$\mathbb{E}(\alpha^Y)$, where $Y$ is negative binomial

Question

It is given that $\alpha>0$ and that \begin{equation} \mathbb{P}(Y=y)=\begin{pmatrix} y+k-1\\ y \end{pmatrix} (1-p)^kp^y \end{equation}

are there any ideas how to calculate expected value of $\alpha^Y$?

I have managed to obtain \begin{equation} \mathbb{E}(\alpha^Y)=(1-p)^k\Bigg(1+k\alpha p\sum_{z=0}^{\infty}\begin{pmatrix} z+k\\ z \end{pmatrix}(\alpha p)^z \Bigg) \end{equation} but the series converges if and only if $|\alpha p| <1$, thus I cannot proceed further with this expression. Any suggestions?