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What is the Mandelbrot set?

Answer 1

Ezra Adams

On the complex plane the Mandelbrot set is the set of complex numbers #c in CC# such that the sequence defined as:

#z_0 = 0#

#z_(n+1) = z_n^2+c#

is limited in module.

In other words, we have:

#z_0 = 0#

#z_1 = c#

#z_2 = c^2+c#

etc... and if #lim_n norm (z_n) < oo# then #c in M#

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Answer 2

Axel Alvarez

The Mandelbrot set is a set of complex numbers ( c ) for which the function ( f_c(z) = z^2 + c ) does not diverge when iterated from ( z = 0 ). In other words, for each complex number ( c ) in the Mandelbrot set, the sequence ( z_0 = 0 ), ( z_{n+1} = z_n^2 + c ) remains bounded as ( n ) approaches infinity. The Mandelbrot set is named after the mathematician Benoit B. Mandelbrot, who studied and popularized it in the 1970s and 1980s.

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.