Calculate the Sum of Series: \sum_{n=1}^{\infty}n(n+1)x^n for Homework

[azatkgz]

Homework Statement

Find the sum of the following series

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

The Attempt at a Solution

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

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


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

 

[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