Representation of Functions as Power Series

In summary, the conversation discusses finding a power series representation for the function f(x)=(1+x)/(1-x) and determining the interval of convergence. The problem is related to the form of 1/(1-x) but the numerator is causing confusion. The suggestion to break it apart into the sum of two fractions is given. After doing the division, the result is -1+3/(1-x) and the idea of a geometric series is mentioned as a potential solution.
  • #1
mateomy
307
0
Find a power series representation for the function and determine the interval of convergence.

[tex]
f(x)=\frac{1+x}{1-x}
[/tex]

This is one of the few problems in this section that I am getting stuck on. I know that I can relate it to the form of;

[tex]
\frac{1}{1-(x)}
[/tex]

...but after that I don't know what to do with the numerator. Do I move the whole thing over to the right side of the Sigma? So lost, so utterly, utterly lost...-Thanks!
 
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  • #2
Try breaking it apart into the sum of two fractions.
 
  • #3
Will do. (So bummed that evaded me.)

Thanks a ton!
 
  • #4
If you go ahead and do the division, you get
(x+1)/(1- x)= -1+ 3/(1- x). Now think "geometric series".
 

Related to Representation of Functions as Power Series

What is the concept of "Representation of Functions as Power Series"?

The concept of "Representation of Functions as Power Series" refers to the process of expressing a function as an infinite sum of terms, each containing a variable raised to a different power. This representation allows for the approximation of a function using a finite number of terms, making it easier to work with and analyze.

What are the benefits of representing a function as a power series?

Representing a function as a power series allows for easier manipulation and analysis of the function. It also provides a way to approximate the function with a finite number of terms, which can be useful in calculations and solving equations. Additionally, power series can be used to study the behavior of functions at specific points, such as the behavior of a function at its zeros or at points of discontinuity.

What are the common techniques for representing a function as a power series?

The most common techniques for representing a function as a power series include the Taylor series, Maclaurin series, and Laurent series. The Taylor series is used to represent a function as a power series around a specific point, while the Maclaurin series is a special case of the Taylor series where the point of expansion is at 0. The Laurent series is used for functions with singularities, where the series includes both positive and negative powers of the variable.

How do you determine the convergence of a power series?

The convergence of a power series can be determined using various tests, such as the ratio test, root test, and the comparison test. These tests evaluate the behavior of the coefficients in the series and determine if the series converges or diverges. It is important to note that the convergence of a power series may also depend on the value of the variable, so it is necessary to specify the interval of convergence for the series.

What are the real-life applications of representing functions as power series?

The representation of functions as power series has various real-life applications in fields such as physics, engineering, and economics. In physics, power series are used to approximate the behavior of natural phenomena, such as oscillations and electromagnetic fields. In engineering, power series are used to model and analyze complex systems. In economics, power series are used to represent economic data and make predictions based on trends.

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