Power Series and Functions

Power Series and Functions - Power Series and Functions We...

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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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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.

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Power Series and Functions - Power Series and Functions We...

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