$a_{n}=\frac{1}{s^{n}}$
$n=0\text{ or }s\neq 0$

$\left\{\begin{matrix}n\in \mathrm{C}\text{, }&a_{n}=1\text{ and }s=1\\n=\frac{2\pi n_{1}i}{\ln(s)}-\log_{s}\left(a_{n}\right)\text{, }n_{1}\in \mathrm{Z}\text{, }&a_{n}\neq 0\text{ and }s\neq 1\text{ and }s\neq 0\end{matrix}\right.$

$a_{n}=\frac{1}{s^{n}}$
$s>0\text{ or }\left(Denominator(n)\text{bmod}2=1\text{ and }s<0\right)$

$\left\{\begin{matrix}n=-\log_{s}\left(a_{n}\right)\text{, }&a_{n}>0\text{ and }s\neq 1\text{ and }s>0\\n\in \mathrm{R}\text{, }&\left(s=-1\text{ and }a_{n}=-1\text{ and }Denominator(n)\text{bmod}2=1\text{ and }Numerator(n)\text{bmod}2=1\right)\text{ or }\left(s=1\text{ and }a_{n}=1\right)\end{matrix}\right.$