6d: What are the bounds of the Mandelbrot set? When does it diverge?
Description
This article is from the Fractal FAQ, by Ermel Stepp stepp@muvms6.mu.wvnet.edu with numerous contributions by others.
6d: What are the bounds of the Mandelbrot set? When does it diverge?
The Mandelbrot set lies within |c|<=2. If |z| exceeds 2, the z sequence
diverges. Proof: if |z|>2, then |z^2+c|>= |z^2|-|c|> 2|z|-|c|. If
|z|>=|c|, then 2|z|-|c|> |z|. So, if |z|>2 and |z|>=c, |z^2+c|>|z|, so the
sequence is increasing. (It takes a bit more work to prove it is unbounded
and diverges.) Also, note that |z1=c, so if |c|>2, the sequence diverges.
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