So, right now I am writing my master thesis and I need to find a reference for a formula I found in a paper: $$ \sum_{k=1}^{\infty}k^{-\alpha}(1-\varepsilon)^k\sim b+c\Gamma(1-\alpha)\varepsilon^{\alpha-1}\text{ as }\varepsilon\downarrow0\text{ for }\alpha\in(1,2) $$ where $b,c$ are some constants. Unfortunately I have not been able to find a reference for this result, does somebody have an idea where I can find this?

Furthermore I would like to extend this result in the following sense: Let $(a_k)_{k\in\mathbb{N}_0}$ be a probability sequence, i.e. $\sum_{k\in\mathbb{N}_0}a_k=1$ such that $a_k=k^{-\alpha+o(1)}$ as $k\to\infty$. Is it possible to show that then $$ \sum_{k=1}^{\infty}a_k(1-\varepsilon)^k\sim 1+c\Gamma(1-\alpha)\varepsilon^{\alpha-1}\text{ as }\varepsilon\downarrow 0? $$ As you may have guessed by now, the subject of my thesis is probability theory and I dont know much about polylogarithm functions and its asymptotic behavior, so I thought it would be a good idea to ask here and hope that someone could help me out with this problem. Thank you for your help!