# Asymptotic behavior of the polylogarithm function and generalisation

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!

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.