How Can You Solve the Geometric Series Expansion for \(\sum n^2 x^n\)?

[Andronicus1717]

I need to find the solution to the geometric series expansion of the form...

\sumn^2*x^n , for n=0,1,2,...

most resources I've found only have answers for n*x^n or n*x^(n-1). I have no idea how to calculate this, so I was wondering if there's a book out there that has massive lists of geometric series or is there method for easily calculating this?

 

[Mute]

A common way to try to find closed forms for series like these is to try to relate them to a series you already know. For example, note that

n^2 x^n = \left(x\frac{d}{dx}\right)^2 x^n.

So,

\sum_{n=0}^{\infty} n^2 x^n = \sum_{n=0}^{\infty}\left(x\frac{d}{dx}\right)^2 x^n = \left(x\frac{d}{dx}\right)^2 \sum_{n=0}^\infty x^n.

(Swapping the sum and derivative isn't always allowed, but in this case it should be ok).

In case you're not sure,

\left(x\frac{d}{dx}\right)^2f(x) = x\frac{d}{dx}\left(x\frac{df}{dx}\right).

 

[Stephen Tashi]

Andronicus1717,

Do you want to sum infinite series or finite series? I don't know of any reference book that gives a large table of finite summations. (I wonder why there aren't several such books.)

If you want to know the method for doing finite summations in closed form, the field you should look up is The Calculus of Finite Differences.

I think this web page talks about the type of sums you are asking about. http://2000clicks.com/MathHelp/SeriesPolynomialGeometric.aspx

You might find more information by searching for "polynomial geometric series".

 

[Andronicus1717]

Thank you, your responses have been very helpful!

Stephen Tashi: The infinite series is what I was looking for. Next time I will try and better use the equation syntax.

 

[JSuarez]

most resources I've found only have answers for n*x^n or n*x^(n-1)

By the way, have you considered multiplying the latter by x, and then differentiating again?