# The Mandelbrot set is defined using an iterative function

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• ## For a given value c, it turns out that the magnitude of xt+1 will do one of two things:

• It will always be smaller than 2 (no matter how large t gets), or
• It will eventually diverge (i.e., xt will go to ∞ as t goes to ∞).

• ## But we have no easy way to know whether the Mandelbrot series diverges for a given value of c!

• If we compute the Mandelbrot series for some value c and the magnitude of xt ever becomes greater than 2, that value c is definitely not in the Mandelbrot set. (It is a property of the series that if xt is greater than 2, then subsequent values will always increase.)
• But a Mandelbrot series may remain below 2 for arbitrarily long before diverging, and the only way to tell if it will diverge is to compute the sequence for long enough.

• ## One can zoom as far as your computing power and patience holds up: the NetLogo model can do up to a one-billion zoom, depending on the region.

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