Series Convergence/Divergence
In the previous section we spent some time getting familiar
with series and we briefly defined convergence and divergence. Before worrying about convergence and
divergence of a series we wanted to make sure that we’ve started to get
comfortable with the notation involved in series and some of the
various manipulations of series that we will, on occasion, need to be able to
do.
As noted in the previous section most of what we were doing
there won’t be done much in this chapter.
So, it is now time to start talking about the convergence and divergence
of a series as this will be a topic that we’ll be dealing with to one extent or
another in almost all of the remaining sections of this chapter.
So, let’s recap just what an infinite series is and what it
means for a series to be convergent or divergent. We’ll start with a sequence and again note that we’re starting the
sequence at only for the sake of convenience and it can,
in fact, be anything.
Next we define the partial sums of the series as,
and these form a new sequence, .
An infinite series, or just series here since almost every
series that we'll be looking at will be an infinite series, is then the limit of
the partial sums. Or,
If the sequence of partial sums is a convergent sequence (i.e. its limit exists and is finite)
then the series is also called convergent
and in this case if then, . Likewise, if the sequence of partial sums is
a divergent sequence (i.e. its limit
doesn’t exist or is plus or minus infinity) then the series is also called divergent.
Let’s take a look at some series and see if we can determine
if they are convergent or divergent and see if we can determine the value of
any convergent series we find.
Example 1 Determine
if the following series is convergent or divergent. If it converges determine its value.
Solution
To determine if the series is convergent we first need to
get our hands on a formula for the general term in the sequence of partial
sums.
This is a known series and its value can be shown to be,
Don’t worry if you didn’t know this formula (I’d be
surprised if anyone knew it…) as you won’t be required to know it in my
course.
So, to determine if the series is convergent we will first
need to see if the sequence of partial sums,
is convergent or divergent. That’s not terribly difficult in this
case. The limit of the sequence terms
is,
Therefore, the sequence of partial sums diverges to and so the series also diverges.

So, as we saw in this example we had to know a fairly
obscure formula in order to determine the convergence of this series. In general finding a formula for the general
term in the sequence of partial sums is a very difficult process. In fact after the next section we’ll not be
doing much with the partial sums of series due to the extreme difficulty faced
in finding the general formula. This
also means that we’ll not be doing much work with the value of series since in
order to get the value we’ll also need to know the general formula for the
partial sums.
We will continue with a few more examples however, since
this is technically how we determine convergence and the value of a
series. Also, the remaining examples
we’ll be looking at in this section will lead us to a very important fact about
the convergence of series.
So, let’s take a look at a couple more examples.
Example 2 Determine
if the following series converges or diverges. If it converges determine its sum.
Solution
This is actually one of the few series in which we are
able to determine a formula for the general term in the sequence of partial
fractions. However, in this section we
are more interested in the general idea of convergence and divergence and so
we’ll put off discussing the process for finding the formula until the next
section.
The general formula for the partial sums is,
and in this case we have,
The sequence of partial sums converges and so the series
converges also and its value is,

Example 3 Determine
if the following series converges or diverges. If it converges determine its sum.
Solution
In this case we really don’t need a general formula for
the partial sums to determine the convergence of this series. Let’s just write down the first few partial
sums.
So, it looks like the sequence of partial sums is,
and this sequence diverges since doesn’t exist. Therefore, the series also diverges.

Example 4 Determine
if the following series converges or diverges. If it converges determine its sum.
Solution
Here is the general formula for the partial sums for this
series.
Again, do not worry about knowing this formula. This is not something that you’ll ever be
asked to know in my class.
In this case the limit of the sequence of partial sums is,
The sequence of partial sums is convergent and so the
series will also be convergent. The
value of the series is,

As we already noted, do not get excited about determining
the general formula for the sequence of partial sums. There is only going to be one type of series
where you will need to determine this formula and the process in that case
isn’t too bad. In fact, you already know
how to do most of the work in the process as you’ll see in the next section.
So, we’ve determined the convergence of four series
now. Two of the series converged and two
diverged. Let’s go back and examine the
series terms for each of these. For each
of the series let’s take the limit as n
goes to infinity of the series terms (not the partial sums!!).
Notice that for the two series that converged the series
term itself was zero in the limit. This
will always be true for convergent series and leads to the following theorem.
Theorem
Proof
Be careful to not misuse this theorem! This theorem gives us a requirement for
convergence but not a guarantee of convergence.
In other words, the converse is NOT true. If the series may actually diverge! Consider the following two series.
In both cases the series terms are zero in the limit as n goes to infinity, yet only the second
series converges. The first series
diverges. It will be a couple of
sections before we can prove this, so at this point please believe this and
know that you’ll be able to prove the convergence of these two series in a
couple of sections.
Again, as noted above, all this theorem does is give us a
requirement for a series to converge. In
order for a series to converge the series terms must go to zero in the
limit. If the series terms do not go to
zero in the limit then there is no way the series can converge since this would
violate the theorem.
This leads us to the first of many tests for the
convergence/divergence of a series that we’ll be seeing in this chapter.
Divergence Test
Again, do NOT misuse this test. This test only says that a series is
guaranteed to diverge if the series terms don’t go to zero in the limit. If the series terms do happen to go to zero
the series may or may not converge!
Again, recall the following two series,
One of the more common mistakes that students make when they
first get into series is to assume that if then will converge.
There is just no way to guarantee this so be careful!
Let’s take a quick look at an example of how this test can
be used.
Example 5 Determine
if the following series is convergent or divergent.
Solution
With almost every series we’ll be looking at in this
chapter the first thing that we should do is take a look at the series terms
and see if they go to zero or not. If
it’s clear that the terms don’t go to zero use the Divergence Test and be
done with the problem.
That’s what we’ll do here.
The limit of the series terms isn’t zero and so by the
Divergence Test the series diverges.

