Is there a known way to evaluate sums of the form
$$\sum_{p\text{ prime}} x^{p},$$ and are there any restrictions on the value of $x$ (e.g., $|x|<1$ for typical geometric series)?
EDIT: The second question was extremely obvious. Of course $|x|<1$ is required. I honestly don't know how I missed that.
On
The restriction can be seen in one or two ways. The first is comparison to the geometric series. The second is that the series can be defined as $\sum\limits_{n=0}^\infty a_nz^n,$ where $a_n=1$ if $n$ is prime and $0$ otherwise. Then, the radius of convergence is defined as $$\frac{1}{R}=\limsup_{n\rightarrow\infty}|a_n|^{1/n}=\lim_{n\rightarrow\infty} 1^{1/n}=1.$$
Partial answer. If $N \ge 2, x \ge 2$ then
$$ \lim_{N \to \infty} \sup \frac{\sum_{n \le N} x^n}{\sum_{p \le N} x^p} = \frac{x}{x-1} $$ which gives $$ {\sum_{p \le N} x^p} \approx (1-1/x){\sum_{n \le N} x^n} $$
converting the problem from a power series over primes to a much more managable power series over natural numbers.