The geometric series is usually defined as $\sum_{k=0}^{\infty} a \cdot x^{k}$ where $x$ is on the interval $]-1;1[$, which includes $0$.
My Problem is that substituting $x=0$ for the first Term of the sum gives $a \cdot 0^{0} $. This is an indeterminate form and therefor undefined which means that $x=0$ should not be part of the domain.
I know that the both-sided $lim_{x \rightarrow 0} ( a\cdot x^{0} )$, as well as $\frac{a}{1-x}$ at $x=0$, are equal to $a$. Both shouldn't be reason enough for including $x=0$ in the domain. Under most circumstances having the Geometric Series at $x=0$ equate to $a$ is a good thing, but there must be a better reason for doing so.
Generally when dealing with power series (and indeed we do the same when dealing with polynomials), we define $0^0=1$. This is for various reasons, including that we want continuity. Furthermore, we wish for our summation notation to formalize the following notion:
$$\sum_{j=0}^\infty a_j x^j=a_0+a_1x+a_2x^2+\cdots.$$
In particular, we only get this if we define $0^0=1$, as we wish to have
$$a_0x^0=a_0$$
for the first term, for all $x$, including $0$.