Root locus of discrete time system
The root locus plot for positive K is 


You can see that intersections the unit circle at z=±1j. The Closed-loop pole is 
$$\frac{-1}{G(z)H(z)} = -(z+0.5)(z-0.5)$$
For z=±1j we have \$K=-(j+0.5)(j-0.5)=-(-1-.25)=1.25\$

For negative K the root locus is

This intersects the unit circle at z=±1. For that, we have  \$K=-(1+0.5)(1-0.5) = -0.75\$ 

So, you're region of convergence is from \$-0.75 &lt; K &lt; 1.25\$