Re Write Infinite summation

Question

I want to re write the following summation: $$ \sum^{\infty}_{n=0}a_n(n+1)nx^{n-1}$$ I want to give it a form in terms of $x^{n+1}$, is this possible?

Answer 1

Since $$(n+1)nx^{n-1} = \frac{d}{dx}(n+1)x^n = \frac{d^2}{dx^2}x^{n+1}$$ we can rewrite the original sum as $$\sum_{n=0}^\infty a_n \frac{d^2}{dx^2}x^{n+1} = \frac{d^2}{dx^2}\sum_{n=0}^\infty a_n x^{n+1}$$

Answer 2

We could shift the index $n$ to start from $n=-2$.

We obtain \begin{align*} \sum_{n=-2}^\infty a_{n+2}(n+3)(n+2)x^{n+1} \end{align*}