How to show that the sequence $z_n$ is unbounded if $|c|>2?$ $(z_0 = 0, z_{n+1} = z_n^2 + c)$.

Asked Sep 11 '23 at 14:30

Active Sep 11 '23 at 15:17

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I want to show that the sequence $z_n$ is unbounded if $|c|>2$, but I can't find a nice way to do that. $(z_0 = 0, z_{n+1} = z_n^2 + c)$.

The only usable tool seems to be the reverse triangle inequality. By using that I've found that $|z_2|>|c|\epsilon$, where $\epsilon>1$ and that $|z_3|>|c|(|c|\epsilon^2-1)$. How to continue the argument is unclear.

edited Sep 11 '23 at 15:17

cbx

asked Sep 11 '23 at 14:30

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See https://math.stackexchange.com/a/3468273/42969 or https://math.stackexchange.com/q/424143/42969 – Martin R Sep 11 '23 at 14:34
Found with Approach0 – Martin R Sep 11 '23 at 14:46

How to show that the sequence $z_n$ is unbounded if $|c|>2?$ $(z_0 = 0, z_{n+1} = z_n^2 + c)$.

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