The divergence test is the first test of many tests that we
will be looking at over the course of the next several sections. You will need to keep track of all these
tests, the conditions under which they can be used and their conclusions all in
one place so you can quickly refer back to them as you need to.
Next we should briefly revisit arithmetic of series and
convergence/divergence. As we saw in the
previous section if and are both convergent series then so are and . Furthermore, these series will have the
following sums or values.
We’ll see an example of this in the next section after we
get a few more examples under our belt.
At this point just remember that a sum of convergent series is
convergent and multiplying a convergent series by a number will not change
its convergence.
We need to be a little careful with these facts when it
comes to divergent series. In the first
case if is divergent then will also be divergent (provided c isn’t zero of course) since
multiplying a series that is infinite in value or doesn’t have a value by a
finite value (i.e. c) won’t change
the fact that the series has an infinite or no value. However, it is possible to have both and be divergent series and yet have be a convergent series.
Now, since the main topic of this section is the convergence
of a series we should mention a stronger type of convergence. A series is said to converge absolutely if also converges. Absolute convergence is stronger than convergence in the sense that a series that is
absolutely convergent will also be convergent, but a series that is convergent
may or may not be absolutely convergent.
In fact if converges and diverges the series is called conditionally convergent.
At this point we don’t really have the tools at hand to
properly investigate this topic in detail nor do we have the tools in hand to
determine if a series is absolutely convergent or not. So we’ll not say anything more about this
subject for a while. When we finally
have the tools in hand to discuss this topic in more detail we will revisit
it. Until then don’t worry about it. The idea is mentioned here only because we
were already discussing convergence in this section and it ties into the last
topic that we want to discuss in this section.
In the previous section after we’d introduced the idea of an
infinite series we commented on the fact that we shouldn’t think of an infinite
series as an infinite sum despite the fact that the notation we use for
infinite series seems to imply that it is an infinite sum. It’s now time to briefly discuss this.
First, we need to introduce the idea of a rearrangement. A rearrangement of a
series is exactly what it might sound like, it is the same series with the
terms rearranged into a different order.
For example, consider the following infinite series.
A rearrangement of this series is,
The issue we need to discuss here is that for some series
each of these arrangements of terms can have different values despite the
fact that they are using exactly the same terms.
Here is an example of this.
It can be shown that,
Since this series converges we know that if we multiply it
by a constant c its value will also
be multiplied by c. So, let’s multiply this by to get,
Now, let’s add in a zero between each term as follows.
Note that this won’t change the value of the series
because the partial sums for this series will be the partial sums for the (2)
except that each term will be repeated.
Repeating terms in a series will not affect its limit however and so
both (2)
and (3)
will be the same.
We know that if two series converge we can add them by
adding term by term and so add (1) and (3)
to get,
Now, notice that the terms of (4) are
simply the terms of (1)
rearranged so that each negative term comes after two positive terms. The values however are definitely different
despite the fact that the terms are the same.
Note as well that this is not one of those “tricks” that you
see occasionally where you get a contradictory result because of a hard to spot
math/logic error. This is a very real result and we’ve not made any logic mistakes/errors.
Here is a nice set of facts that govern this idea of when a
rearrangement will lead to a different value of a series.
Facts
Again, we do not have the tools in hand yet to determine if
a series is absolutely convergent and so don’t worry about this at this
point. This is here just to make sure
that you understand that we have to be very careful in thinking of an infinite
series as an infinite sum. There are
times when we can (i.e. the series is
absolutely convergent) and there are times when we can’t (i.e. the series is conditionally convergent).
As a final note, the fact above tells us that the series,
must be conditionally convergent since two rearrangements
gave two separate values of this series.
Eventually it will be very simple to show that this series conditionally
convergent.