Finding the Closed Form of a Power Series

In summary, the conversation discusses how to find the closed form of the series \sum_{n=0}^{\infty} n^2x^n by using the theorem for finding derivatives of power series and potentially applying the geometric series identity. The conversation concludes with suggestions to manipulate the series for g'(x) and g''(x) to find the desired closed form.
  • #1
Yagoda
46
0

Homework Statement


Using that [itex]\frac{1}{1-x} = \sum_{n=0}^{\infty} x^n [/itex] for |x|<1 and that
[itex] f'(x) =\sum_{n=0}^{\infty} (n+1)a_{n+1}(x-x_0)^n [/itex], write [itex] \sum_{n=0}^{\infty} n^2x^n [/itex] in closed form.


Homework Equations





The Attempt at a Solution

In this series, [itex]a_n = n^2 [/itex] and [itex]x_0 = 0 [/itex]. Applying the theorem I get that [itex] f'(x) =\sum_{n=0}^{\infty} (n+1)(n+1)^2(x)^n = \sum_{n=0}^{\infty} (n+1)^3(x)^n[/itex]. I know I want to try to apply the sum of the geometric series and then integrate to get f(x) (or maybe those things in reverse order), but the [itex](n+1)^3[/itex] is giving me trouble.
 
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  • #2
I think you failed to define f(x) - is that 1/(1 - x) or a general function ##f(x) = \sum_{n = 0}^\infty a_n x^n## ?

Consider this:
$$\sum_{n = 0}^\infty n^2 x^n = \sum_{n = 1}^\infty n^2 x^n = \sum_{n = 0}^\infty (n + 1)^2 x^{n + 1} = x \sum_{n = 0}^\infty (n + 1) (n + 1) x^n$$

and try taking it from there.
 
  • #3
I am considering f(x) to be the closed form of the series that I'm looking for so I use the theorem to find its derivative f'(x) and then hopefully find f itself. So what I've got so far is that given f'(x), [itex]f(x) =\sum_{n=0}^{\infty} (n+1)^2(x)^n [/itex], which tell us that [itex]f(x) =\sum_{n=0}^{\infty} (n+1)^2(x)^n =\sum_{n=0}^{\infty} n^2x^n [/itex] since that was the original series, if that makes sense.
Maybe this is an elementary question, but is there a way to apply the geometric series identity to your expression to get a closed form or does some other type manipulation that I'm not seeing have to be done?
 
Last edited:
  • #4
Yagoda said:
I am considering f(x) to be the closed form of the series that I'm looking for so I use the theorem to find its derivative f'(x) and then hopefully find f itself. So what I've got so far is that given f'(x), [itex]f(x) =\sum_{n=0}^{\infty} (n+1)^2(x)^n [/itex], which tell us that [itex]f(x) =\sum_{n=0}^{\infty} (n+1)^2(x)^n =\sum_{n=0}^{\infty} n^2x^n [/itex] since that was the original series, if that makes sense.
Maybe this is an elementary question, but is there a way to apply the geometric series identity to your expression to get a closed form or does some other type manipulation that I'm not seeing have to be done?

[tex] n^2 x^n = x \frac{d}{dx}\left( x \frac{d}{dx} x^n \right). [/tex]
Alternatively, if ##g(x) = \sum x^n,## look at the series for ##g'(x)## and ##g''(x).## Can you see how to get ##\sum n^2 x^n## from those two series?
 

1. What is a closed form of power series?

A closed form of power series is a mathematical expression that can be written in a finite number of terms without the use of an infinite series. It is a way of representing a function as a polynomial in terms of a variable, typically x.

2. How is a closed form of power series different from an infinite series?

The main difference between a closed form of power series and an infinite series is that the former can be written in a finite number of terms, while the latter has an infinite number of terms. This means that a closed form of power series can be evaluated and calculated more easily than an infinite series.

3. What is the purpose of using a closed form of power series?

The purpose of using a closed form of power series is to represent a function in a simpler and more manageable form. It allows for easier calculations and can provide insights into the behavior and properties of the function.

4. How do you determine the closed form of a power series?

The closed form of a power series can be determined by using various techniques such as Taylor series, Maclaurin series, and geometric series. These methods involve finding the coefficients of the terms in the series and using them to write the function in a polynomial form.

5. Can all functions be represented by a closed form of power series?

No, not all functions can be represented by a closed form of power series. Some functions may have singularities or infinitely many terms, making it impossible to write them in a finite form. Additionally, some functions may not have a known closed form representation.

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