When we first talked about series convergence we briefly
mentioned a stronger type of convergence but didn’t do anything with it because
we didn’t have any tools at our disposal that we could use to work problems
involving it. We now have some of those
tools so it’s now time to talk about absolute convergence in detail.
First, let’s go back over the definition of absolute
convergence.
Definition
We also have the following fact about absolute convergence.
Fact
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If  is absolutely convergent then it is also
convergent.
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Proof
This fact is one of the ways in which absolute convergence
is a “stronger” type of convergence.
Series that are absolutely convergent are guaranteed to be
convergent. However, series that are
convergent may or may not be absolutely convergent.
Let’s take a quick look at a couple of examples of absolute
convergence.
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Example 1 Determine
if each of the following series are absolute convergent, conditionally
convergent or divergent.
(a) [Solution]
(b) [Solution]
(c) [Solution]
Solution
(a)
This is the alternating harmonic series and we saw in the
last section that it is a convergent series so we don’t need to check that
here. So, let’s see if it is an
absolutely convergent series. To do
this we’ll need to check the convergence of.
This is the harmonic series and we know from the integral
test section that it is divergent.
Therefore, this series is not absolutely convergent. It is however conditionally convergent
since the series itself does converge.
[Return to Problems]
(b)
In this case let’s just check absolute convergence first
since if it’s absolutely convergent we won’t need to bother checking
convergence as we will get that for free.
This series is convergent by the p-series test and so the series is absolute convergent. Note that this does say as well that it’s a
convergent series.
[Return to Problems]
(c)
In this part we need to be a little careful. First, this is NOT an alternating series
and so we can’t use any tools from that section.
What we’ll do here is check for absolute convergence first
again since that will also give convergence.
This means that we need to check the convergence of the following
series.
To do this we’ll need to note that
and so we have,
Now we know that
converges by the p-series
test and so by the Comparison Test we also know that
converges.
Therefore the original series is absolutely convergent
(and hence convergent).
[Return to Problems]
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Let’s close this section off by recapping a topic we saw earlier. When we first discussed the convergence of
series in detail we noted that we can’t think of series as an infinite sum
because some series can have different sums if we rearrange their terms. In fact, we gave two rearrangements of an
Alternating Harmonic series that gave two different values. We closed that section off with the following
fact,
Facts
Now that we’ve got the tools under our belt to determine
absolute and conditional convergence we can make a few more comments about
this.
First, as we showed above in Example 1a an Alternating
Harmonic is conditionally convergent and so no matter what value we chose there
is some rearrangement of terms that will give that value. Note as well that this fact does not tell us
what that rearrangement must be only that it does exist.
Next, we showed in Example 1b that,
is absolutely convergent and so no matter how we rearrange
the terms of this series we’ll always get the same value. In fact, it can be shown that the value of
this series is,