Alternating Series Test
The last two tests that we looked at for series convergence have required that all the
terms in the series be positive. Of course there are many series out there that have
negative terms in them and so we now need to start looking at tests for these kinds of
series.
The test that we are going to look into in this section will be a test for alternating
series. An
alternating series
is any series,
, for which the series terms
can be written in one of the following two forms.
There are many other ways to deal with the alternating sign, but they can all be
written as one of the two forms above. For instance,
There are of course many others, but they all follow the same basic pattern of
reducing to one of the first two forms given. If you should happen to run into a
different form than the first two, don’t worry about converting it to one of those
forms, just be aware that it can be and so the test from this section can be used.
Alternating Series Test
Suppose that we have a series
and either
or
where
for
all
n
. Then if,
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1.
and,
2.
is a decreasing sequence
the series
is convergent.
A
proof
of this test is at the end of the section.
There are a couple of things to note about this test. First, unlike the Integral Test and
the Comparison/Limit Comparison Test, this test will only tell us when a series
converges and not if a series will diverge.
Secondly, in the second condition all that we need to require is that the series
terms,
will be eventually decreasing. It is possible for the first few terms of a
series to increase and still have the test be valid. All that is required is that eventually
we will have
for all
n
after some point.
To see why this is consider the following series,
Let’s suppose that for
is not decreasing and
that for
is decreasing. The series can then be
written as,
The first series is a finite sum (no matter how large
N
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 Fall '08
 prellis

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