Find the sum of a power series

Question

I want to find the sum of a given power series:

$$\sum_{n=0}^\infty(n+4)x^{n-3}$$

I'm trying to find the sum through slow integration or differentiation of a series. But, unfortunately, I can't find the right combination for this, although it's probably a simple one. I know there are a bunch of other ways to find the sum of a power series, but I'm interested in solving this example through slow integration/differentiation.

Answer 1

\begin{align} \sum_{n=0}^\infty(n+4)x^{n-3}&=x^{-6}\sum_{n=0}^\infty(n+4)x^{n+3}\\ &=x^{-6}\frac d{dx}\left(\sum_{n=0}^\infty x^{n+4}\right)\\ &=x^{-6}\frac d{dx}\left(\frac{x^4}{1-x}\right).\\ \end{align}