For all $s>0$ define for $\epsilon\in(0,1)$ the function:
\begin{equation}
g(\epsilon)=\sum\_{k=0}^{\infty}(1+k)^s(\sqrt{1-\epsilon})^k.
\end{equation}
Prove that $\exists C>0$ and $\phi(s)$ such that:
\begin{equation}
g(\epsilon)\leq C \epsilon^{\phi(s)}.
\end{equation} 



An interpolation inequality.
$$
g(\epsilon)=\sum\_{k\ge 1} k^se^{\frac {k-1}2\ln(1-\epsilon)}
\lesssim \int\_0^{+\infty} x^s e^{-a\epsilon x} dx=
 \int\_0^{+\infty}x^s e^{-ax}dx\epsilon^{-s-1}
$$
where $a$ is a fixed constant. So
$$
C=\int\_0^{+\infty}x^s e^{-ax}dx,\quad \phi(s)=-s-1.
$$


