Evaluating Sum of $\dfrac{i}{(-x)^i}$

Question

I would like to ask if the expression below can be simplified using standard summation properties? Or should I dive into much deeper concepts like the power series? Thank you.

$$\sum_{i=1}^{n-1} \dfrac{i}{(-x)^i}$$

Answer 1

\begin{align*} \sum_{i=1}^{n-1}\frac{i}{(-x)^i} &= \sum_{i=2}^{n-1}(i-1)(-x)^{-i} + \sum_{i=1}^{n-1}(-x)^{-i} = \frac{d}{dx}\sum_{i=2}^{n-1}(-x)^{-i+1} + \sum_{i=1}^{n-1}(-x)^{-i}\\ &= \frac{d}{dx}\sum_{i=1}^{n-2}\left(-\frac 1 x\right)^{i} + \sum_{i=1}^{n-1}\left(-\frac 1x\right)^{i} \end{align*}