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In this section we are going to take a brief look at three
special series. Actually, special may
not be the correct term. All three have
been named which makes them special in some way, however the main reason that
we’re going to look at two of them in this section is that they are the only
types of series that we’ll be looking at for which we will be able to get
actual values for the series. The third
type is divergent and so won’t have a value to worry about.
In general, determining the value of a series is very
difficult and outside of these two kinds of series that we’ll look at in this
section we will not be determining the value of series in this chapter.
So, let’s get started.
Geometric Series
A geometric series is any series that can be written in the
form,
or, with an index
shift the geometric series will often be written as,
These are identical series and will have identical values,
provided they converge of course.
If we start with the first form it can be shown that the
partial sums are,
The series will converge provided the partial sums form a
convergent sequence, so let’s take the limit of the partial sums.
Now, from Theorem 3
from the Sequences section we know that the limit above will exist and be
finite provided 
. However, note that we can’t let since this will give division by zero. Therefore, this will exist and be finite
provided and in this case the limit is zero and so we
get,
Therefore, a geometric series will converge if ,
which is usually written ,
its value is,
Note that in using this formula we’ll need to make sure that
we are in the correct form. In other
words, if the series starts at then the exponent on the r must be n. Likewise if the series starts at then the exponent on the r must be .
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Example 1 Determine
if the following series converge or diverge.
If they converge give the value of the series.
(a) [Solution]
(b) [Solution]
Solution
(a)
This series doesn’t really look like a geometric
series. However, notice that both
parts of the series term are numbers raised to a power. This means that it can be put into the form
of a geometric series. We will just
need to decide which form is the correct form. Since the series starts at we will want the exponents on the numbers to
be .
It will be fairly easy to get this into the correct
form. Let’s first rewrite things
slightly. One of the n’s in the exponent has a negative in
front of it and that can’t be there in the geometric form. So, let’s first get rid of that.
Now let’s get the correct exponent on each of the
numbers. This can be done using simple
exponent properties.
Now, rewrite the term a little.
So, this is a geometric series with and . Therefore, since we know the series will converge and its
value will be,
[Return to Problems]
(b)
Again, this doesn’t look like a geometric series, but it
can be put into the correct form. In
this case the series starts at so we’ll need the exponents to be n on the terms. Note that this means we’re going to need to
rewrite the exponent on the numerator a little
So, we’ve got it into the correct form and we can see that
and . Also note that and so this series diverges.
[Return to Problems]
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Back in the Series
Basics section we talked about stripping
out terms from a series, but didn’t really provide any examples of how this
idea could be used in practice. We can
now do some examples.
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Example 2 Use
the results from the previous example to determine the value of the following
series.
(a) [Solution]
(b) [Solution]
Solution
(a)
In this case we could just acknowledge that this is a
geometric series that starts at and so we could put it into the correct form
and be done with it. However, this
does provide us with a nice example of how to use the idea of stripping out
terms to our advantage.
Let’s notice that if we strip out the first term from this
series we arrive at,
From the previous example we know the value of the new
series that arises here and so the value of the series in this example is,
[Return to Problems]
(b)
In this case we can’t strip out terms from the given
series to arrive at the series used in the previous example. However, we can start with the series used
in the previous example and strip terms out of it to get the series in this
example. So, let’s do that. We will strip out the first two terms from
the series we looked at in the previous example.
We can now use the value of the series from the previous
example to get the value of this series.
[Return to Problems]
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Notice that we didn’t discuss the convergence of either of
the series in the above example. Here’s
why. Consider the following series
written in two separate ways (i.e. we
stripped out a couple of terms from it).
Let’s suppose that we know is a convergent series. This means that it’s got a finite value and
adding three finite terms onto this will not change that fact. So the value of is also finite and so is convergent.
Likewise, suppose that is convergent.
In this case if we subtract three finite values from this value we will
remain finite and arrive at the value of . This is now a finite value and so this series
will also be convergent.
