You can navigate through this E-Book using the menu to the left. For E-Books that have a Chapter/Section organization each option in the menu to the left indicates a chapter and will open a menu showing the sections in that chapter. Alternatively, you can navigate to the next/previous section or chapter by clicking the links in the boxes at the very top and bottom of the material.
Also, depending upon the E-Book, it will be possible to download the complete E-Book, the chapter containing the current section and/or the current section. You can do this be clicking on the E-Book, Chapter, and/or the Section link provided below.
For those pages with mathematics on them you can, in most cases, enlarge the mathematics portion by clicking on the equation. Click the enlarged version to hide it.
We’ve spent quite a bit of time talking about series now and
with only a couple of exceptions we’ve spent most of that time talking about
how to determine if a series will converge or not. It’s now time to start looking at some
specific kinds of series and we’ll eventually reach the point where we can talk
about a couple of application of series.
In this section we are going to start talking about power
series. A power series about a, or just power
series, is any series that can be written in the form,
where a and cn are numbers. The cn’s
are often called the coefficients of
the series. The first thing to notice
about a power series is that it is a function of x. That is different from
any other kind of series that we’ve looked at to this point. In all the prior sections we’ve only allowed
numbers in the series and now we are allowing variables to be in the series as
well. This will not change how things
work however. Everything that we know
about series still holds.
In the discussion of power series convergence is still a
major question that we’ll be dealing with.
The difference is that the convergence of the series will now depend
upon the value of x that we put into
the series. A power series may converge
for some values of x and not for
other values of x.
Before we get too far into power series there is some
terminology that we need to get out of the way.
First, as we will see in our examples, we will be able to
show that there is a number R so that
the power series will converge for, 
and will diverge for . This number is called the radius of convergence for the
series. Note that the series may or may
not converge if . What happens at these points will not change
the radius of convergence.
Secondly, the interval of all x’s, including the end points if need be, for which the power
series converges is called the interval
of convergence of the series.
These two concepts are fairly closely tied together. If we know that the radius of convergence of
a power series is R then we have the
following.
The interval of convergence must then contain the interval since we know that the power series will
converge for these values. We also know
that the interval of convergence can’t contain x’s in the ranges and since we know the power series diverges for
these value of x. Therefore, to completely identify the interval
of convergence all that we have to do is determine if the power series will
converge for or . If the power series converges for one or both
of these values then we’ll need to include those in the interval of
convergence.
Before getting into some examples let’s take a quick look at
the convergence of a power series for the case of . In this case the power series becomes,
and so the power series converges. Note that we had to strip out the first term
since it was the only non-zero term in the series.
It is important to note that no matter what else is
happening in the power series we are guaranteed to get convergence for . The series may not converge for any other
value of x, but it will always
converge for .
Let’s work some examples.
We’ll put quite a bit of detail into the first example and then not put
quite as much detail in the remaining examples.
|
Example 1 Determine
the radius of convergence and interval of convergence for the following power
series.
Solution
Okay, we know that this power series will converge for ,
but that’s it at this point. To
determine the remainder of the x’s
for which we’ll get convergence we can use any of the tests that we’ve
discussed to this point. After
application of the test that we choose to work with we will arrive at
condition(s) on x that we can use
to determine which values of x for
which the power series will converge and which values of x for which the power series will diverge. From this we can get the radius of
convergence and most of the interval of convergence (with the possible
exception of the endpoints.
With all that said, the best tests to use here are almost
always the ratio or root test. Most of
the power series that we’ll be looking at are set up for one or the
other. In this case we’ll use the
ratio test.
Before going any farther with the limit let’s notice that
since x is not dependent on the
limit and so it can be factored out of the limit. Notice as well that in doing this well need
to keep the absolute value bars on it since we need to make sure everything
stays positive and x could well be
a value that will make things negative.
The limit is then,
So, the ratio test tells us that if the series will converge, if the series will diverge, and if we don’t know what will happen. So, we have,
We’ll deal with the case in a bit. Notice that we now have the radius of
convergence for this power series.
These are exactly the conditions required for the radius of
convergence. The radius of convergence
for this power series is .
Now, let’s get the interval of convergence. We’ll get most (if not all) of the interval
by solving the first inequality from above.
So, most of the interval of validity is given by . All we need to do is determine if the power
series will converge or diverge at the endpoints of this interval. Note that these values of x will correspond to the value of x that will give .
The way to determine convergence at these points is to
simply plug them into the original power series and see if the series
converges or diverges using any test necessary.
:
In this case the series is,
This series is divergent by the Divergence Test since .
:
In this case the series is,
This series is also divergent by the Divergence Test since
doesn’t exist.
So, in this case the power series will not converge for
either endpoint. The interval of
convergence is then,
|
In the previous example the power series didn’t converge for
either end point of the interval.
Sometimes that will happen, but don’t always expect that to happen. The power series could converge at either
both of the end points or only one of the end points.
|
Example 2 Determine
the radius of convergence and interval of convergence for the following power
series.
Solution
Let’s jump right into the ratio test.
So we will get the following convergence/divergence information
from this.
We need to be careful here in determining the interval of
convergence. The interval of convergence requires and . In other words, we need to factor a 4 out of
the absolute value bars in order to get the correct radius of
convergence. Doing this gives,
So, the radius of convergence for this power series is .
Now, let’s find the interval of convergence. Again, we’ll first solve the inequality
that gives convergence above.
Now check the end points.
:
The series here is,
This is the alternating harmonic series and we know that
it converges.
:
The series here is,
This is the harmonic series and we know that it diverges.
So, the power series converges for one of the end points,
but not the other. This will often
happen so don’t get excited about it when it does. The interval of convergence for this power
series is then,
|
We now need to take a look at a couple of special cases with
radius and intervals of convergence.