=1
(−1)
k+1 a k ,
whose index starts at k = 1, is also alternating. As we will soon see, there are several very
nice results that hold about 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. Thus, alternating series require a
different test. To investigate this idea, we return to the example in Preview Activity 8.4.
Activity 8.16.
Remember that, by definition, a series converges if and only if its corresponding
sequence of partial sums converges.
(a) Complete Table 8.7 by calculating the first few partial sums (to 10 decimal
places) of the alternating series
∞
k=1
(−1)
k+1 1
k
.
(b) Plot the sequence of partial sums from part (a) in the plane. What do you
notice about this sequence?
⊳
Activity 8.16 exemplifies the general behavior that any convergent alternating series
will demonstrate. In this example, 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.