<s>[INST] Find the sum of the power series $\sum_{n=1}^\infty n*(n+1)*x^n$ $\sum_{n=1}^\infty n*(n+1)*x^n$
Hello everyone,  I need help in solving the question above.
I started with $\frac{1}{1-x} = \sum_{n=0}^\infty x^n$, if differentiated it once so it became $\frac{1}{(1-x)^2} =\sum_{n=1}^\infty nx^{n-1} $ but from here I don't know how to continue.
Thanks. 
 [/INST] Using 
$$\sum_{n=0}^{\infty} x^{n} = \frac{1}{1-x}$$
then
$$\sum_{n} n \, x^{n+1} = x^2 \, \frac{d}{dx} \, \frac{1}{1-x}$$
and 
$$\sum_{n} n \, (n+1) \, x^n = \frac{d}{dx} \left( x^{2} \, \frac{d}{dx} \frac{1}{1-x} \right).$$
After some work it is found that:
\begin{align}
\sum_{n=0}^{\infty} x^{n} &= \frac{1}{1-x} \\
\sum_{n=0}^{\infty} n \, x^{n} &= \frac{x}{1-x} \\
\sum_{n=0}^{\infty} n(n+1) \, x^{n} &= \frac{2 x \, (x^2 - x + 1)}{(1-x)^{3}} = \frac{2 x (1+x^3)}{(1-x)^2 \, (1-x^2)}. 
\end{align}
</s>