Asymptotic behavior of the polylogarithm function and generalisation

Question

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!

Answer 1

See for example this reference: $$\sum_{k=1}^{\infty}k^{-\alpha}(1-\varepsilon)^k=\text{Li}_{\alpha}(1-\epsilon)=$$ $$\qquad=\varepsilon^{\alpha} \left[\frac{\Gamma (1-\alpha)}{\varepsilon}+\tfrac{1}{2} (\alpha-1) \Gamma (1-\alpha)+\tfrac{1}{24} \left(3 \alpha^2-\alpha-2\right) \Gamma (1-\alpha) \varepsilon+O\left(\varepsilon^2\right)\right]$$ $$\qquad+\left[\zeta (\alpha)-\zeta (\alpha-1) \varepsilon+O\left(\varepsilon^2\right)\right].$$

Concerning the generalization, note that the finite sum $\sum_{k=1}^Na_k(1-\varepsilon)^k$ has for small $\varepsilon$ the expansion $b+c\varepsilon+O(\varepsilon^2)$, while $\sum_{k=N+1}^\infty k^{-\alpha}(1-\varepsilon)^k$ has the expansion $b'+\Gamma(1-\alpha)\varepsilon^{\alpha-1}+O(\varepsilon)$. So for $1<\alpha<2$ the large-$k$ behavior of $a_k$ dominates.