So far we have looked mainly at series consisting of positive terms, and we have derived and used the comparison tests and ratio test for these. But many series have positive and negative terms, and we also need to look at these. This page discusses a particular case of these, alternating series. Some aspects of alternating series are easy and you will see why. Other aspects are rather frightening! To put this into context you will need to also read the pages on absolute convergence that come later.
An alternating series is one of the form where is a sequence of positive terms.
For example, the series is an alternating series. The next theorem shows that this series converges.
Alternating Series Test.
Let be an alternating series, where the sequence is a decreasing sequence of positive terms converging to 0. Then converges.
Let be arbitrary. We consider the partial sums and .
By bracketing one way,
(since is a decreasing sequence) so that is a sum of positive termsand hence the sequence is monotonic increasing. Also,
so for all . Thus is bounded and by the Monotone Convergence Theorem converges to some limit .
Now consider . On the one hand we have
so is monotonic decreasing, and on the other we have
which is a sum of positive terms hence positive. Thus is decreasing and bounded below by 0, and hence by the Monotone Convergence Theorem converges to some .
Finally we show . Note that as , as is a subsequence of the null sequence . But by continuity of we also have so by uniqueness of limits, so . Even numbered terms and odd numbered terms of converge to the same limit . This means by the Theorem on Covering by Subsequences that as , as required.
The series and converge by the alternating series test, even though the corresponding terms of positive terms, and , do not converge. (One is the harmonic series; the other can be proved divergent by comparison with the harmonic series.)
We saw the series a long time ago at the very beginning of the course, where it caused some consternation because its value seemed to be greater than a half, but on rearranging its terms it got a value less than a quarter.
A series which is convergent but for which the series diverges is said to be conditionally convergent.
So for example,
is conditionally convergent. The thing we observed
about rearrangments of terms for this series is, unfortunately, in fact true
more generally for all conditionally convergent series.
Conditionally convergent series are particularly badly behaved,
should be avoided if possible, and always need great care. The results
we have proved about this one are correct though:
according to the definitions we have given,
the sum of the series
does exist and is greater than a half. But a rearrangement of the
series can make the sum less than a half, or in fact, any number you
like. To avoid such
bad cases you need to look at
absolutely convergent series, which are
the subject of the next web page.
You have seen the definition of alternating series, the alternating series test, and some examples. You have also seen how many such series behave very badly with respect to rearrangement of their terms.