Finding the Sum of an Infinite Series: \sum_{0}^{\infty} \frac {n^2} {3^n}

[wilcofan3]

Homework Statement

Find the sum $$\sum_{0}^{\infty} \frac {n^2} {3^n}$$

Homework Equations

The Attempt at a Solution

I don't know how to go about finding this sum, I have a guess of what it will be just by adding the first ten terms or so, but how do I find an actual approximation?

 

[John Creighto]

Try multiplying three geometric series together and see how close it is to the above series. I think it's simmilar to repeated roots in differential equations.

 

[wilcofan3]

Try multiplying three geometric series together and see how close it is to the above series. I think it's simmilar to repeated roots in differential equations.

The only similar series I see here are:

$$\sum_{0}^{\infty} (\frac {1} {3})^n$$

$$\sum_{0}^{\infty} n^2$$

 

[foxjwill]

Try looking at the second derivative of $$\sum_0^\infty x^n$$.

 

[lurflurf]

homogeneous differentiation

sum=[(xD)^2](1/(1-x))|x=1/3

that is
sum=g(1/3)
when
f(x)=1/(1-x)
and
g(x)=x[x*f'(x)]'=(x^2)*f''(x)+x*f'(x)