Given a particular expansion, find the function in powers of x

In summary, any expansion can be written as a function in powers of x, as polynomials can always be expressed as power series. To find the function, identify the pattern and replace coefficients with powers of x. Any value of x can be used, but it is recommended to use simpler values. Writing a function in powers of x allows for easier manipulation and understanding of the function's behavior, but there may be limitations such as undefined values at certain points.
  • #1
eclayj
20
0
I am given this expansion:

x + 2^2 x^2 + 3^2 x^3 + 4^2 x^4 + ... + k^2 x^k + ...

To get the function that yields this expansion, I cannot figure out the manipulation to (x)/(1-x)
 
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  • #2
That's probably because it is not true. To take a simple example, if x= 1/2, that sum becomes [itex](1/2)+ 4(1/4)+ 9(1/8)+ 16(1/16)+ ...[/itex]
which is already larger than 1/2+ 1+ 9/8+ 1= 3 and 5/8. Since every term is positive, the sum (if it converges) is larger than 3 and 5/8. But (1/2)(1- 1/2)= (1/2)/(1/2)= 1.

In fact, 1/(1- x) is the sum of the geometric series [itex]\sum_{n=0}^\infty x^n[/itex] so x/(1- x) is [itex]\sum_{n=0}^\infty x^{n+1}[/itex].
 

1. Can any expansion be represented as a function in powers of x?

Yes, any expansion can be written as a function in powers of x. This is because the expansion is essentially a polynomial expression, and polynomials can always be written in the form of a power series.

2. How do I find the function in powers of x from a given expansion?

To find the function in powers of x from a given expansion, you need to first identify the pattern or rule in the expansion. Once you have identified the pattern, you can write the function by replacing the coefficients with the corresponding powers of x.

3. Can I use any value of x in the expansion to find the function?

Yes, you can use any value of x in the expansion to find the function. However, it is recommended to use values of x that are easy to work with, such as 0, 1, or -1.

4. What is the purpose of writing a function in powers of x?

Writing a function in powers of x allows for easier manipulation and evaluation of the expression. It also helps in identifying the behavior of the function and its relationship with the variable x.

5. Are there any limitations to writing a function in powers of x?

There are some limitations to writing a function in powers of x, such as the function not being defined for certain values of x or the function being undefined at x=0. It is important to check for these limitations when working with functions in powers of x.

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