Partial sums for a power series

Question

I'm having trouble finding the formula for the partial sums of this series,

$$\sum_{n=1}^{\infty\:}{nz^n}$$

where $z$ is a complex number. I'm not looking for the answer just a nudge in the right direction.

Answer 1

The terms in the sum are close to the derivative of a simple power of $z$. For the sum of those, you can use the formula for the geometric series.

Answer 2

Hint

\begin{align*} \frac{1 - z^{N+1}}{1-z} &= 1 + z + z^2 + \cdots + z^N \\ \frac{d}{dz} \left( \frac{1 - z^{N+1}}{1-z} \right) &= 1 + 2z + \cdots + Nz^{N-1} \\ z \frac{d}{dz} \left( \frac{1 - z^{N+1}}{1-z} \right) &= \quad ? \\ \end{align*}