Z Transform - ROC values
The fact that evaluating the function
$$
 \begin{align}
   \frac{z}{z-a}
 \end{align}
$$
yields a finite value for some given values of $z$ and $a$, does not mean that the original series (from $\mathcal{Z}$-transform definition)
$$
 \begin{align}
   X(z) = \sum_{n=0}^\infty(az^{-1})^n
 \end{align}
$$
converges at all for those same values of $z$ and $a$. Before we determine the region of convergence, the only thing we can say is that if the series converges, then it will converge to $\frac{z}{z-a}$.

In the case $z=5$ and $a=10$, looking at the sequence
$$
 \begin{align}
   \left\{\sum_{n=0}^N (az^{-1})^n \right\}
 \end{align}
$$
as $N$ increases yields:
$$
  \{1, 3, 7, 15, 31, 63, ...\}
$$
which as you may notice does not converge to -1, and in fact diverges as $N\rightarrow\infty$. The region of convergence tells you where this sequence converges (and in this case it is $|z|&gt;|a|$).