Why in geometric formula definitions do they not exclude the negatives in the domain or use an absolute value? In resources like Wikipedia the following equation holds for $k>0$.
\begin{equation}
\sum_{n=0}^{k} [x^n]=\frac{1-x^{k+1}}{1-x}\hspace{1cm}x\neq1
\end{equation}
If you put the right hand side in a limit it holds for $k\geq0$. But why do we  rarely show the bounds of the k value and why not put the k  in an absolute value on the right hand side (Other than the aversion to absolute values)?
 A: The answers come from what Rob Arthan said in the comments.
People tend to leave out constraints that are assumed.
As for the second part of the question being too general in an equation overcomplicates it and can have the affect of confusing the reader.
A: Your question

But why do we rarely show the bounds of the k value

is one about conventions and expectations which is a common situation.
For example, to use an authoritative source, the
DLMF section 1.2.11 states explicitly

$$a+ax+ax^2+\cdots+ax^{n-1}=\frac{a(1-x^n)}{1-x},\qquad x\ne1. $$

and also in a mouse popup states in reference to
DLMF section 1.1 Special Notation that

$m,n\quad$ nonnegative integers, unless specified otherwise.

Notice that in the summation the DLMF decided to use an explicit sum
with $\,\cdots\,$ instead of using summation notation. Both choices
are valid and, in some cases, both are used for clarity.
A much less popular choice would have been to use a convention such that
$$ s_n := \sum_{k=0}^n a_k \quad \text{ where }\quad s_n = s_{n-1} +
 a_n,\quad s_0=a_0 $$ and which is supposed to be true for all
integer $\,n.\,$ For example, your case of
$$\sum_{k=0}^n x^k = \frac{1-x^{n+1}}{1-x}\quad\text{ where }\quad x\ne1 $$ is true for all integer $\,n\,$ by this convention and would be
assumed as such in this context by this convention without being
specifically mentioned explicitly. No need to use absolute value for $n$.
Your further question

and why not put the k in an absolute value on the right hand side

is simply because, in the common convention, $\,k\,$ is always assumed
to be nonnegative. Therefore, the equation is only assumed to be valid
for those $\,k\,$ although it may be valid outside that range using an
alternative convention.
