A geometric series is one having a common ratio, right? Something like:
$$\sum_{n=0}^{\infty} ar^n$$
And a power series is one with the form:
$$\sum_{n=0}^{\infty} c_n(x-a)^n$$
Initially, I thought a geometric power series was one which had a rational function in $x-a$; something like $\frac{x}{2}$ with $a=0$, or maybe $\frac{x-a}{n}$.
But, you can think of the expression $(x-a)$ as a rational function of the form:
$$\frac{(x-a)}{1}$$
So, does that mean all power series (for $x$ within a radius of convergence) are geometric series? It seems so, but I don't want any misconceptions in my understanding, so I thought it best to ask.
A geometric series is characterized by its fixed ratio $r$ and starting term $a$. Therefore, we can represent a geometric series in a number of ways.
$a + ar + ar^{2} + \cdots = \sum\limits_{n = 0}^{\infty} ar^{n}$
A power series is a polynomial characterized by increasing powers of a variable centered at some value $c$ multiplied by coefficients.
$a_0(x - c)^{0} + a_1(x - c)^{1} + a_2(x - c)^{2} + \cdots = \sum\limits_{n = 0}^{\infty}a_n(x - c)^n$.
Look at the relationship between the two mathematical structures. From this, we can conclude that a geometric series is a power series in the specific case that the power series has constant coefficients and $c = 0$. Because a geometric series is a power series, it inherits the usual properties such as radius of convergence.