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
For example, the series
Alternating Series Test.
By bracketing one way,
2 is monotonic increasing. Also,
2 for all 2 is bounded and by the Monotone Convergence Theorem
converges to some limit
2 is monotonic decreasing, and on the other
which is a sum of positive terms hence positive. Thus (
decreasing and bounded below by 0,
and hence by the Monotone Convergence Theorem converges to some
Finally we show
as 2 2
We saw the series
So for example,
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.