In other words, if we have two series and they differ only
by the presence, or absence, of a finite number of finite terms they will
either both be convergent or they will both be divergent. The difference of a few terms one way or the
other will not change the convergence of a series. This is an important idea and we will use it
several times in the following sections to simplify some of the tests that
we’ll be looking at.
Telescoping Series
It’s now time to look at the second of the three series in
this section. In this portion we are
going to look at a series that is called a telescoping series. The name in this case comes from what happens
with the partial sums and is best shown in an example.
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Example 3 Determine
if the following series converges or diverges. If it converges find its value.
Solution
We first need the partial sums for this series.
Now, let’s notice that we can use partial fractions on the
series term to get,
I’ll leave the details of the partial fractions to
you. By now you should be fairly adept
at this since we spent a fair amount of time doing partial fractions back in the Integration Techniques
chapter. If you need a refresher you
should go back and review that section.
So, what does this do for us? Well, let’s start writing out the terms of
the general partial sum for this series using the partial fraction form.
Notice that every term except the first and last term
canceled out. This is the origin of
the name telescoping series.
This also means that we can determine the convergence of
this series by taking the limit of the partial sums.
The sequence of partial sums is convergent and so the
series is convergent and has a value of
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In telescoping series be careful to not assume that
successive terms will be the ones that cancel.
Consider the following example.
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Example 4 Determine
if the following series converges or diverges. If it converges find its value.
Solution
As with the last example we’ll leave the partial fractions
details to you to verify. The partial
sums are,
In this case instead of successive terms canceling a term
will cancel with a term that is farther down the list. The end result this time is two initial and
two final terms are left. Notice as
well that in order to help with the work a little we factored the out of the series.
The limit of the partial sums is,
So, this series is convergent (because the partial sums
form a convergent sequence) and its value is,
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Note that it’s not always obvious if a series is telescoping
or not until you try to get the partial sums and then see if they are in fact
telescoping. There is no test that will
tell us that we’ve got a telescoping series right off the bat. Also note that just because you can do
partial fractions on a series term does not mean that the series will be a
telescoping series. The following
series, for example, is not a telescoping series despite the fact that we can
partial fraction the series terms.
In order for a series to be a telescoping we must get terms
to cancel and all of these terms are positive and so none will cancel.
Next, we need to go back and address an issue that was first
raised in the previous section. In that section we stated that the sum or
difference of convergent series was also convergent and that the presence of a
multiplicative constant would not affect the convergence of a series. Now that we have a few more series in hand
let’s work a quick example showing that.
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Example 5 Determine
the value of the following series.
Solution
To get the value of this series all we need to do is
rewrite it and then use the previous results.
We didn’t discuss the convergence of this series because
it was the sum of two convergent series and that guaranteed that the original
series would also be convergent.
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Harmonic Series
This is the third and final series that we’re going to look
at in this chapter. Here is the harmonic
series.
The harmonic series is divergent and we’ll need to wait
until the next section to show that.
This series is here because it’s got a name and so I wanted to put it
here with the other two named series that we looked at in this section. We’re also going to use the harmonic series
to illustrate a couple of ideas about divergent series that we’ve already
discussed for convergent series. We’ll
do that with the following example.
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Example 6 Show
that each of the following series are divergent.
(a)
(b)
Solution
(a)
To see that this series is divergent all we need to do is
use the fact that we can factor a constant out of a series as follows,
Now, is divergent and so five times this will
still not be a finite number and so the series has to be divergent. In other words, if we multiply a divergent
series by a constant it will still be divergent.
(b)
In this case we’ll start with the harmonic series and
strip out the first three terms.
In this case we are subtracting a finite number from a
divergent series. This subtraction
will not change the divergence of the series.
We will either have infinity minus a finite number, which is still
infinity, or a series with no value minus a finite number, which will still
have no value.
Therefore, this series is divergent.
Just like with convergent series, adding/subtracting a
finite number from a divergent series is not going to change the fact the
convergence of the series.
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So, some general rules about the convergence/divergence of a
series are now in order. Multiplying a
series by a constant will not change the convergence/divergence of the series and
adding or subtracting a constant from a series will not change the
convergence/divergence of the series.
These are nice ideas to keep in mind.