Sum of Series.

1. Nov 8, 2007

azatkgz

1. The problem statement, all variables and given/known data
Find the sum of the following series

$$\sum_{n=1}^{\infty}n(n+1)x^n$$

3. The attempt at a solution

$$x\sum_{n=1}^{\infty}n(n+1)x^{n-1}$$

$$x\int_{0}^{x}(\int_{0}^{x}f(t)dt)dt=x(x^2+x^3+x^4+x^5+\cdots)=x\frac{x^2}{1-x}$$

$$x\frac{d^2}{dx^2}\frac{x^2}{1-x}=\frac{2x}{1-x}$$

2. Nov 8, 2007

Midy1420

I am not really sure of how to do this but looking at your last line, I think you messed up taking the second derivative of x^2/(1-x) since you need to use the quotient rule, the (1-x) should be squared on each derivative