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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,
(1)
Turning this around we can see that we can represent the function
(2)
with the power series
(3)
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View Full Document 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.
Example 1
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This note was uploaded on 11/10/2011 for the course MATH 136 taught by Professor Prellis during the Fall '08 term at Rutgers.
 Fall '08
 prellis
 Power Series

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