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Power Series and
Functions
We opened the last section by saying that we were going to
start thinking about applications of series and then promptly spent the section
talking about convergence again. It’s
now time to actually start with the applications of series.
With this section we will start talking about how to
represent functions with power series.
The natural question of why we might want to do this will be answered in
a couple of sections once we actually learn how to do this.
Let’s start off with one that we already know how to do,
although when we first ran across this series we didn’t think of it as a power
series nor did we acknowledge that it represented a function.
Recall that the geometric series is
Don’t forget as well that if 
the series diverges.
Now, if we take and this becomes,
Turning this around we can see that we can represent the
function
with the power series
This provision is important.
We can clearly plug any number other than into the function, however, we will only get a
convergent power series if . This means the equality in (1)
will only hold if . For any other value of x the equality won’t hold.
Note as well that we can also use this to acknowledge that the radius of
convergence of this power series is and the interval of convergence is .
This idea of convergence is important here. We will be representing many functions as
power series and it will be important to recognize that the representations
will often only be valid for a range of x’s
and that there may be values of x
that we can plug into the function that we can’t plug into the power series
representation.
In this section we are going to concentrate on representing
functions with power series where the functions can be related back to (2).
In this way we will hopefully become familiar with some of
the kinds of manipulations that we will sometimes need to do when working with
power series.
So, let’s jump into a couple of examples.
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Example 1 Find
a power series representation for the following function and determine its
interval of convergence.
Solution
What we need to do here is to relate this function back to
(2). This is actually easier than it might
look. Recall that the x in (2)
is simply a variable and can represent anything. So, a quick rewrite of gives,
and so the in holds the same place as the x in (2). Therefore, all we need to do is replace the
x in (3)
and we’ve got a power series representation for .
Notice that we replaced both the x in the power series and in the interval of convergence.
All we need to do now is a little simplification.
So, in this case the interval of convergence is the same
as the original power series. This
usually won’t happen. More often than
not the new interval of convergence will be different from the original
interval of convergence.
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Example 2 Find
a power series representation for the following function and determine its
interval of convergence.
Solution
This function is similar to the previous function. The difference is the numerator and at
first glance that looks to be an important difference. Since (2)
doesn’t have an x in the numerator
it appears that we can’t relate this function back to that.
However, now that we’ve worked the first example this one
is actually very simple since we can use the result of the answer from that
example. To see how to do this let’s
first rewrite the function a little.
Now, from the first example we’ve already got a power
series for the second term so let’s use that to write the function as,
Notice that the presence of x’s outside of the series will NOT affect its convergence and so
the interval of convergence remains the same.
The last step is to bring the coefficient into the series
and we’ll be done. When we do this
make sure and combine the x’s as
well. We typically only want a single x in a power series.
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As we saw in the previous example we can often use previous
results to help us out. This is an
important idea to remember as it can often greatly simplify our work.
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Example 3 Find
a power series representation for the following function and determine its
interval of convergence.
Solution
So, again, we’ve got an x in the numerator. So, as
with the last example let’s factor that out and see what we’ve got left.
If we had a power series representation for
we could get a power series representation for .
So, let’s find one.
We’ll first notice that in order to use (4)
we’ll need the number in the denominator to be a one. That’s easy enough to get.
Now all we need to do to get a power series representation
is to replace the x in (3)
with .
Doing this gives,
Now let’s do a little simplification on the series.
The interval of convergence for this series is,
Okay, this was the work for the power series
representation for let’s now find a power series representation
for the original function. All we need
to do for this is to multiply the power series representation for by x
and we’ll have it.
The interval of convergence doesn’t change and so it will
be .
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So, hopefully we now have an idea on how to find the power
series representation for some functions.
Admittedly all of the functions could be related back to (2)
but it’s a start.
We now need to look at some further manipulation of power
series that we will need to do on occasion.
We need to discuss differentiation and integration of power series.
Let’s start with differentiation of the power series,
Now, we know that if we differentiate a finite sum of terms
all we need to do is differentiate each of the terms and then add them back
up. With infinite sums there are some
subtleties involved that we need to be careful with, but are somewhat beyond
the scope of this course.
Nicely enough for us however, it is known that if the power
series representation of has a radius of convergence of then the term by term differentiation of the
power series will also have a radius of convergence of R and (more importantly) will in fact be the power series
representation of provided we say within the radius of
convergence.
Again, we should make the point that if we aren’t dealing
with a power series then we may or may not be able to differentiate each term
of the series to get the derivative of the series.
So, what all this means for is that,
Note the initial value of this series. It has been changed from to . This is an acknowledgement of the fact that
the derivative of the first term is zero and hence isn’t in the
derivative. Notice however, that since
the n=0 term of the above series is
also zero, we could start the series at if it was required for a particular
problem. In general however, this won’t
be done in this class.
We can now find formulas for higher order derivatives as
well now.
Once again, notice that the initial value of n changes with each differentiation in
order to acknowledge that a term from the original series differentiated to
zero.
Let’s now briefly talk about integration. Just as with the differentiation, when we’ve
got an infinite series we need to be careful about just integration term by
term. Much like with derivatives it
turns out that as long as we’re working with power series we can just integrate
the terms of the series to get the integral of the series itself. In other words,
Notice that we pick up a constant of integration, C, that is outside the series here.
Let’s summarize the differentiation and integration ideas before
moving on to an example or two.
Fact
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If has a radius of convergence of then,
and both of these also have a radius of convergence of R.
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