Answer
The point $~~-0.5~i~~$ is not in the Mandelbrot set.
Work Step by Step
$z = -0.5~i = -\frac{i}{2}$
We can perform the calculation $z^2+z$:
$z^2+z = (-\frac{i}{2})^2+(-\frac{i}{2})$
$z^2+z = -\frac{1}{4}-\frac{~i}{2}$
We can perform the calculation $(z^2+z)^2+z$:
$z^2+z = (-\frac{1}{4}-\frac{~i}{2})^2-\frac{i}{2}$
$z^2+z = (-\frac{3}{16}+\frac{i}{4})-\frac{i}{2}$
$z^2+z = -\frac{3}{16}-\frac{i}{4}$
We can perform the calculation $[(z^2+z)^2+z]^2+z$:
$z^2+z = (-\frac{3}{16}-\frac{i}{4})^2-\frac{i}{2}$
$z^2+z = (-\frac{7}{256}+\frac{3i}{32})-\frac{i}{2}$
$z^2+z = -\frac{7}{256}-\frac{13i}{32}$
The absolute value is less than 2, and if we continue doing this calculation, the absolute value seems to be getting smaller and smaller.
Therefore, the point $~~-0.5~i~~$ is not in the Mandelbrot set.
