My thinking is that all geometric series are power series, since the form of a term of a power series is $c_n(x-a)^n$. If you look at $c_n$ as the starting constant and $x$ as the common ratio (and $a$ being $0$), then it matches the formula for a term of a geometric series, $ar^{n-1}$.
But if $a$ is zero, that makes the power series centered around zero. Thus my conclusion that every geometric series is a power series centered at zero. Is this correct?