Power Series Representation of a Function Help

Then, to center the series at c=-2, we can write (5/7)/(1-(x/7)) as (5/9)/(1-((x+2)/9)). Finally, we can use the formula for the geometric series, a/(1-r), to get the power series for f(x), which is the sum of 5(x+2)^n/9^(n+1) from 0 to infinity. In summary, to find the power series for f(x)=5/(7-x), centered at c=-2, we can use the formula for the geometric series and center it at x=-
  • #1
DCASH88
7
0
1. Homework Statement : Find the power series for the function f(x)=5/(7-x), centered at c=-2.
2. Homework Equations : a/(1-r)
3. The Attempt at a Solution : I know that I need to divide by seven to get (5/7)/(1-(x/7)) and then rewrite in the form the sum of (a)(r)^n. I tried adding 2 to x/7 so i had the sum of (5/7)((x/7)+2)^n from 0 to infinity but this was wrong. The professor wrote that the correct answer is the sum of 5(x+2)^n/9^(n+1) from 0 to infinity but I don't know how to get to that answer mainly because I don't know how to account for it being centered at -2 any help would be appreciated.

Homework Statement


Homework Equations


The Attempt at a Solution

 
Last edited:
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  • #2
You want to express it as a power series in powers of (x+2), that's how you center it at x=-2. 5/(7-x)=5/(9-(x+2)). Try expanding the second form.
 
  • #3
Thank you that was very helpful

Daniel
 

FAQ: Power Series Representation of a Function Help

What is a power series representation of a function?

A power series representation of a function is an infinite polynomial expression that can be used to represent a function as a sum of powers of a variable. It is useful for approximating functions and solving differential equations.

How is a power series representation of a function calculated?

A power series representation of a function is calculated by finding the coefficients of each power of the variable in the infinite polynomial expression. This is typically done using the Taylor series expansion.

What is the purpose of using a power series representation of a function?

The purpose of using a power series representation of a function is to approximate the behavior of a function, especially when the function is difficult to evaluate directly. It can also be used to solve differential equations or to find the value of a function at a specific point.

What are some common examples of power series representations of functions?

Some common examples of power series representations of functions include the Maclaurin series, which is a special case of the Taylor series, and the geometric series, which is used to represent geometric sequences.

What are the limitations of using a power series representation of a function?

One of the limitations of using a power series representation of a function is that it may not converge for all values of the variable. This means that the series may only be valid for a certain range of values, and may not accurately represent the function outside of that range.

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